text:
الرئيسية Precalculus with Limits
Precalculus with Limits
Ron Larson
الناشر: Brooks Cole
الصفحات: 1151
تحميل (pdf, 43.04 MB)
اقرأ الكتاب مباشرة
cos2201
graph2085
exercises1908
graphing utility684
polynomial524
trigonometric473
graphs467
use a graphing434
inverse399
sketch395
zeros371
triangle361
cot322
utility to graph313
intercept288
matrices269
polar260
parabola257
approximate249
exponential248
numbered exercises244
logarithmic230
asymptote221
simplify221
linear equations214
coordinate203
verify201
trigonometric functions198
ellipse190
vertices185
formulas171
nth169
sine167
coefficient167
denominator166
asymptotes166
multiply165
tangent164
geometric163
rectangular161
calculator155
arithmetic154
intercepts151
ORBIT (LITT)
Encyclopedia of Plague and Pestilence: From Ancient Times to the Present (Facts on File Library of World History)
George C. Kohn
GRAPHS OF PARENT FUNCTIONS
Linear Function
Absolute Value Function
f x  x
f x  mx  b
Square Root Function
f x
x, x < 0
f(x) = ⏐x⏐
(− mb , 0( (− mb , 0(
f(x) = mx + b,
m>0
(0, b)
m<0
(0, 0)
Domain:  ,
Range: 0,
Intercept: 0, 0
Decreasing on  , 0
Increasing on 0,
Even function
y-axis symmetry
Domain: 0,
Greatest Integer Function
f x  x
Quadratic (Squaring) Function
f x  ax2
Cubic Function
f x  x3
f(x) = [[x]]
−3 −2 −1
Range: the set of integers
x-intercepts: in the interval 0, 1
y-intercept: 0, 0
Constant between each pair of
consecutive integers
Jumps vertically one unit at
each integer value
−2 −1
ax 2 ,
a>0
Range:  ,
x-intercept: bm, 0
y-intercept: 0, b
Increasing when m > 0
Decreasing when m < 0
f(x) = ax 2 , a < 0
Range a > 0: 0,
Range a < 0 :  , 0
Decreasing on  , 0 for a > 0
Increasing on 0,  for a > 0
Increasing on  , 0 for a < 0
Decreasing on 0,  for a < 0
Relative minimum a > 0,
relative maximum a < 0,
or vertex: 0, 0
f(x) = x 3
Increasing on  ,
Odd function
Origin symmetry
Rational (Reciprocal) Function
Exponential Function
Logarithmic Function
f x  ax, a > 0, a  1
f x  loga x, a > 0, a  1
f(x) = a −x
f(x) = a x
f(x) = loga x
Domain:  , 0 傼 0, )
Range:  , 0 傼 0, )
No intercepts
Decreasing on  , 0 and 0,
Vertical asymptote: y-axis
Horizontal asymptote: x-axis
for f x  ax
Decreasing on  ,
Reflection of graph of f x  ax
in the line y  x
Sine Function
f x  sin x
Cosine Function
f x  cos x
Tangent Function
f x  tan x
f(x) = sin x
f(x) = cos x
−π
f(x) = tan x
2π
Range: 1, 1
Period: 2
x-intercepts: n, 0
x-intercepts:
 n , 0
Period:
Vertical asymptotes:
x   n
Domain: all x
Cosecant Function
f x  csc x
Secant Function
f x  sec x
f(x) = csc x =
sin x
Cotangent Function
f x  cot x
f(x) = sec x =
cos x
f(x) = cot x =
tan x
Domain: all x  n
Range:  , 1 傼 1,
Vertical asymptotes: x  n
Inverse Sine Function
f x  arcsin x
Inverse Cosine Function
f x  arccos x
Inverse Tangent Function
f x  arctan x
f(x) = arccos x
f(x) = arcsin x
Domain: 1, 1
Range:  ,
f(x) = arctan x
y-intercept: 0,
Horizontal asymptotes:
y±
The Behrend College
With the assistance of
David C. Falvo
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Precalculus with Limits, Second Edition
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1 2 3 4 5 6 7 13 12 11 10 09
A Word from the Author (Preface) vii
Functions and Their Graphs
Rectangular Coordinates 2
Graphs of Equations 13
Linear Equations in Two Variables 24
Functions 39
Analyzing Graphs of Functions 54
A Library of Parent Functions 66
Transformations of Functions 73
Combinations of Functions: Composite Functions 83
Inverse Functions 92
1.10 Mathematical Modeling and Variation 102
Chapter Summary 114
Review Exercises 116
Chapter Test 121
Proofs in Mathematics 122
Problem Solving 123
Polynomial and Rational Functions
Quadratic Functions and Models 126
Polynomial Functions of Higher Degree 136
Polynomial and Synthetic Division 150
Complex Numbers 159
Zeros of Polynomial Functions 166
Rational Functions 181
Nonlinear Inequalities 194
Exponential Functions and Their Graphs 216
Logarithmic Functions and Their Graphs 227
Properties of Logarithms 237
Exponential and Logarithmic Equations 244
Exponential and Logarithmic Models 255
Cumulative Test for Chapters 1–3 274
Radian and Degree Measure 280
Trigonometric Functions: The Unit Circle 292
Right Triangle Trigonometry 299
Trigonometric Functions of Any Angle 310
Graphs of Sine and Cosine Functions 319
Graphs of Other Trigonometric Functions 330
Inverse Trigonometric Functions 341
Applications and Models 351
Analytic Trigonometry
Using Fundamental Identities 372
Verifying Trigonometric Identities 380
Solving Trigonometric Equations 387
Sum and Difference Formulas 398
Multiple-Angle and Product-to-Sum Formulas 405
Additional Topics in Trigonometry
Law of Sines 428
Law of Cosines 437
Vectors in the Plane 445
Vectors and Dot Products 458
Trigonometric Form of a Complex Number 468
Systems of Equations and Inequalities
Linear and Nonlinear Systems of Equations 494
Two-Variable Linear Systems 505
Multivariable Linear Systems 517
Partial Fractions 530
Systems of Inequalities 538
Linear Programming 549
Matrices and Determinants
Matrices and Systems of Equations 570
Operations with Matrices 584
The Inverse of a Square Matrix 599
The Determinant of a Square Matrix 608
Applications of Matrices and Determinants 616
Sequences, Series, and Probability
Sequences and Series 640
Arithmetic Sequences and Partial Sums 651
Geometric Sequences and Series 661
Mathematical Induction 671
The Binomial Theorem 681
Counting Principles 689
Probability 699
Topics in Analytic Geometry
10.1 Lines 726
10.2 Introduction to Conics: Parabolas 733
10.3 Ellipses 742
10.4 Hyperbolas 751
10.5 Rotation of Conics 761
10.6 Parametric Equations 769
10.7 Polar Coordinates 777
10.8 Graphs of Polar Equations 783
10.9 Polar Equations of Conics 791
Analytic Geometry in Three Dimensions
11.1 The Three-Dimensional Coordinate System 810
11.2 Vectors in Space 817
11.3 The Cross Product of Two Vectors 824
11.4 Lines and Planes in Space 831
Limits and an Introduction to Calculus
12.1 Introduction to Limits 850
12.2 Techniques for Evaluating Limits 861
12.3 The Tangent Line Problem 871
12.4 Limits at Infinity and Limits of Sequences 881
12.5 The Area Problem 890
Cumulative Test for Chapters 10–12 904
Appendix A Review of Fundamental Concepts of Algebra
Real Numbers and Their Properties A1
Exponents and Radicals A14
Polynomials and Factoring A27
Rational Expressions A39
Solving Equations A49
Linear Inequalities in One Variable A63
Errors and the Algebra of Calculus A73
Answers to Odd-Numbered Exercises and Tests
Index of Applications (web)
Appendix B Concepts in Statistics (web)
Representing Data
Measures of Central Tendency and Dispersion
Least Squares Regression
A Word from
Welcome to the Second Edition of Precalculus with Limits! We are proud to offer you
a new and revised version of our textbook. With the Second Edition, we have listened
to you, our users, and have incorporated many of your suggestions for improvement.
In this edition, we continue to offer instructors and students a text that is pedagogically
sound, mathematically precise, and still comprehensible. There are many changes in the
mathematics, art, and design; the more significant changes are noted here.
• New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In
Real Life, and In Careers. In Mathematics describes an important mathematical
topic taught in the chapter. In Real Life tells students where they will encounter this
topic in real-life situations. In Careers relates application exercises to a variety of
• New Study Tips and Warning/Cautions Insightful information is given to
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information or suggestions for learning the topic. The Warning/Caution points out
common mathematical errors made by students.
• New Algebra Helps Algebra Help directs students to sections of the textbook
where they can review algebra skills needed to master the current topic.
• New Side-by-Side Examples Throughout the text, we present solutions to many
examples from multiple perspectives—algebraically, graphically, and numerically.
The side-by-side format of this pedagogical feature helps students to see that a problem
can be solved in more than one way and to see that different methods yield the same
result. The side-by-side format also addresses many different learning styles.
A Word from the Author
• New Capstone Exercises Capstones are conceptual problems that synthesize key
topics and provide students with a better understanding of each section's
concepts. Capstone exercises are excellent for classroom discussion or test prep, and
teachers may find value in integrating these problems into their reviews of the
• New Chapter Summaries The Chapter Summary now includes an explanation
and/or example of each objective taught in the chapter.
• Revised Exercise Sets The exercise sets have been carefully and extensively
examined to ensure they are rigorous and cover all topics suggested by our users.
Many new skill-building and challenging exercises have been added.
For the past several years, we've maintained an independent website—
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Thousands of students using our textbooks have visited the site for practice and help
with their homework. For the Second Edition, we were able to use information from
CalcChat.com, including which solutions students accessed most often, to help guide
the revision of the exercises.
I hope you enjoy the Second Edition of Precalculus with Limits. As always, I
welcome comments and suggestions for continued improvements.
I would like to thank the many people who have helped me prepare the text and the
supplements package. Their encouragement, criticisms, and suggestions have been
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Thank you to all of the instructors who took the time to review the changes in this
edition and to provide suggestions for improving it. Without your help, this book would
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University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo,
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Florida, for their significant contributions to previous editions of this text.
I would also like to thank the staff at Larson Texts, Inc. who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and
typesetting the supplements.
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love, patience, and support. Also, a special thanks goes to R. Scott O'Neil. If you have
suggestions for improving this text, please feel free to write to me. Over the past two
decades I have received many useful comments from both instructors and students, and
I value these comments very highly.
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Functions and
Their Graphs
Rectangular Coordinates
Graphs of Equations
Linear Equations in Two Variables
Analyzing Graphs of Functions
A Library of Parent Functions
Transformations of Functions
Inverse Functions
Combinations of Functions:
Composite Functions
Mathematical Modeling and Variation
Functions show how one variable is related
to another variable.
Functions are used to estimate values,
simulate processes, and discover relationships. For instance, you can model the
enrollment rate of children in preschool
and estimate the year in which the rate
will reach a certain number. Such an
estimate can be used to plan measures
for meeting future needs, such as hiring
additional teachers and buying more
books. (See Exercise 113, page 64.)
Jose Luis Pelaez/Getty Images
IN CAREERS
There are many careers that use functions. Several are listed below.
• Financial analyst
Exercise 95, page 51
• Tax preparer
Example 3, page 104
• Biologist
• Oceanographer
Exercise 83, page 112
1.1 RECTANGULAR COORDINATES
What you should learn
The Cartesian Plane
• Plot points in the Cartesian plane.
• Use the Distance Formula to find
the distance between two points.
• Use the Midpoint Formula to find
the midpoint of a line segment.
• Use a coordinate plane to model
and solve real-life problems.
Just as you can represent real numbers by points on a real number line, you can
represent ordered pairs of real numbers by points in a plane called the rectangular
coordinate system, or the Cartesian plane, named after the French mathematician
René Descartes (1596–1650).
The Cartesian plane is formed by using two real number lines intersecting at right
