Y-intercepts: Let calculate so y-intercept is (0 ; x = 0, f(0), d) X -intercepts: let = 0. Factorise by: grouping of terms OR using the factor f(x) f(x) theorem OR synthetic division Step 3: Stationary points Determine (x). Let (x). = 0 and solve for The solutions are the of the x. x-coordinates stationary points, say or x = p x = t Now find and These give you the y-coordinates of the stationary f(p) f(t). point. Step 4: Sketch the graph by: Plotting the intercepts (x and y-intercepts) Stationary points Join them to form the graph Worked example: Use the method outlined above, to sketch the graph of y = = + 3x + 2. f(x) –x3 Solution: Shape - < 0 a Intercepts: Y-intercept. Let = 0, y = 2 ; (0; 2) x X-intercepts: Let 0 : Then 0 = + 3x + 2. y= –x3 = + 3(–1) + 2 = 0. f(–1) –1(–1)3 Thus, (x + 1) is a factor of + 3x + 2. –x3 Find the other factors by inspection or division. = + 3x + 2 f(x) –x3 = (x +1)(–x2 + + 2) x 0 = + 1)(x2 2) x –(x – – 0 = +1)(x + 1) –(x –2)(x So, are given by: (–1 ; 0) ; (–1 ; 0) and (2 ; 0) x-intercepts 3.Stationary points: = + 3x + 2. f(x) –x3 (x) = + 3 –3x2 (x) = 0 + 3 = 0 –3x2 = 0 –3(x2 –1) (x 1)(x + 1) = 0 – = 1 OR = 1 x x –
Y-intercepts: Let x = 0, calculate f(0), so y-intercept is (0 ; d) X -intercepts: let f(x) = 0. Factorise f(x) by: grouping of terms OR using the factor theorem OR synthetic division Step 3: Stationary points Determine (x). Let (x). = 0 and solve for x. The solutions are the x-coordinates of the stationary points, say x = p or x = t Now find f(p) and f(t). These give you the y-coordinates of the stationary point. Step 4: Sketch the graph by: Plotting the intercepts (x and y-intercepts) Stationary points Join them to form the graph Worked example: Use the method outlined above, to sketch the graph of y = f(x) = –x3 + 3x + 2. Solution: Shape - a < 0 Intercepts: Y-intercept. Let x = 0, y = 2 ; (0; 2) X-intercepts: Let y= 0 : Then 0 = –x3 + 3x + 2. f(–1) = –1(–1)3 + 3(–1) + 2 = 0. Thus, (x + 1) is a factor of –x3 + 3x + 2. Find the other factors by inspection or division. f(x) = –x3 + 3x + 2 = (x +1)(–x2 + x + 2) 0 = –(x + 1)(x2 – x – 2) 0 = –(x +1)(x –2)(x + 1) So, x-intercepts are given by: (–1 ; 0) ; (–1 ; 0) and (2 ; 0) 3.Stationary points: f(x) = –x3 + 3x + 2. (x) = –3x2 + 3 (x) = 0 –3x2 + 3 = 0 –3(x2 –1) = 0 (x – 1)(x + 1) = 0 x = 1 OR x = – 1
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