# 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):
$\displaystyle \displaystyle \frac{g(b) - g(a)}{b - a}$

For example, average rate of change between x=0 and x=1 is
$\displaystyle \displaystyle \frac{g(1) - g(0)}{1 - 0} = \frac{(4(1)^2 -3(1) +1) - (4(0)^2 -3(0) +1)}{1 - 0} = 1$

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 $\displaystyle f(x) = 4x^2 - 3x + 1$ ??? you called it $\displaystyle g(x)$.

if so,

$\displaystyle f(x+h) - f(x)$

the numerator should be ...

$\displaystyle 4(x+h)^2 - 3(x+h) + 1 - (4x^2 - 3x + 1)$

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 ...

$\displaystyle 4(x + h)^2 - 3(x + h) + 1 - (4x^2 - 3x + 1) =$

$\displaystyle 4(x^2 + 2xh + h^2) - 3(x + h) + 1 - (4x^2 - 3x + 1) =$

$\displaystyle 4x^2 + 8xh + 4h^2 - 3x - 3h + 1 - 4x^2 + 3x - 1 =$

$\displaystyle 8xh + 4h^2 - 3h =$

$\displaystyle h(8x + 4h - 3)$