angles, as shown in Figure 1.1. The horizontal real number line is usually called the
x-axis, and the vertical real number line is usually called the y-axis. The point of
intersection of these two axes is the origin, and the two axes divide the plane into four
parts called quadrants.
Why you should learn it
The Cartesian plane can be used to
represent relationships between two
variables. For instance, in Exercise 70
on page 11, a graph represents the
minimum wage in the United States
from 1950 to 2009.
Quadrant II
Quadrant I
Directed distance
(Vertical
number line)
Quadrant III
(Horizontal
y distance
Quadrant IV
© Ariel Skelly/Corbis
Each point in the plane corresponds to an ordered pair (x, y) of real numbers x and
y, called coordinates of the point. The x-coordinate represents the directed distance
from the y-axis to the point, and the y-coordinate represents the directed distance from
the x-axis to the point, as shown in Figure 1.2.
from y-axis
(−1, 2)
(−2, −3)
from x-axis
The notation x, y denotes both a point in the plane and an open interval on the real
number line. The context will tell you which meaning is intended.
Plotting Points in the Cartesian Plane
Plot the points 1, 2, 3, 4, 0, 0, 3, 0, and 2, 3.
To plot the point 1, 2, imagine a vertical line through 1 on the x-axis and a
horizontal line through 2 on the y-axis. The intersection of these two lines is the point
1, 2. The other four points can be plotted in a similar way, as shown in Figure 1.3.
Now try Exercise 7.
Section 1.1
The beauty of a rectangular coordinate system is that it allows you to see relationships between two variables. It would be difficult to overestimate the importance of
Descartes's introduction of coordinates in the plane. Today, his ideas are in common use
in virtually every scientific and business-related field.
Subscribers, N
From 1994 through 2007, the numbers N (in millions) of subscribers to a cellular
telecommunication service in the United States are shown in the table, where t
represents the year. Sketch a scatter plot of the data. (Source: CTIA-The Wireless
Association)
To sketch a scatter plot of the data shown in the table, you simply represent each pair
of values by an ordered pair t, N  and plot the resulting points, as shown in Figure 1.4.
For instance, the first pair of values is represented by the ordered pair 1994, 24.1.
Note that the break in the t-axis indicates that the numbers between 0 and 1994 have
been omitted.
Year, t
Sketching a Scatter Plot
Subscribers to a Cellular
Telecommunication Service
Now try Exercise 25.
In Example 2, you could have let t  1 represent the year 1994. In that case, the
horizontal axis would not have been broken, and the tick marks would have been
labeled 1 through 14 (instead of 1994 through 2007).
T E C H N O LO G Y
The scatter plot in Example 2 is only one way to represent the data graphically.
You could also represent the data using a bar graph or a line graph. If you have
access to a graphing utility, try using it to represent graphically the data given in
Example 2.
The Pythagorean Theorem and the Distance Formula
a2 + b2 = c2
The following famous theorem is used extensively throughout this course.
For a right triangle with hypotenuse of length c and sides of lengths a and b, you
have a 2  b2  c 2, as shown in Figure 1.5. (The converse is also true. That is, if
a 2  b2  c 2, then the triangle is a right triangle.)
Suppose you want to determine the distance d between two points x1, y1 and
x2, y2 in the plane. With these two points, a right triangle can be formed, as shown in
Figure 1.6. The length of the vertical side of the triangle is y2  y1 , and the length of
the horizontal side is x2  x1 . By the Pythagorean Theorem, you can write
(x1, y1 )
d 2  x2  x1 2  y2  y1
y 2 − y1
x 2 − x1
The Distance Formula
The distance d between the points x1, y1 and x2, y2  in the plane is
x2  x12   y2  y12.
x2  x1 2  y2  y1 2
This result is the Distance Formula.
(x1, y2 ) (x2, y2 )
Finding a Distance
Find the distance between the points 2, 1 and 3, 4.
Algebraic Solution
Let x1, y1  2, 1 and x2, y2   3, 4. Then apply the
Distance Formula.
x2  x12   y2  y12
3  2  4  1
Distance Formula
Substitute for
x1, y1, x2, and y2.
Graphical Solution
Use centimeter graph paper to plot the points A2, 1 and
B3, 4. Carefully sketch the line segment from A to B. Then
use a centimeter ruler to measure the length of the segment.
Use a calculator.
Distance checks.
So, the distance between the points is about 5.83 units. You
can use the Pythagorean Theorem to check that the distance is
d 2  32  52
Substitute for d.
 34   32  52
The line segment measures about 5.8 centimeters, as shown
in Figure 1.7. So, the distance between the points is about
5.8 units.
Verifying a Right Triangle
Show that the points 2, 1, 4, 0, and 5, 7 are vertices of a right triangle.
d1 = 45
The three points are plotted in Figure 1.8. Using the Distance Formula, you can find the
lengths of the three sides as follows.
d2 = 5
d1
5  2 2  7  1 2
5  4  7  0
d12  d22  45  5  50  d32
you can conclude by the Pythagorean Theorem that the triangle must be a right triangle.
You can review the techniques
for evaluating a radical in
Appendix A.2.
The Midpoint Formula
To find the midpoint of the line segment that joins two points in a coordinate plane,
you can simply find the average values of the respective coordinates of the two
endpoints using the Midpoint Formula.
The midpoint of the line segment joining the points x1, y1 and x 2, y 2  is given
by the Midpoint Formula
Midpoint
x1  x 2 y1  y2
For a proof of the Midpoint Formula, see Proofs in Mathematics on page 122.
Finding a Line Segment's Midpoint
Find the midpoint of the line segment joining the points 5, 3 and 9, 3.
Let x1, y1  5, 3 and x 2, y 2   9, 3.
x1  x2 y1  y2
5  9 3  3
 2, 0
Midpoint Formula
Substitute for x1, y1, x2, and y2.
The midpoint of the line segment is 2, 0, as shown in Figure 1.9.
Now try Exercise 47(c).
Finding the Length of a Pass
A football quarterback throws a pass from the 28-yard line, 40 yards from the sideline.
The pass is caught by a wide receiver on the 5-yard line, 20 yards from the same
sideline, as shown in Figure 1.10. How long is the pass?
You can find the length of the pass by finding the distance between the points 40, 28
and 20, 5.
Football Pass
Distance (in yards)
(20, 5)
40  20 2  28  5 2
(40, 28)
So, the pass is about 30 yards long.
In Example 6, the scale along the goal line does not normally appear on a football
field. However, when you use coordinate geometry to solve real-life problems, you are
free to place the coordinate system in any way that is convenient for the solution of the
Estimating Annual Revenue
Barnes & Noble had annual sales of approximately $5.1 billion in 2005, and $5.4 billion
in 2007. Without knowing any additional information, what would you estimate the
2006 sales to have been? (Source: Barnes & Noble, Inc.)
Sales (in billions of dollars)
One solution to the problem is to assume that sales followed a linear pattern. With this
assumption, you can estimate the 2006 sales by finding the midpoint of the line
segment connecting the points 2005, 5.1 and 2007, 5.4.
Barnes & Noble Sales
(2007, 5.4)
(2006, 5.25)
2005  2007 5.1  5.4
Substitute for x1, x2, y1 and y2.
 2006, 5.25
So, you would estimate the 2006 sales to have been about $5.25 billion, as shown in
Figure 1.11. (The actual 2006 sales were about $5.26 billion.)
Translating Points in the Plane
The triangle in Figure 1.12 has vertices at the points 1, 2, 1, 4, and 2, 3. Shift
the triangle three units to the right and two units upward and find the vertices of the
shifted triangle, as shown in Figure 1.13.
Paul Morrell
Much of computer graphics,
including this computer-generated
goldfish tessellation, consists of
transformations of points in a
coordinate plane. One type of
transformation, a translation, is
illustrated in Example 8. Other
types include reflections, rotations,
and stretches.
(1, −4)
To shift the vertices three units to the right, add 3 to each of the x-coordinates. To shift
the vertices two units upward, add 2 to each of the y-coordinates.
Original Point
Translated Point
1  3, 2  2  2, 4
The figures provided with Example 8 were not really essential to the solution.
Nevertheless, it is strongly recommended that you develop the habit of including
sketches with your solutions—even if they are not required.
CLASSROOM DISCUSSION
Extending the Example Example 8 shows how to translate points in a coordinate
plane. Write a short paragraph describing how each of the following transformed
points is related to the original point.
Transformed Point
ⴚx, y
x, ⴚy
ⴚx, ⴚy
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
1. Match each term with its definition.
(a) x-axis
(i) point of intersection of vertical axis and horizontal axis
(b) y-axis
(ii) directed distance from the x-axis
(c) origin
(iii) directed distance from the y-axis
(d) quadrants
(iv) four regions of the coordinate plane
(e) x-coordinate
(v) horizontal real number line
(f) y-coordinate
(vi) vertical real number line
In Exercises 2– 4, fill in the blanks.
2. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the
________ plane.
3. The ________ ________ is a result derived from the Pythagorean Theorem.
4. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane
is also known as using the ________ ________.
SKILLS AND APPLICATIONS
In Exercises 5 and 6, approximate the coordinates of the
points.
In Exercises 7–10, plot the points in the Cartesian plane.
3, 8, 0.5, 1, 5, 6, 2, 2.5
1,  13 ,  34, 3, 3, 4,  43,  32
In Exercises 11–14, find the coordinates of the point.
11. The point is located three units to the left of the y-axis
and four units above the x-axis.
12. The point is located eight units below the x-axis and
four units to the right of the y-axis.
13. The point is located five units below the x-axis and the
coordinates of the point are equal.
14. The point is on the x-axis and 12 units to the left of the
y-axis.
In Exercises 15–24, determine the quadrant(s) in which
x, y is located so that the condition(s) is (are) satisfied.
x > 0 and y < 0
x  4 and y > 0
y < 5
x < 0 and y > 0
xy > 0
x < 0 and y < 0
x > 2 and y  3
x > 4
xy < 0
In Exercises 25 and 26, sketch a scatter plot of the data shown
in the table.
25. NUMBER OF STORES The table shows the number y
of Wal-Mart stores for each year x from 2000 through
2007. (Source: Wal-Mart Stores, Inc.)
Year, x
Number of stores, y
26. METEOROLOGY The table shows the lowest temperature on record y (in degrees Fahrenheit) in Duluth,
Minnesota for each month x, where x  1 represents
January. (Source: NOAA)
Month, x
Temperature, y
In Exercises 43–46, show that the points form the vertices of
the indicated polygon.
Right triangle: 4, 0, 2, 1, 1, 5
Right triangle: 1, 3), 3, 5, 5, 1
Isosceles triangle: 1, 3, 3, 2, 2, 4
 23, 3, 1, 54
57. FLYING DISTANCE An airplane flies from Naples,
Italy in a straight line to Rome, Italy, which is
120 kilometers north and 150 kilometers west of Naples.
How far does the plane fly?
58. SPORTS A soccer player passes the ball from a point
that is 18 yards from the endline and 12 yards from the
sideline. The pass is received by a teammate who is
42 yards from the same endline and 50 yards from the
same sideline, as shown in the figure. How long is the
pass?
In Exercises 39– 42, (a) find the length of each side of the
right triangle, and (b) show that these lengths satisfy the
Pythagorean Theorem.
48. 1, 12, 6, 0
50. 7, 4, 2, 8
52. 2, 10, 10, 2
54.  13,  13 ,  16,  12
56. 16.8, 12.3, 5.6, 4.9
53.  12, 1,  52, 43
55. 6.2, 5.4, 3.7, 1.8
12, 43 , 2, 1
4.2, 3.1, 12.5, 4.8
9.5, 2.6, 3.9, 8.2
SALES In Exercises 59 and 60, use the Midpoint Formula to
estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc.
in 2005, given the sales in 2003 and 2007. Assume that the
sales followed a linear pattern. (Source: Big Lots, Inc.;
Dollar Tree Stores, Inc.)
59. Big Lots
Sales (in millions)
In Exercises 47–56, (a) plot the points, (b) find the distance
between the points, and (c) find the midpoint of the line
segment joining the points.
In Exercises 27–38, find the distance between the points.
In Exercises 61–64, the polygon is shifted to a new position
in the plane. Find the coordinates of the vertices of the
polygon in its new position.
(−3, 6) 7 (−1, 3)
5 6 units
(−2, − 4)
