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

Answer:

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:

$\displaystyle m=\frac{27-18}{6-5}=9$

Our answer is D.