Question

This table represents a quadratic function with a vertex at (1, 2). What is the

average rate of change for the interval from x = 5 to x = 6?

Answer 1

D: 9

Step-by-Step Explanation:

The average rate is synonymous with the slope. Since we want to find the average rate of change from x = 5 to x = 6, we will use the two points (5, 18) and (6, ?). We will need to find ? first.

Since the table represents a quadratic function and we are given the vertex, we can use the vertex form of a quadratic:

`\displaystyle f(x)=a(x-h)^2+k`

Where (h, k) is the vertex.

The vertex is (1, 2). Hence:

`f(x)=a(x-1)^2+2`

To determine a, pick a sample point from the table and solve for a. We can use (2, 3). Hence:

`(3)=a((2)-1)^2+2`

Solve for a:

`1=a(1)^2\Rightarrow a=1`

Hence, our function is:

`f(x)=(x-1)^2+2`

Evaluate the function when x = 6:

`\displaystyle f(6)=(6-1)^2+2=27`

So, our two points are (5, 18) and (6, 27).

Again, to find the average rate of change between x= 5 and x = 6, find the slope between their two points. Hence:

`\displaystyle m=(27-18)/(6-5)=9`

Our answer is D.