63. Original coordinates of vertices: 7, 2,2, 2,
Shift: eight units upward, four units to the right
64. Original coordinates of vertices: 5, 8, 3, 6, 7, 6,
Shift: 6 units downward, 10 units to the left
RETAIL PRICE In Exercises 65 and 66, use the graph,
which shows the average retail prices of 1 gallon of whole
milk from 1996 to 2007. (Source: U.S. Bureau of Labor
Statistics)
(in dollars per gallon)
FIGURE FOR
Cost of 30-second TV spot
(in thousands of dollars)
(a) Estimate the percent increase in the average cost of
a 30-second spot from Super Bowl XXXIV in 2000
to Super Bowl XXXVIII in 2004.
(b) Estimate the percent increase in the average cost of
to Super Bowl XLII in 2008.
68. ADVERTISING The graph shows the average costs of
a 30-second television spot (in thousands of dollars)
during the Academy Awards from 1995 to 2007.
(Source: Nielson Monitor-Plus)
65. Approximate the highest price of a gallon of whole milk
shown in the graph. When did this occur?
66. Approximate the percent change in the price of milk
from the price in 1996 to the highest price shown in the
graph.
during the Super Bowl from 2000 to 2008. (Source:
Nielson Media and TNS Media Intelligence)
a 30-second spot in 1996 to the cost in 2002.
69. MUSIC The graph shows the numbers of performers
who were elected to the Rock and Roll Hall of Fame
from 1991 through 2008. Describe any trends in the
data. From these trends, predict the number of
performers elected in 2010. (Source: rockhall.com)
Number elected
Minimum wage (in dollars)
70. LABOR FORCE Use the graph below, which shows
the minimum wage in the United States (in dollars)
from 1950 to 2009. (Source: U.S. Department of
Labor)
Pieces of mail, y
(a) Which decade shows the greatest increase in
minimum wage?
(b) Approximate the percent increases in the minimum
wage from 1990 to 1995 and from 1995 to 2009.
(c) Use the percent increase from 1995 to 2009 to
predict the minimum wage in 2013.
(d) Do you believe that your prediction in part (c) is
reasonable? Explain.
71. SALES The Coca-Cola Company had sales of $19,805
million in 1999 and $28,857 million in 2007. Use the
Midpoint Formula to estimate the sales in 2003. Assume
that the sales followed a linear pattern. (Source: The
Coca-Cola Company)
72. DATA ANALYSIS: EXAM SCORES The table shows
the mathematics entrance test scores x and the final
examination scores y in an algebra course for a sample
of 10 students.
(a) Sketch a scatter plot of the data.
(b) Find the entrance test score of any student with a
final exam score in the 80s.
(c) Does a higher entrance test score imply a higher
final exam score? Explain.
73. DATA ANALYSIS: MAIL The table shows the number
y of pieces of mail handled (in billions) by the U.S.
Postal Service for each year x from 1996 through 2008.
(Source: U.S. Postal Service)
TABLE FOR
(b) Approximate the year in which there was the greatest
decrease in the number of pieces of mail handled.
(c) Why do you think the number of pieces of mail
handled decreased?
74. DATA ANALYSIS: ATHLETICS The table shows the
numbers of men's M and women's W college basketball
teams for each year x from 1994 through 2007.
(Source: National Collegiate Athletic Association)
Year,
teams, M
teams, W
(a) Sketch scatter plots of these two sets of data on the
same set of coordinate axes.
(b) Find the year in which the numbers of men's and
women's teams were nearly equal.
(c) Find the year in which the difference between the
numbers of men's and women's teams was the greatest. What was this difference?
75. A line segment has x1, y1 as one endpoint and xm, ym
as its midpoint. Find the other endpoint x2, y2  of the
line segment in terms of x1, y1, xm, and ym.
76. Use the result of Exercise 75 to find the coordinates of
the endpoint of a line segment if the coordinates of the
other endpoint and midpoint are, respectively,
(a) 1, 2, 4, 1 and (b) 5, 11, 2, 4.
77. Use the Midpoint Formula three times to find the three
points that divide the line segment joining x1, y1 and
x2, y2  into four parts.
78. Use the result of Exercise 77 to find the points that
divide the line segment joining the given points into
four equal parts.
(a) 1, 2, 4, 1 (b) 2, 3, 0, 0
79. MAKE A CONJECTURE Plot the points 2, 1,
3, 5, and 7, 3 on a rectangular coordinate
system. Then change the sign of the x-coordinate of
each point and plot the three new points on the same
rectangular coordinate system. Make a conjecture about
the location of a point when each of the following
occurs.
(a) The sign of the x-coordinate is changed.
(b) The sign of the y-coordinate is changed.
(c) The signs of both the x- and y-coordinates are
80. COLLINEAR POINTS Three or more points are
collinear if they all lie on the same line. Use the steps
below to determine if the set of points A2, 3, B2, 6,
C6, 3 and the set of points A8, 3, B5, 2, C2, 1
are collinear.
(a) For each set of points, use the Distance Formula to
find the distances from A to B, from B to C, and
from A to C. What relationship exists among these
distances for each set of points?
(b) Plot each set of points in the Cartesian plane. Do all
the points of either set appear to lie on the same
line?
(c) Compare your conclusions from part (a) with the
conclusions you made from the graphs in part (b).
Make a general statement about how to use the
Distance Formula to determine collinearity.
TRUE OR FALSE? In Exercises 81 and 82, determine
whether the statement is true or false. Justify your answer.
81. In order to divide a line segment into 16 equal parts, you
would have to use the Midpoint Formula 16 times.
82. The points 8, 4, 2, 11, and 5, 1 represent the
vertices of an isosceles triangle.
83. THINK ABOUT IT When plotting points on the
rectangular coordinate system, is it true that the scales
on the x- and y-axes must be the same? Explain.
84. CAPSTONE Use the plot of the point x0 , y0  in the
figure. Match the transformation of the point with the
correct plot. Explain your reasoning. [The plots are
labeled (i), (ii), (iii), and (iv).]
(x0 , y0 )
(a) x0, y0
(c) x0, 12 y0
(b) 2x0, y0
(d) x0, y0
85. PROOF Prove that the diagonals of the parallelogram
in the figure intersect at their midpoints.
(b , c)
(a + b , c)
(a, 0)
1.2 GRAPHS OF EQUATIONS
• Sketch graphs of equations.
• Find x- and y-intercepts of graphs of
equations.
• Use symmetry to sketch graphs of
• Find equations of and sketch graphs
of circles.
• Use graphs of equations in solving
real-life problems.
The graph of an equation can help
you see relationships between real-life
quantities. For example, in Exercise 87
on page 23, a graph can be used to
estimate the life expectancies of
children who are born in 2015.
The Graph of an Equation
In Section 1.1, you used a coordinate system to represent graphically the relationship
between two quantities. There, the graphical picture consisted of a collection of points
in a coordinate plane.
Frequently, a relationship between two quantities is expressed as an equation in
two variables. For instance, y  7  3x is an equation in x and y. An ordered pair
a, b is a solution or solution point of an equation in x and y if the equation is true
when a is substituted for x and b is substituted for y. For instance, 1, 4 is a solution of
y  7  3x because 4  7  31 is a true statement.
In this section you will review some basic procedures for sketching the graph of an
equation in two variables. The graph of an equation is the set of all points that are
solutions of the equation.
Determining Solution Points
Determine whether (a) 2, 13 and (b) 1, 3 lie on the graph of y  10x  7.
y  10x  7
13  102  7
Write original equation.
Substitute 2 for x and 13 for y.
2, 13 is a solution.
The point 2, 13 does lie on the graph of y  10x  7 because it is a solution point
of the equation.
3  101  7
Substitute 1 for x and 3 for y.
1, 3 is not a solution.
© John Griffin/The Image Works
The point 1, 3 does not lie on the graph of y  10x  7 because it is not a
solution point of the equation.
The basic technique used for sketching the graph of an equation is the
point-plotting method.
Sketching the Graph of an Equation by Point Plotting
1. If possible, rewrite the equation so that one of the variables is isolated on one
side of the equation.
When evaluating an expression
or an equation, remember to
follow the Basic Rules of
Algebra. To review these rules,
see Appendix A.1.
2. Make a table of values showing several solution points.
3. Plot these points on a rectangular coordinate system.
4. Connect the points with a smooth curve or line.
Sketching the Graph of an Equation
Sketch the graph of
y  7  3x.
Because the equation is already solved for y, construct a table of values that consists of
several solution points of the equation. For instance, when x  1,
y  7  31
which implies that 1, 10 is a solution point of the graph.
y  7  3x
From the table, it follows that
1, 10, 0, 7, 1, 4, 2, 1, 3, 2, and 4, 5
are solution points of the equation. After plotting these points, you can see that they
appear to lie on a line, as shown in Figure 1.14. The graph of the equation is the line
that passes through the six plotted points.
(− 1, 10)
(3, − 2)
y  x 2  2.
Because the equation is already solved for y, begin by constructing a table of values.
y  x2  2
One of your goals in this course
is to learn to classify the basic
shape of a graph from its
equation. For instance, you will
learn that the linear equation in
Example 2 has the form
Next, plot the points given in the table, as shown in Figure 1.15. Finally, connect the
points with a smooth curve, as shown in Figure 1.16.
y  mx  b
and its graph is a line. Similarly,
the quadratic equation in
y  ax  bx  c
y = x2 − 2
and its graph is a parabola.
The point-plotting method demonstrated in Examples 2 and 3 is easy to use, but it
has some shortcomings. With too few solution points, you can misrepresent the graph
of an equation. For instance, if only the four points
2, 2, 1, 1, 1, 1, and 2, 2
in Figure 1.15 were plotted, any one of the three graphs in Figure 1.17 would be
reasonable.
To graph an equation involving x and y on a graphing utility, use the
following procedure.
1. Rewrite the equation so that y is isolated on the left side.
2. Enter the equation into the graphing utility.
No x-intercepts; one y-intercept
3. Determine a viewing window that shows all important features of the graph.
4. Graph the equation.
Intercepts of a Graph
It is often easy to determine the solution points that have zero as either the x-coordinate
or the y-coordinate. These points are called intercepts because they are the points at
which the graph intersects or touches the x- or y-axis. It is possible for a graph to have
no intercepts, one intercept, or several intercepts, as shown in Figure 1.18.
Note that an x-intercept can be written as the ordered pair x, 0 and a
y-intercept can be written as the ordered pair 0, y. Some texts denote the x-intercept
as the x-coordinate of the point a, 0 [and the y-intercept as the y-coordinate of the
point 0, b] rather than the point itself. Unless it is necessary to make a distinction, we
will use the term intercept to mean either the point or the coordinate.
Three x-intercepts; one y-intercept
Finding Intercepts
One x-intercept; two y-intercepts
1. To find x-intercepts, let y be zero and solve the equation for x.
2. To find y-intercepts, let x be zero and solve the equation for y.
Finding x- and y-Intercepts
Find the x- and y-intercepts of the graph of y  x3  4x.
Let y  0. Then
0  x3  4x  xx2  4
has solutions x  0 and x  ± 2.
y = x 3 − 4x 4
Let x  0. Then
x-intercepts: 0, 0, 2, 0, 2, 0
y  03  40
has one solution, y  0.
See Figure 1.19.
Graphs of equations can have symmetry with respect to one of the coordinate axes or
with respect to the origin. Symmetry with respect to the x-axis means that if the
Cartesian plane were folded along the x-axis, the portion of the graph above the x-axis
would coincide with the portion below the x-axis. Symmetry with respect to the y-axis
or the origin can be described in a similar manner, as shown in Figure 1.20.
(−x, y)
(x, −y)
(−x, −y)
x-axis symmetry
Knowing the symmetry of a graph before attempting to sketch it is helpful, because
then you need only half as many solution points to sketch the graph. There are three basic
types of symmetry, described as follows.
Graphical Tests for Symmetry
