text: # Thread: Help with average rate of change
  
  1. ## Help with average rate of change
  
  Hi there.
  
  For g(x)=4x² -3x +1 Find the rate of change for:
  
  A) x=1 and x=3
  B) (x, f(x)) and (x+h, f(x+h)
  
  I dont know how and would appreciate an explanation. Thanks so much.
  
  2. Perhaps you just need the definition for average rate of change?
  
  Between x=a and x=b, the average rate of change is (change in g(x) divided by the change in x):
  $
  \displaystyle \frac{g(b) - g(a)}{b - a}
  $
  
  For example, average rate of change between x=0 and x=1 is
  $
  \displaystyle \frac{g(1) - g(0)}{1 - 0} = \frac{(4(1)^2 -3(1) +1) - (4(0)^2 -3(0) +1)}{1 - 0} = 1
  $
  
  3. So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
  Which becomes 12
  
  Also, I am confused about B).
  
  So for I got it to f(x+h)-f(x)/(x+h)-x
  
  Then I simplified it to 3(x+h)²-3x²/h
  
  Did I do it correctly?
  
  4. Originally Posted by theridon
  So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
  Which becomes 12
  
  Also, I am confused about B).
  
  So for I got it to [f(x+h)-f(x)]/(x+h)-x brackets are important!
  
  Then I simplified it to 3(x+h)²-3x²/h - where did that 3 come from?
  
  Did I do it correctly?
  
  5. Sorry about the brackets, thats what I meant. I think you plug it into 4x² -3x +1, correct?
  
  This is how I got there:
  1. [f(x+h)-f(x)]/(x+h)-x
  2. [[3(x+h)²+1]-(3x²+1)]/h
  2. 3(x+h)²-3x²/h
  
  6. Originally Posted by theridon
  Sorry about the brackets, thats what I meant. I think you plug it into 4x² -3x +1, correct?
  
  This is how I got there:
  1. [f(x+h)-f(x)]/(x+h)-x
  2. [[3(x+h)²+1]-(3x²+1)]/h
  2. 3(x+h)²-3x²/h
  yes, but that's not what you did.
  
  first of all, is $f(x) = 4x^2 - 3x + 1$ ??? you called it $g(x)$.
  
  if so,
  
  $f(x+h) - f(x)$
  
  the numerator should be ...
  
  $4(x+h)^2 - 3(x+h) + 1 - (4x^2 - 3x + 1)$
  
  7. Oh I missed the first part. So then it would become:
  [4x²+8xh-3x-3h+1-4x²+3x-1]/h
  
  (4h²+8hx)/h
  
  4h+8x
  
  Which simplifies to 4(h+2x)
  
  I believe its correct
  
  8. Originally Posted by theridon
  Oh I missed the first part. So then it would become:
  [4x²+8xh-3x-3h+1-4x²+3x-1]/h
  
  (4h²+8hx)/h
  
  4h+8x
  
  Which simplifies to 4(h+2x)
  
  I believe its correct
  numerator again ...
  
  $4(x + h)^2 - 3(x + h) + 1 - (4x^2 - 3x + 1) =$
  
  $4(x^2 + 2xh + h^2) - 3(x + h) + 1 - (4x^2 - 3x + 1) =$
  
  $4x^2 + 8xh + 4h^2 - 3x - 3h + 1 - 4x^2 + 3x - 1 =$
  
  $8xh + 4h^2 - 3h =$
  
  $h(8x + 4h - 3)$
explanation: This document is a forum thread where a user asks for help with calculating the average rate of change for a given function. The user provides a specific function and asks for the rate of change over certain intervals. Another user responds with a definition and an example, and the original poster attempts to solve the problem, leading to further clarifications and corrections from the responder.
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distilabel_metadata.content: # Thread: Help with average rate of change
  
  1. ## Help with average rate of change
  
  Hi there.
  
  For g(x)=4x² -3x +1 Find the rate of change for:
  
  A) x=1 and x=3
  B) (x, f(x)) and (x+h, f(x+h)
  
  I dont know how and would appreciate an explanation. Thanks so much.
  
  2. Perhaps you just need the definition for average rate of change?
  
  Between x=a and x=b, the average rate of change is (change in g(x) divided by the change in x):
  $
  \displaystyle \frac{g(b) - g(a)}{b - a}
  $
  
  For example, average rate of change between x=0 and x=1 is
  $
  \displaystyle \frac{g(1) - g(0)}{1 - 0} = \frac{(4(1)^2 -3(1) +1) - (4(0)^2 -3(0) +1)}{1 - 0} = 1
  $
  
  3. So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
  Which becomes 12
  
  Also, I am confused about B).
  
  So for I got it to f(x+h)-f(x)/(x+h)-x
  
  Then I simplified it to 3(x+h)²-3x²/h
  
  Did I do it correctly?
  
  4. Originally Posted by theridon
  So, for the first part it would be: (4-3+1)-(36-9+1)/1-3
  Which becomes 12
  
  Also, I am confused about B).
  
  So for I got it to [f(x+h)-f(x)]/(x+h)-x brackets are important!
  
  Then I simplified it to 3(x+h)²-3x²/h - where did that 3 come from?
  
  Did I do it correctly?
  
  5. Sorry about the brackets, thats what I meant. I think you plug it into 4x² -3x +1, correct?
  
  This is how I got there:
  1. [f(x+h)-f(x)]/(x+h)-x
  2. [[3(x+h)²+1]-(3x²+1)]/h
  2. 3(x+h)²-3x²/h
  
  6. Originally Posted by theridon
  Sorry about the brackets, thats what I meant. I think you plug it into 4x² -3x +1, correct?
  
  This is how I got there:
  1. [f(x+h)-f(x)]/(x+h)-x
  2. [[3(x+h)²+1]-(3x²+1)]/h
  2. 3(x+h)²-3x²/h
  yes, but that's not what you did.
  
  first of all, is $f(x) = 4x^2 - 3x + 1$ ??? you called it $g(x)$.
  
  if so,
  
  $f(x+h) - f(x)$
  
  the numerator should be ...
  
  $4(x+h)^2 - 3(x+h) + 1 - (4x^2 - 3x + 1)$
  
  7. Oh I missed the first part. So then it would become:
  [4x²+8xh-3x-3h+1-4x²+3x-1]/h
  
  (4h²+8hx)/h
  
  4h+8x
  
  Which simplifies to 4(h+2x)
  
  I believe its correct
  
  8. Originally Posted by theridon
  Oh I missed the first part. So then it would become:
  [4x²+8xh-3x-3h+1-4x²+3x-1]/h
  
  (4h²+8hx)/h
  
  4h+8x
  
  Which simplifies to 4(h+2x)
  
  I believe its correct
  numerator again ...
  
  $4(x + h)^2 - 3(x + h) + 1 - (4x^2 - 3x + 1) =$
  
  $4(x^2 + 2xh + h^2) - 3(x + h) + 1 - (4x^2 - 3x + 1) =$
  
  $4x^2 + 8xh + 4h^2 - 3x - 3h + 1 - 4x^2 + 3x - 1 =$
  
  $8xh + 4h^2 - 3h =$
  
  $h(8x + 4h - 3)$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: This document is a forum thread where a user asks for help with calculating the average rate of change for a given function. The user provides a specific function and asks for the rate of change over certain intervals. Another user responds with a definition and an example, and the original poster attempts to solve the problem, leading to further clarifications and corrections from the responder.
  
  Label: question-and-answer