Question

These tables represents a quadratic function with a vertex at (0,3) what is the average rate of change for the interval from x=8 to x=9

Answer 1

There are a few ways to do it; the easiest is just to follow the pattern of taking away two from the average.

The next lines of the table are

6 to 7  -13

7 to 8   -15

8 to 9   -17

Answer: C. -17

Let's find the equation and do it that way.

We have a vertex at (0,3) so we can fill out the vertex form a bit.  In general for vertex (p,q) it's

y = a(x-p)^2+q

We have

f(x) = ax^2 + 3

f(1) = 2

a+3 =2

a = -1

So we found our equation,

y = -x^2 + 3

Let's check it at x=5, y=-5^2+3=-25+3=-22, good

We want the rate of change from 8 to 9, which is

$r = \dfrac{f(9)-f(8)}{9-8}=f(9)-f(8)=-9^2 - -8^2 = -17$

Answer: -17 again, that checks