1. A graph is symmetric with respect to the x-axis if, whenever x, y is on the
graph, x, y is also on the graph.
2. A graph is symmetric with respect to the y-axis if, whenever x, y is on the
3. A graph is symmetric with respect to the origin if, whenever x, y is on the
(− 1, −1)
You can conclude that the graph of y  x 2  2 is symmetric with respect to the
y-axis because the point x, y is also on the graph of y  x2  2. (See the table below
and Figure 1.21.)
1.21 y-axis symmetry
Algebraic Tests for Symmetry
1. The graph of an equation is symmetric with respect to the x-axis if replacing y
with y yields an equivalent equation.
2. The graph of an equation is symmetric with respect to the y-axis if replacing x
with x yields an equivalent equation.
3. The graph of an equation is symmetric with respect to the origin if replacing x
with x and y with y yields an equivalent equation.
Test y  2x3 for symmetry with respect to both axes and the origin.
x-axis:
y = 2x 3 1
y-axis:
Replace y with y. Result is not an equivalent equation.
x−
y  2x3
y  2x
Replace y with y and x with x.
Equivalent equation
Of the three tests for symmetry, the only one that is satisfied is the test for origin
symmetry (see Figure 1.22).
Replace x with x.
Simplify. Result is not an equivalent equation.
Testing for Symmetry
Using Symmetry as a Sketching Aid
Use symmetry to sketch the graph of x  y 2  1.
Of the three tests for symmetry, the only one that is satisfied is the test for x-axis symmetry because x  y2  1 is equivalent to x  y2  1. So, the graph is symmetric
with respect to the x-axis. Using symmetry, you only need to find the solution points
above the x-axis and then reflect them to obtain the graph, as shown in Figure 1.23.
In Example 7, x  1 is an
absolute value expression. You
can review the techniques for
evaluating an absolute value
expression in Appendix A.1.
Sketch the graph of y  x  1 .
This equation fails all three tests for symmetry and consequently its graph is not
symmetric with respect to either axis or to the origin. The absolute value sign indicates
that y is always nonnegative. Create a table of values and plot the points, as shown in
Figure 1.24. From the table, you can see that x  0 when y  1. So, the y-intercept is
0, 1. Similarly, y  0 when x  1. So, the x-intercept is 1, 0.
y = ⏐x − 1⏐
(−2, 3) 4
(1, 0) 2
y x1
Throughout this course, you will learn to recognize several types of graphs from
their equations. For instance, you will learn to recognize that the graph of a seconddegree equation of the form
y  ax 2  bx  c
Center: (h, k)
is a parabola (see Example 3). The graph of a circle is also easy to recognize.
Radius: r
Point on
circle: (x, y)
Consider the circle shown in Figure 1.25. A point x, y is on the circle if and only if its
distance from the center h, k is r. By the Distance Formula,
x  h2   y  k2  r.
By squaring each side of this equation, you obtain the standard form of the equation
of a circle.
Standard Form of the Equation of a Circle
The point x, y lies on the circle of radius r and center (h, k) if and only if
x  h 2   y  k 2  r 2.
WARNING / CAUTION
Be careful when you are finding
h and k from the standard
equation of a circle. For
instance, to find the correct h
and k from the equation of the
circle in Example 8, rewrite the
quantities x  12 and  y  22
using subtraction.
From this result, you can see that the standard form of the equation of a
circle with its center at the origin, h, k  0, 0, is simply
x 2  y 2  r 2.
 y  22   y  22
So, h  1 and k  2.
The radius of the circle is the distance between 1, 2 and 3, 4.
x  h2   y  k2
3  1 2  4  22
Substitute for x, y, h, and k.
Finding the Equation of a Circle
The point 3, 4 lies on a circle whose center is at 1, 2, as shown in Figure 1.26.
Write the standard form of the equation of this circle.
x  12  x  12,
Circle with center at origin
Using h, k  1, 2 and r
20, the equation of the circle is
x  h2   y  k2  r 2
Equation of circle
x  1   y  2   20
x  1 2   y  2 2  20.
Substitute for h, k, and r.
In this course, you will learn that there are many ways to approach a problem. Three
common approaches are illustrated in Example 9.
You should develop the habit
of using at least two approaches
to solve every problem. This
helps build your intuition and
helps you check that your
answers are reasonable.
A Numerical Approach: Construct and use a table.
A Graphical Approach: Draw and use a graph.
An Algebraic Approach: Use the rules of algebra.
Recommended Weight
The median recommended weight y (in pounds) for men of medium frame who are 25
to 59 years old can be approximated by the mathematical model
y  0.073x 2  6.99x  289.0, 62
where x is the man's height (in inches).
Company)
(Source: Metropolitan Life Insurance
a. Construct a table of values that shows the median recommended weights for men
with heights of 62, 64, 66, 68, 70, 72, 74, and 76 inches.
b. Use the table of values to sketch a graph of the model. Then use the graph to
estimate graphically the median recommended weight for a man whose height is
71 inches.
c. Use the model to confirm algebraically the estimate you found in part (b).
Weight, y
a. You can use a calculator to complete the table, as shown at the left.
b. The table of values can be used to sketch the graph of the equation, as shown in
Figure 1.27. From the graph, you can estimate that a height of 71 inches
corresponds to a weight of about 161 pounds.
Height, x
Height (in inches)
c. To confirm algebraically the estimate found in part (b), you can substitute 71 for x
in the model.
y  0.073(71)2  6.99(71)  289.0  160.70
So, the graphical estimate of 161 pounds is fairly good.
VOCABULARY: Fill in the blanks.
1. An ordered pair a, b is a ________ of an equation in x and y if the equation is true when a is substituted for
x, and b is substituted for y.
The set of all solution points of an equation is the ________ of the equation.
The points at which a graph intersects or touches an axis are called the ________ of the graph.
A graph is symmetric with respect to the ________ if, whenever x, y is on the graph, x, y is also on the graph.
The equation x  h2   y  k2  r 2 is the standard form of the equation of a ________ with center ________
and radius ________.
When you construct and use a table to solve a problem, you are using a ________ approach.
In Exercises 7–14, determine whether each point lies on the
graph of the equation.
y 5x
y  x 2  3x  2
y4 x2
y x1 2
2x  y  3  0
x2  y2  20
y  13x3  2x 2
2,  3  (b)
18. y  5  x 2
In Exercises 19–22, graphically estimate the x- and y-intercepts
of the graph. Verify your results algebraically.
19. y  x  32
22. y2  4  x
−4 −3 −2 −1
In Exercises 23–32, find the x- and y-intercepts of the graph
17. y  x 2  3x
21. y  x  2
16. y  4 x  1
20. y  16  4x 2
15. y  2x  5
In Exercises 15–18, complete the table. Use the resulting
solution points to sketch the graph of the equation.
y  5x  6
y  2x3  4x 2
y2  6  x
y   x  10
y  x 4  25
y2  x  1
In Exercises 33– 40, use the algebraic tests to check for
symmetry with respect to both axes and the origin.
33. x 2  y  0
35. y  x 3
37. y  2
39. xy  10  0
34. x  y 2  0
36. y  x 4  x 2  3
38. y
40. xy  4
In Exercises 41– 44, assume that the graph has the indicated
type of symmetry. Sketch the complete graph of the equation.
To print an enlarged copy of the graph, go to the website
www.mathgraphs.com.
66. y  6  x x
68. y  2  x
In Exercises 69–76, write the standard form of the equation
of the circle with the given characteristics.
65. y  x x  6
Center: 0, 0; Radius: 4
Center: 1, 2; Solution point: 0, 0
Endpoints of a diameter: 0, 0, 6, 8
In Exercises 45–56, identify any intercepts and test for
symmetry. Then sketch the graph of the equation.
y  x 2  2x
x  y2  1
y1 x
In Exercises 57–68, use a graphing utility to graph the equation.
Use a standard setting. Approximate any intercepts.
57. y  3  2x
59. y  x 2  4x  3
60. y  x 2  x  2
In Exercises 77– 82, find the center and radius of the circle,
and sketch its graph.
x 2  y 2  25
x  12   y  32  9
x  12 2  y  12 2  94
x  22   y  32  169
78. x 2  y 2  16
80. x 2   y  1 2  1
83. DEPRECIATION A hospital purchases a new magnetic
resonance imaging (MRI) machine for $500,000. The
depreciated value y (reduced value) after t years is given
by y  500,000  40,000t, 0 t 8. Sketch the
84. CONSUMERISM You purchase an all-terrain vehicle
(ATV) for $8000. The depreciated value y after t years
is given by y  8000  900t, 0 t 6. Sketch the
85. GEOMETRY A regulation NFL playing field (including
the end zones) of length x and width y has a perimeter
of 3463 or 3 yards.
(a) Draw a rectangle that gives a visual representation
of the problem. Use the specified variables to label
the sides of the rectangle.
(b) Show that the width of the rectangle is
 x and its area is A  x
(c) Use a graphing utility to graph the area equation. Be
sure to adjust your window settings.
(d) From the graph in part (c), estimate the dimensions
of the rectangle that yield a maximum area.
(e) Use your school's library, the Internet, or some
other reference source to find the actual dimensions
and area of a regulation NFL playing field and
compare your findings with the results of part (d).
indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility.
86. GEOMETRY A soccer playing field of length x and
width y has a perimeter of 360 meters.
(b) Show that the width of the rectangle is y  180  x
and its area is A  x180  x.
and area of a regulation Major League Soccer field
and compare your findings with the results of
part (d).
87. POPULATION STATISTICS The table shows the life
expectancies of a child (at birth) in the United States for
selected years from 1920 to 2000. (Source: U.S.
National Center for Health Statistics)
Life Expectancy, y
(e) Do you think this model can be used to predict the
life expectancy of a child 50 years from now?
88. ELECTRONICS The resistance y (in ohms) of 1000 feet
of solid copper wire at 68 degrees Fahrenheit can be
approximated by the model
 0.37, 5
where x is the diameter of the wire in mils (0.001 inch).
(Source: American Wire Gage)
(a) Complete the table.
(b) Use the table of values in part (a) to sketch a graph
of the model. Then use your graph to estimate the
resistance when x  85.5.
(c) Use the model to confirm algebraically the estimate
you found in part (b).
(d) What can you conclude in general about the
relationship between the diameter of the copper
wire and the resistance?
A model for the life expectancy during this period is
y  0.0025t 2  0.574t  44.25, 20
where y represents the life expectancy and t is the time
in years, with t  20 corresponding to 1920.
(a) Use a graphing utility to graph the data from the
table and the model in the same viewing window.
How well does the model fit the data? Explain.
(b) Determine the life expectancy in 1990 both graphically and algebraically.
(c) Use the graph to determine the year when life
expectancy was approximately 76.0. Verify your
answer algebraically.
(d) One projection for the life expectancy of a child
born in 2015 is 78.9. How does this compare with
the projection given by the model?
89. THINK ABOUT IT Find a and b if the graph of
y  ax 2  bx 3 is symmetric with respect to (a) the
y-axis and (b) the origin. (There are many correct
answers.)
90. CAPSTONE Match the
the given characteristic.
(i) y  3x3  3x (ii)
(iii) y  3x  3
(v) y  3x  3 (vi)
equation or equations with
y  x  32
y3x
Symmetric with respect to the y-axis
Three x-intercepts
Symmetric with respect to the x-axis
2, 1 is a point on the graph
Symmetric with respect to the origin
Graph passes through the origin
1.3 LINEAR EQUATIONS IN TWO VARIABLES
• Use slope to graph linear equations
in two variables.
• Find the slope of a line given two
points on the line.
• Write linear equations in two
variables.
• Use slope to identify parallel and
perpendicular lines.
• Use slope and linear equations in
two variables to model and solve
Linear equations in two variables can
be used to model and solve real-life
problems. For instance, in Exercise 129
on page 36, you will use a linear
equation to model student enrollment
at the Pennsylvania State University.
Using Slope
The simplest mathematical model for relating two variables is the linear equation in
two variables y  mx  b. The equation is called linear because its graph is a line. (In
mathematics, the term line means straight line.) By letting x  0, you obtain
y  m0  b
Substitute 0 for x.
 b.
So, the line crosses the y-axis at y  b, as shown in Figure 1.28. In other words, the
y-intercept is 0, b. The steepness or slope of the line is m.
The slope of a nonvertical line is the number of units the line rises (or falls) vertically
for each unit of horizontal change from left to right, as shown in Figure 1.28 and
Figure 1.29.
y = mx + b
m units,
Courtesy of Pennsylvania State University
Positive slope, line rises.
Negative slope, line falls.
A linear equation that is written in the form y  mx  b is said to be
written in slope-intercept form.
The Slope-Intercept Form of the Equation of a Line
The graph of the equation
is a line whose slope is m and whose y-intercept is 0, b.
Once you have determined the slope and the y-intercept of a line, it is a relatively
simple matter to sketch its graph. In the next example, note that none of the lines is
vertical. A vertical line has an equation of the form
x  a.
Vertical line
The equation of a vertical line cannot be written in the form y  mx  b because the
slope of a vertical line is undefined, as indicated in Figure 1.30.
Graphing a Linear Equation
Sketch the graph of each linear equation.
1.30 Slope is undefined.
a. y  2x  1
b. y  2
c. x  y  2
a. Because b  1, the y-intercept is 0, 1. Moreover, because the slope is m  2, the
line rises two units for each unit the line moves to the right, as shown in Figure 1.31.
b. By writing this equation in the form y  0x  2, you can see that the
y-intercept is 0, 2 and the slope is zero. A zero slope implies that the line is
horizontal—that is, it doesn't rise or fall, as shown in Figure 1.32.
c. By writing this equation in slope-intercept form
y  x  2
Subtract x from each side.
Write in slope-intercept form.
you can see that the y-intercept is 0, 2. Moreover, because the slope is m  1,
the line falls one unit for each unit the line moves to the right, as shown in Figure 1.33.
y = 2x + 1
y=2
m = −1
y = −x + 2
When m is positive, the line rises.
When m is 0, the line is horizontal.
When m is negative, the line falls.
Finding the Slope of a Line
Given an equation of a line, you can find its slope by writing the equation in slopeintercept form. If you are not given an equation, you can still find the slope of a line.
For instance, suppose you want to find the slope of the line passing through the points
x1, y1 and x2, y2 , as shown in Figure 1.34. As you move from left to right along this
line, a change of  y2  y1 units in the vertical direction corresponds to a change of
x2  x1 units in the horizontal direction.
(x 2, y 2 )
y2 − y1
(x 1, y 1)
y2  y1  the change in y  rise
x2  x1  the change in x  run
The ratio of  y2  y1 to x2  x1 represents the slope of the line that passes through
the points x1, y1 and x2, y2 .
Slope
change in y
change in x
y2  y1
x2  x1
The Slope of a Line Passing Through Two Points
The slope m of the nonvertical line through x1, y1 and x2, y2  is
where x1  x2.
When this formula is used for slope, the order of subtraction is important. Given
two points on a line, you are free to label either one of them as x1, y1 and the other
as x2, y2 . However, once you have done this, you must form the numerator and
denominator using the same order of subtraction.
For instance, the slope of the line passing through the points 3, 4 and 5, 7 can be
calculated as
or, reversing the subtraction order in both the numerator and denominator, as
Finding the Slope of a Line Through Two Points
Find the slope of the line passing through each pair of points.
a. 2, 0 and 3, 1
b. 1, 2 and 2, 2
c. 0, 4 and 1, 1
d. 3, 4 and 3, 1
a. Letting x1, y1  2, 0 and x2, y2   3, 1, you obtain a slope of
To find the slopes in Example 2,
you must be able to evaluate
rational expressions. You can
review the techniques for
evaluating rational expressions
in Appendix A.4.
x2  x1 3  2 5
b. The slope of the line passing through 1, 2 and 2, 2 is
  0.
2  1 3
c. The slope of the line passing through 0, 4 and 1, 1 is
 5.
d. The slope of the line passing through 3, 4 and 3, 1 is
Because division by 0 is undefined, the slope is undefined and the line is vertical.
In Figures 1.35 to 1.38, note the
relationships between slope and
the orientation of the line.
a. Positive slope: line rises
from left to right
b. Zero slope: line is horizontal
c. Negative slope: line falls
d. Undefined slope: line is
(− 2, 0)
Slope is
undefined.
Writing Linear Equations in Two Variables
If x1, y1 is a point on a line of slope m and x, y is any other point on the line, then
y  y1
 m.
x  x1
This equation, involving the variables x and y, can be rewritten in the form
y  y1  mx  x1
which is the point-slope form of the equation of a line.
Point-Slope Form of the Equation of a Line
The equation of the line with slope m passing through the point x1, y1 is
y  y1  mx  x1.
The point-slope form is most useful for finding the equation of a line. You should
remember this form.
y = 3x − 5
Find the slope-intercept form of the equation of the line that has a slope of 3 and passes
through the point 1, 2.
Use the point-slope form with m  3 and x1, y1  1, 2.
Using the Point-Slope Form
y  2  3x  1
Point-slope form
Substitute for m, x1, and y1.
The slope-intercept form of the equation of the line is y  3x  5. The graph of this
line is shown in Figure 1.39.
The point-slope form can be used to find an equation of the line passing through
two points x1, y1 and x2, y2 . To do this, first find the slope of the line
When you find an equation of
the line that passes through two
given points, you only need to
substitute the coordinates of one
of the points in the point-slope
form. It does not matter which
point you choose because both
points will yield the same result.
, x1  x2
and then use the point-slope form to obtain the equation
y  y1
x  x1.
Two-point form
This is sometimes called the two-point form of the equation of a line.
Parallel and Perpendicular Lines
Slope can be used to decide whether two nonvertical lines in a plane are parallel,
perpendicular, or neither.
1. Two distinct nonvertical lines are parallel if and only if their slopes are equal.
That is, m1  m2.
2. Two nonvertical lines are perpendicular if and only if their slopes are
negative reciprocals of each other. That is, m1  1m2.
2x − 3y = 5
Finding Parallel and Perpendicular Lines
Find the slope-intercept forms of the equations of the lines that pass through the point
2, 1 and are (a) parallel to and (b) perpendicular to the line 2x  3y  5.
y = − 23 x + 2
By writing the equation of the given line in slope-intercept form
y = 23 x −
2x  3y  5
3y  2x  5
Subtract 2x from each side.
you can see that it has a slope of m  23, as shown in Figure 1.40.
a. Any line parallel to the given line must also have a slope of 23. So, the line through
2, 1 that is parallel to the given line has the following equation.
y  1  23x  2
3 y  1  2x  2
On a graphing utility, lines will
not appear to have the correct
slope unless you use a viewing
window that has a square setting.
For instance, try graphing the
lines in Example 4 using the
standard setting ⴚ10 x 10
and ⴚ10 y 10. Then reset
the viewing window with the
square setting ⴚ9 x 9 and
ⴚ6 y 6. On which setting
do the lines y ⴝ 23 x  53 and
y ⴝ ⴚ 32 x ⴙ 2 appear to be
perpendicular?
3y  3  2x  4
Write in point-slope form.
Multiply each side by 3.
Distributive Property
b. Any line perpendicular to the given line must have a slope of  32 because  32
is the negative reciprocal of 3 . So, the line through 2, 1 that is perpendicular to
the given line has the following equation.
y  1   32x  2
y   32x  2
Notice in Example 4 how the slope-intercept form is used to obtain information about
the graph of a line, whereas the point-slope form is used to write the equation of
a line.
In real-life problems, the slope of a line can be interpreted as either a ratio or a rate. If
the x-axis and y-axis have the same unit of measure, then the slope has no units and is
a ratio. If the x-axis and y-axis have different units of measure, then the slope is a rate
or rate of change.
Using Slope as a Ratio
The maximum recommended slope of a wheelchair ramp is 12
. A business is installing a
wheelchair ramp that rises 22 inches over a horizontal length of 24 feet. Is the ramp
steeper than recommended? (Source: Americans with Disabilities Act Handbook)
The horizontal length of the ramp is 24 feet or 1224  288 inches, as shown in
Figure 1.41. So, the slope of the ramp is
vertical change
 0.076.
horizontal change 288 in.
Because 12
 0.083, the slope of the ramp is not steeper than recommended.
Now try Exercise 115.
A kitchen appliance manufacturing company determines that the total cost in dollars of
producing x units of a blender is
Cost (in dollars)
C  25x  3500.
C = 25x + 3500
Cost equation
Describe the practical significance of the y-intercept and slope of this line.
Marginal cost:
m = $25
Fixed cost: $3500
Using Slope as a Rate of Change
1.42 Production cost
The y-intercept 0, 3500 tells you that the cost of producing zero units is $3500. This
is the fixed cost of production—it includes costs that must be paid regardless of the
number of units produced. The slope of m  25 tells you that the cost of producing
each unit is $25, as shown in Figure 1.42. Economists call the cost per unit the
marginal cost. If the production increases by one unit, then the "margin," or extra
amount of cost, is $25. So, the cost increases at a rate of $25 per unit.
Most business expenses can be deducted in the same year they occur. One
exception is the cost of property that has a useful life of more than 1 year. Such costs
must be depreciated (decreased in value) over the useful life of the property. If the
same amount is depreciated each year, the procedure is called linear or straight-line
depreciation. The book value is the difference between the original value and the total
amount of depreciation accumulated to date.
Straight-Line Depreciation
A college purchased exercise equipment worth $12,000 for the new campus
fitness center. The equipment has a useful life of 8 years. The salvage value at the end
of 8 years is $2000. Write a linear equation that describes the book value of the
equipment each year.
Let V represent the value of the equipment at the end of year t. You can represent the
initial value of the equipment by the data point 0, 12,000 and the salvage value of the
equipment by the data point 8, 2000. The slope of the line is
2000  12,000
 $1250
which represents the annual depreciation in dollars per year. Using the point-slope
form, you can write the equation of the line as follows.
V  12,000  1250t  0
V  1250t  12,000
The table shows the book value at the end of each year, and the graph of the equation
is shown in Figure 1.43.
Useful Life of Equipment
Value (in dollars)
(0, 12,000)
V = −1250t +12,000
Value, V
(8, 2000)
Number of years
1.43 Straight-line depreciation
In many real-life applications, the two data points that determine the line are often
given in a disguised form. Note how the data points are described in Example 7.
Predicting Sales
The sales for Best Buy were approximately $35.9 billion in 2006 and $40.0 billion in
2007. Using only this information, write a linear equation that gives the sales (in
billions of dollars) in terms of the year. Then predict the sales for 2010. (Source: Best
Buy Company, Inc.)
Let t  6 represent 2006. Then the two given values are represented by the data points
6, 35.9 and 7, 40.0. The slope of the line through these points is
y = 4.1t + 11.3
 4.1.
(10, 52.3)
Using the point-slope form, you can find the equation that relates the sales y and the
year t to be
(7, 40.0)
y  35.9  4.1t  6
y  4.1t  11.3.
Year (6 ↔ 2006)
According to this equation, the sales for 2010 will be
y  4.110  11.3  41  11.3  $52.3 billion. (See Figure 1.44.)
The prediction method illustrated in Example 8 is called linear extrapolation.
Note in Figure 1.45 that an extrapolated point does not lie between the given points.
When the estimated point lies between two given points, as shown in Figure 1.46, the
procedure is called linear interpolation.
Because the slope of a vertical line is not defined, its equation cannot be written in
slope-intercept form. However, every line has an equation that can be written in the
Ax  By  C  0
Linear extrapolation
where A and B are not both zero. For instance, the vertical line given by x  a can be
represented by the general form x  a  0.
Summary of Equations of Lines
1. General form:
2. Vertical line:
3. Horizontal line:
4. Slope-intercept form: y  mx  b
5. Point-slope form:
6. Two-point form:
Linear interpolation
In Exercises 1–7, fill in the blanks.
The simplest mathematical model for relating two variables is the ________ equation in two variables y  mx  b.
For a line, the ratio of the change in y to the change in x is called the ________ of the line.
Two lines are ________ if and only if their slopes are equal.
Two lines are ________ if and only if their slopes are negative reciprocals of each other.
When the x-axis and y-axis have different units of measure, the slope can be interpreted as a ________.
The prediction method ________ ________ is the method used to estimate a point on a line when the point
does not lie between the given points.
7. Every line has an equation that can be written in ________ form.
8. Match each equation of a line with its form.
(a) Ax  By  C  0
(i) Vertical line
(b) x  a
(ii) Slope-intercept form
(c) y  b
(iii) General form
(d) y  mx  b
(iv) Point-slope form
(e) y  y1  mx  x1
(v) Horizontal line
In Exercises 9 and 10, identify the line that has each slope.
9. (a) m  3
(b) m is undefined.
(c) m  2
In Exercises 11 and 12, sketch the lines through the point
with the indicated slopes on the same set of coordinate axes.
11. 2, 3
(a) 0 (b) 1
(c) 2 (d) 3
(a) 3 (b) 3 (c) 2 (d) Undefined
In Exercises 13–16, estimate the slope of the line.
10. (a) m  0
(b) m   4
In Exercises 17–28, find the slope and y-intercept (if possible)
of the equation of the line. Sketch the line.
5x  2  0
7x  6y  30
y   2x  6
3y  5  0
In Exercises 29–40, plot the points and find the slope of the
line passing through the pair of points.
 2 ,  3 ,  2,  3
1.75, 8.3, 2.25, 2.6
12, 0, 0, 8
0, 10, 4, 0
 78, 34 ,  54, 14
In Exercises 41–50, use the point on the line and the slope m
of the line to find three additional points through which the
line passes. (There are many correct answers.)
2, 1, m  0
8, 1, m is undefined.
10, 6, m  1
1, 6, m   12
In Exercises 51– 64, find the slope-intercept form of the
equation of the line that passes through the given point and
has the indicated slope m. Sketch the line.
51. 0, 2, m  3
55. 4, 0, m   13
52. 0, 10, m  1
56. 8, 2, m  14
10, 4, m is undefined.
62.  2, 2 , m  0
4, 52 , m  0
5.1, 1.8, m  5
64. 2.3, 8.5, m  2.5
In Exercises 65–78, find the slope-intercept form of the
equation of the line passing through the points. Sketch the
2, 12 ,  12, 54
 101 ,  35 , 109 ,  95
1, 0.6, 2, 0.6
73, 8, 73, 1
1, 1, 6,  23
34, 32 ,  43, 74
1.5, 2, 1.5, 0.2
In Exercises 79– 82, determine whether the lines are parallel,
79. L1: y  3 x  2
L2: y
81. L1: y  12 x  3
L2: y   12 x  1
80. L1: y  4x  1
L2: y  4x  7
82. L1: y   45 x  5
L2: y  54 x  1
In Exercises 83– 86, determine whether the lines L1
and L2 passing through the pairs of points are parallel,
83. L1: 0, 1, 5, 9
L2: 0, 3, 4, 1
L2: 0, 1, 5, 73
In Exercises 87–96, write the slope-intercept forms of the
equations of the lines through the given point (a) parallel to
the given line and (b) perpendicular to the given line.
4x  2y  3, 2, 1
3x  4y  7,  3, 8
y  3  0, 1, 0
x  4  0, 3, 2
x  y  4, 2.5, 6.8
6x  2y  9, 3.9, 1.4
x  y  7, 3, 2
5x  3y  0,  78, 34
In Exercises 97–102, use the intercept form to find the
equation of the line with the given intercepts. The intercept
form of the equation of a line with intercepts a, 0 and
0, b is
1 ⴝ 1, a ⴝ 0, b ⴝ 0.
97. x-intercept: 2, 0
99. x-intercept:  16, 0
y-intercept: 0,  23
101. Point on line: 1, 2
x-intercept: c, 0
y-intercept: 0, c, c  0
x-intercept: d, 0
y-intercept: 0, d, d  0
x-intercept: 3, 0
x-intercept:  23, 0
GRAPHICAL ANALYSIS In Exercises 103–106, identify
any relationships that exist among the lines, and then use a
graphing utility to graph the three equations in the same
viewing window. Adjust the viewing window so that the slope
appears visually correct—that is, so that parallel lines appear
parallel and perpendicular lines appear to intersect at right
y  23x
y   12x
y   2x
y   2x  3 (c)
In Exercises 107–110, find a relationship between x and y
such that x, y is equidistant (the same distance) from the
107. 4, 1, 2, 3
109. 3, 52 , 7, 1
110.  12, 4, 72, 54
111. SALES The following are the slopes of lines
representing annual sales y in terms of time x in years.
Use the slopes to interpret any change in annual sales
for a one-year increase in time.
(a) The line has a slope of m  135.
(b) The line has a slope of m  0.
(c) The line has a slope of m  40.
112. REVENUE The following are the slopes of lines
representing daily revenues y in terms of time x in
days. Use the slopes to interpret any change in daily
revenues for a one-day increase in time.
(b) The line has a slope of m  100.
(c) The line has a slope of m  0.
113. AVERAGE SALARY The graph shows the average
salaries for senior high school principals from 1996
through 2008. (Source: Educational Research
Service)
Salary (in dollars)
(18, 97,486)
(a) Use the slopes of the line segments to determine
the time periods in which the average salary
increased the greatest and the least.
(b) Find the slope of the line segment connecting the
points for the years 1996 and 2008.
(c) Interpret the meaning of the slope in part (b) in the
context of the problem.
114. SALES The graph shows the sales (in billions of
dollars) for Apple Inc. for the years 2001 through 2007.
(Source: Apple Inc.)
(7, 24.01)
(2, 5.74)
the years in which the sales showed the greatest
increase and the least increase.
115. ROAD GRADE You are driving on a road that has a
6% uphill grade (see figure). This means that the slope
of the road is 100
. Approximate the amount of vertical
change in your position if you drive 200 feet.
116. ROAD GRADE From the top of a mountain road, a
surveyor takes several horizontal measurements x and
several vertical measurements y, as shown in the table
(x and y are measured in feet).
(b) Use a straightedge to sketch the line that you think
best fits the data.
(c) Find an equation for the line you sketched in
part (b).
(d) Interpret the meaning of the slope of the line in
part (c) in the context of the problem.
(e) The surveyor needs to put up a road sign that
indicates the steepness of the road. For instance, a
surveyor would put up a sign that states "8%
grade" on a road with a downhill grade that has a
slope of  100
. What should the sign state for the
road in this problem?
RATE OF CHANGE In Exercises 117 and 118, you are given
the dollar value of a product in 2010 and the rate at which
the value of the product is expected to change during the next
5 years. Use this information to write a linear equation that
gives the dollar value V of the product in terms of the year t.
(Let t ⴝ 10 represent 2010.)
2010 Value
117. $2540
118. $156
$125 decrease per year
$4.50 increase per year
119. DEPRECIATION The value V of a molding machine
t years after it is purchased is
V  4000t  58,500, 0
Explain what the V-intercept and the slope measure.
120. COST The cost C of producing n computer laptop
bags is given by
C  1.25n  15,750,
0 < n.
Explain what the C-intercept and the slope measure.
DEPRECIATION A sub shop purchases a used pizza
oven for $875. After 5 years, the oven will have to be
replaced. Write a linear equation giving the value V of
the equipment during the 5 years it will be in use.
DEPRECIATION A school district purchases a
high-volume printer, copier, and scanner for $25,000.
After 10 years, the equipment will have to be replaced.
Its value at that time is expected to be $2000. Write a
linear equation giving the value V of the equipment
during the 10 years it will be in use.
SALES A discount outlet is offering a 20% discount
on all items. Write a linear equation giving the sale
price S for an item with a list price L.
HOURLY WAGE A microchip manufacturer pays its
assembly line workers $12.25 per hour. In addition,
workers receive a piecework rate of $0.75 per unit
produced. Write a linear equation for the hourly wage
W in terms of the number of units x produced per hour.
MONTHLY SALARY A pharmaceutical salesperson
receives a monthly salary of $2500 plus a commission
of 7% of sales. Write a linear equation for the salesperson's monthly wage W in terms of monthly sales S.
BUSINESS COSTS A sales representative of a
company using a personal car receives $120 per day
for lodging and meals plus $0.55 per mile driven.
Write a linear equation giving the daily cost C to the
company in terms of x, the number of miles driven.
CASH FLOW PER SHARE The cash flow per share
for the Timberland Co. was $1.21 in 1999 and $1.46 in
2007. Write a linear equation that gives the cash flow
per share in terms of the year. Let t  9 represent
1999. Then predict the cash flows for the years 2012
and 2014. (Source: The Timberland Co.)
NUMBER OF STORES In 2003 there were 1078
J.C. Penney stores and in 2007 there were 1067 stores.
Write a linear equation that gives the number of stores
in terms of the year. Let t  3 represent 2003. Then
predict the numbers of stores for the years 2012 and
2014. Are your answers reasonable? Explain.
(Source: J.C. Penney Co.)
129. COLLEGE ENROLLMENT The Pennsylvania State
University had enrollments of 40,571 students in 2000
and 44,112 students in 2008 at its main campus in
University Park, Pennsylvania. (Source: Penn State
Fact Book)
(a) Assuming the enrollment growth is linear, find a
linear model that gives the enrollment in terms of
the year t, where t  0 corresponds to 2000.
(b) Use your model from part (a) to predict the enrollments in 2010 and 2015.
(c) What is the slope of your model? Explain its
meaning in the context of the situation.
130. COLLEGE ENROLLMENT The University of Florida
had enrollments of 46,107 students in 2000 and 51,413
students in 2008. (Source: University of Florida)
(a) What was the average annual change in enrollment
from 2000 to 2008?
(b) Use the average annual change in enrollment to
estimate the enrollments in 2002, 2004, and 2006.
(c) Write the equation of a line that represents the
given data in terms of the year t, where t  0
corresponds to 2000. What is its slope? Interpret
the slope in the context of the problem.
(d) Using the results of parts (a)–(c), write a short
paragraph discussing the concepts of slope and
average rate of change.
131. COST, REVENUE, AND PROFIT A roofing
contractor purchases a shingle delivery truck with a
shingle elevator for $42,000. The vehicle requires an
average expenditure of $6.50 per hour for fuel and
maintenance, and the operator is paid $11.50 per hour.
(a) Write a linear equation giving the total cost C of
operating this equipment for t hours. (Include the
purchase cost of the equipment.)
(b) Assuming that customers are charged $30 per hour
of machine use, write an equation for the revenue
R derived from t hours of use.
(c) Use the formula for profit
to write an equation for the profit derived from t
hours of use.
(d) Use the result of part (c) to find the break-even
point—that is, the number of hours this equipment
must be used to yield a profit of 0 dollars.
132. RENTAL DEMAND A real estate office handles an
apartment complex with 50 units. When the rent per
unit is $580 per month, all 50 units are occupied.
However, when the rent is $625 per month, the
average number of occupied units drops to 47. Assume
that the relationship between the monthly rent p and
the demand x is linear.
(a) Write the equation of the line giving the demand x
in terms of the rent p.
(b) Use this equation to predict the number of units
occupied when the rent is $655.
(c) Predict the number of units occupied when the rent
is $595.
133. GEOMETRY The length and width of a rectangular
garden are 15 meters and 10 meters, respectively. A
walkway of width x surrounds the garden.
(a) Draw a diagram that gives a visual representation
of the problem.
(b) Write the equation for the perimeter y of the walkway in terms of x.
(c) Use a graphing utility to graph the equation for the
perimeter.
(d) Determine the slope of the graph in part (c). For each
additional one-meter increase in the width of the
walkway, determine the increase in its perimeter.
134. AVERAGE ANNUAL SALARY The average salaries
(in millions of dollars) of Major League Baseball
players from 2000 through 2007 are shown in the
scatte


text:
Discover Edinburgh Health Informatics
Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.
The MRC Proximity to Discovery scheme awards universities funds to help develop new collaborations, and ways of exchanging knowledge and skills. The awards can be used to support activities that promote the value of academic-industry partnership, and enhance academic and industry researchers' understanding of each other's needs and capabilities. This may be through people exchanges, creation of technology demonstrators, showcase events, commercialisation workshops and 'entrepreneurs in residence' schemes. Such exchanges of knowledge and skills will boost the most fruitful collaborations between UK universities and life science companies.
Total Expenditure April 2006 - March 2020:
Mar 17 - Dec 18
MC_PC_16071
Andrew David Morris
Charles Ffrench-Constant
Health Category:
HRCS Health Uncodeable (100%)
HRCS Research Uncodeable (100%)
University of Edinburgh, United Kingdom (Lead Research Organisation)
State Life Insurance Corporation (Collaboration)
Brainsview Inc (Collaboration)
Andrew David Morris (Principal Investigator)
Charles Ffrench-Constant (Principal Investigator)
Elizabeth Anne Elliot (Co-Investigator)
David Robertson (Co-Investigator)
Description 3 Famous Project - International Diabetes Centre
Organisation Government of China
Title Retinal biomarkers for Alzheimer's disease trial
Description In Association with the European Prevention of Alzheimer's Dementia (EPAD) Consortium a funded trial into the use of the University of Edinburgh's Vampire image analysis software for providing diagnostic insights in Alzheimer's disease.
Impact Work is ongoing.
Title VIVOx Prototype
Description Funding to provide collaborative discussions with Baxter Healthcare and to allow for a concept in prototype regarding the concept of VIVOx, a theoretical approach to increasing blood oxygenation levels (and reduce carbon dioxide) quickly and safely in critical care patients. Outcome to be reported in 2019.
Impact Work is still ongoing.
Title Economic outcomes measurement for Health Technology Assessment
Description Access to the Scottish national data extract of NHS, Social Care and Census data in survivors of cancer and major illness.
Impact The data will allow analysis that quantifies and predicts long term healthcare activity, morbidity and related clinical and economic outcomes. Interested companies include Eisai, Pfizer, Novartis, AstraZeneca, Roche.
Description Accelerating clinical translation of novel brain connectivity analysis
Organisation Brainsview Inc
PI Contribution A collaboration between KidsBrainIT Group based in the Usher Institute at the University of Edinburgh and Brainsview Ltd, a start-up company based in Toronto to develop and accelerate clinical translations / applications of innovative brain wave analytical solution using routinely generated health care data in the children intensive care units. Funding was provided to support a two week person exchange between KidsBrainIT Group and Brainsview, software license and IT equipment for the project and refreshments for an event to discuss the project in more detail.
Collaborator Contribution Brainsview Ltd participated in a 2 week person exchange for the purposes of knowledge transfer.
Description Building health insurance data dictum through academia-industry linkages
Organisation State Life Insurance Corporation
PI Contribution A two month placement of a PhD student in Islamabad to look into the utilization of healthcare in households below the poverty line, with a specific focus on infant and maternal mortality rates. In 2016, the Government of Pakistan launched two large scale social health insurance schemes, covering millions of households below national poverty line and State Life Insurance Corporation were interested in collaborating with the University of Edinburgh to analyse the data.
Collaborator Contribution Working alongside the University of Edinburgh to analyse the data.
Impact Work remains ongoing.
Description Artificial Intelligence Augmented Biomedical Discovery and Therapeutic Delivery of Pluripotency Driven Tissue Rejuvenation in Ageing and Adult Onset Disease.
Results and Impact Two brainstorming workshops, one at the University of Edinburgh and one at University of Agder in Norway, to discuss the use of artificial intelligence to induce and maintain pluripotency to stimulate tissue rejuvenation to reverse the hallmarks of ageing and adult onset disease. The workshops took place in October and December and included attendees from University of Edinburgh, Anzyz Associates, University of Agder and Sorlandet Hospital.
Description Design Informatics Collider Event
Results and Impact Frontiers Meeting in May 18 bringing together academia, patients, industry and NHS to explore ways in which data science and machine learning can improve health care for those with long-term conditions, using asthma as an exemplar. There were 18 attendees including industry representation from Tactuum, CapGemini and NHS Scotland. A paper is currently being written around the discussion on shared decision-making with a look to align the work on asthma and machine learning with the clinical decision support roadmap for Scotland.
Description Imaging Tech Meet Up
Results and Impact Edinburgh is home to world-leading medical researchers who work in partnership with the NHS and the private sector. This close cooperation helps to accelerate medical discoveries and transform them into treatments for patients.
Edinburgh imaging couples world leading multidisciplinary academic imaging expertise with cutting edge research imaging equipment of all modalities and technical expertise alongside the delivery of imaging services. https://www.ed.ac.uk/clinical-sciences/edinburgh-imaging/about
An Edinburgh Imaging Alliance was formed on the 5th December 2017. The Edinburgh Imaging Alliance currently involves Toshiba Medical Europe, Optos, Blackford Analysis Ltd, Heriot-Watt University, NHS Lothian and The University of Edinburgh working together to promote Edinburgh's world leading imaging facilities and expertise in medical imaging, digital health, engineering, physics, biochemistry, statistics and informatics. The Edinburgh Imaging Alliance will catalyse the growth of a medical imaging cluster in Edinburgh utilising key skills in digital health and datasets available.
https://www.ed.ac.uk/clinical-sciences/edinburgh-imaging-alliance
Funding is requested to support an Edinburgh Imaging Alliance "tech meet up". A tech meet up is an event intended for education, networking and community; connecting academic research expertise with the interests and needs of companies. This activity also provides a route for these companies to help shape their future workforce through connection to students and providing input to educational material and training courses
The tech meet model has been used successfully in Edinburgh by Informatics Ventures (IV) http://www.informatics-ventures.com/ . The Edinburgh Imaging Alliance is requesting funding to implement this model with an imaging focus. We have contacted IV for advice and support, and have proposed running an "Imaging" tech meet up using their using their audience as a first step. This first meet up session would be hosted and sponsored by the University using the P2D funds; if this proves to be worthwhile in terms of generating ideas for collaborative activity and encouraging the best talent to Edinburgh then subsequent sessions will be supported by the companies.
Description MHRA drug repurposing data meeting
Results and Impact A meeting with MHRA to discuss the repurposing of orally administered Sulbutiamine for the treatment of chronic alcoholism. Following this a meeting was held with the Head of Licensing at Servier and the proposal is now being discussed internally.
Description My Story Now
Results and Impact My Story Now is a mobile application for young adults who have survived childhood cancer enabling them to access and share (if appropriate) health data concerning their past medical history. This project supported the costs for a final year student to help evaluate the application from a technology perspective.
Description Olink workshop meeting "Applying data science to accelerate precision medicine"
Results and Impact Two University of Edinburgh academics travelled to Sweden to attend the two day workshop hosted by Olink Proteomics. The academics were able to talk to the entire data science team and discuss specifics relating to the assays that produced the data the academics work with. They were also able to get a better understanding of the diversity of research interests of groups working with Olink data, highlighting potential future collaborations
Description One HealthTech (OHT) Edinburgh Hub Event
Results and Impact The recently formed OHT is a grassroots community that promotes equality and diversity in the healthcare technology sector. Funding was provided to run a networking event for Edinburgh-based academic researchers, NHS staff and health-tech companies following on from the launch event in June 2018.
Description Regeneration Innovation Event 2018 - Realising Regenerative Therapies
Results and Impact Regeneration Innovation 2018 - Realising Regenerative Therapies', CRM's annual industry engagement event, was held on 2nd May 2018. It brought together key players to increase translation and commercialisation of research outputs. The event was attended by ca 75 attendees, ranging from CRM junior researchers/principal investigators, to investors and stakeholders from regulatory bodies and Business Development / Knowledge Exchange & Commercialisation (KEC) organisations and pharma.
Description Robotics in Healthcare Event: Current issues and future trends
Results and Impact An AIMDay event to explore existing and anticipated future opportunities and challenges (including social, technical, commercial, and legal) associated with integration of robotics in healthcare.
Description Scaling machine learning within Health Systems in Low and Middle Income Countries.
Results and Impact A workshop held in Nairobi to discuss the scalability of machine learning within health systems in low and middle income countries.
Description University of Edinburgh and Congenica Meeting
Results and Impact Person exchange between Congenica and the University of Edinburgh for the purposes of knowledge transfer to plan a project.
Description University of Edinburgh and Radiation Imaging Technologies, Lda (RI-TE) Collaboration
Results and Impact Funding to allow a University of Edinburgh academic to attend RI-TE', Lda Coimbra testing site for a two week period for image and data collection. Following on from this visit, the aim is now to pursue a collaborative agreement between RI-TE, Lda and the University to Edinburgh in the standardisation of preclinical PET/CT systems across multiple platforms in both Europe and USA.
Description Usher Workshop Series
Results and Impact The Usher Institute at the University of Edinburgh is dedicated to improving the health of individuals and populations locally and globally. The Usher Institute held a series of 3 workshops on health care data:
• Technology for community health
• Mobility and active healthcare
• Internet of Things (IOT) and Health
79 people attended in total, 20 from industry, 22 from policy and 37 from academia.
Description Visit by Congenica
Results and Impact As part of the The Institute of Genetics and Molecular Medicine ongoing promotion of datascience in biomedicine an initial meeting with Congenica to scope potential projects in more depth and engage other stakeholders (NHS clinical genetics services, Farr Institute and Scottish Development International) that will be key to developing a program of collaboration. The meeting provided a great opportunity for networking and led to a further project.
Description Visit to EuroImmun
Results and Impact A University of Edinburgh academic wants to develop an assay for the detection of cervical disease in HPV positive women. Euroimmun are on board and are interested in funding a 9 month research collaboration valued at approx £100k. This funded visit allowed conversations around this collaboration to begin and have since resulted in a signed agreement. The University of Edinburgh also showcased other diagnostic opportunities at the meeting, a CDA has been signed and the academic will visit the EuroImmun offices in the new year.

redpajama_set_name: RedPajamaCommonCrawl


An African prelate who was once the world's youngest bishop, who was on the "hit list" of a corrupt dictator, who recharges himself with three-day fasts, and who now, from a powerful Vatican post, leads the defense of orthodoxy. Doesn't that sound like someone you'd like to know better? In the Catholic Herald, Father Mark Drew obliges with a quick portrait of Cardinal Robert Sarah, author of the challenging new book God or Nothing.
Another fascinating portrait, of a Canadian politician of the mid-20th century. Paul Comtois, who rose to become lieutenant governor of Quebec, had an intense devotion to the Blessed Sacrament. Father Raymond de Souza explains how the manner of his death makes it possible to think of Paul Comtois — a martyr for the Eucharist.
Those of us living in France, Germany, and elsewhere have to ask: has it occurred to Pierre Manent that the REAL challenge is integrating into society a group of people who, unless we are very much mistaken, give every indication of not wanting to be integrated at all.


Love. People have tried to understand it since the beginning of time—yet our culture yearns for love more than ever. Is it possible to understand and, more importantly, to live love in a world desperate for it? Elizabeth Turnage reminds us that God alone can show and help us to understand love—for he not only loves us, but also empowers us to love as well.
In previous studies, Elizabeth has helped us to learn God's story of grace and to live it out ourselves. In this final Living Story book, she teaches us the ultimate joy of living in God's story of grace: loving as God loves us.
Beginning by searching the depths and heights of God's love, she moves on to show us where we fit in and how, by continuing to walk in God's story, we can learn to love too.
If you know God's story of grace and where your story fits in, learn how to take the next step and make your story bigger than it's ever been before!


The wait between paydays can be long for performing artist and freelance caterer Edwina Joesler. "Frequently, I will be waiting not just weeks, but months, for clients to pay up," she says. The Sydneysider's annual income is less than $20,000 and she lives without a credit card.


Looking for answers to your questions about sideshow history? Have some questions about shows or performers from the past? You've come to the right place! Using the form at the bottom of this page you can send in your questions and one of our staff members will do our best to research it, provide you with an answer and possibly publish it on our site!
What Do You Know About Koo-Koo Franks??
Do You Know Whose Show this Was?
Tobacco Road - Banner by ?
So� Say that it's 1939 and I am a sideshow performer?
Little Harvey and His Bull Elmer?
Girl 2 Gorilla Set Up?
Is this Celesta Geyer aka Dolly Dimples?
Could this be Frances O�Connor?
Life Among Lizards Not Always Rosy!
This section is not for questions regarding information on how to perform acts, where to learn acts, etc. It is designed for historical style questions only. All other questions will be disregarded. A name (or alias), state and town is required for your question to be considered.


How do I fix the padding of TeXMaths equation rendering? I often have the problem that equations compiled to SVG get cut off along the top and bottom. Attached image shows an example when using LibreOffice Draw.
Any help is seriously appreciated--I have dealt with this issue numerous times and never managed to find a solution. I think the issue is with using "/usr/bin/dvisvgm" to produce the SVG graphic but I don't know how to reconfigure that; PNG graphics come out correctly, but I want vector.
Updating to TeXMaths 0.46.1 seems to have fixed the problem. If this had not worked, I was going to try to figure out how to add the option "--exact" to the /usr/bin/dvisvgm command as discussed here.


text:
Q: Not getting props in react component I'm not getting props in my Nav component. Odd thing is, 'this.props.history.push' is working in my other components.
The same function is working in my other components, but when I try to call the push function, I'm getting 'err in logout TypeError: Cannot read property 'push' of undefined'. The 'this.props' object is logging as '{}'.
Any help is appreciated, thank you.
import React from 'react'

import logo from 'logo.png'

import css from './Nav.module.scss'

import { Link } from 'react-router-dom'

import Cookies from 'js-cookie'
import axios from 'axios'

class Nav extends React.Component {
  constructor(props) {
    super(props)
    this.state = {
      loggedIn: false
    }
    console.log(this.props)
  }

  _handleLogout = () => {
    // const self = this
    console.log(this.props)
    axios.get('http://localhost:8080/logout', {
      withCredentials: true
    })
    .then(res => {
      console.log(res)
      console.log('logout')
      if (Cookies.get('sid') === undefined) {
        this.props.history.push('/')
      }
      console.log(this.props)
    })
    .catch(err => {
      console.log('err in logout', err)
    })
  }

  render() {
    return (

          <Link to="/">

              somebrand

          </Link>

          {
            Cookies.get('sid') === undefined ?
              <Link to="/login">

                  Login

              </Link>
              :

                Logout

          }

    )
  }
}

export default Nav

My Nav component is only referenced once in my Layout component:
import React from 'react'

import Nav from 'components/Nav/Nav'

import css from './BasicLayout.module.scss'

class Basic extends React.Component {
  render() {
    return (

          {this.props.children}

    )
  }
}

export default Basic

A: history and location are special props injected by React Router's HOC withRouter
import { withRouter } from 'react-router-dom'
class Nav extends React.Component{
    render(){
        const { history, location } = this.props
        return
{`I'm at ${location.pathname}`}

    }
}
export default withRouter(Nav)

It works for functional components as well
export const Component = withRouter(({ history, location })) =>(

{`I'm at ${location.pathname}`}

)

redpajama_set_name: RedPajamaStackExchange


FAST BY NATURE. BORN MERCURIAL.Speed in football isn't just measured in straight lines. You've got to be fast in all directions on the pitch. Fast by nature, the Mercurial player is born with natural speed - it's in their DNA. Here to write the next chapter for Nike's most storied silo, the Mercurial VaporX XII Club GS TF is packed with innovation to take speed to a whole new level.Features Include:Synthetic upper wraps the foot for a close fit.Texture provides touch at high speeds.Rubber outsole provides traction on artificial-turf surfaces.Colour: Total Orange/Black/Total Orange/Volt.