# Math Snap

## PROBLEM

## These tables represent a quadratic function with a vertex at $(0,3)$. What is the average rate of change for the interval from $x=7$ to $x=8$ ? \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 0 & 3 \\ \hline 1 & 2 \\ \hline 2 & -1 \\ \hline 3 & -6 \\ \hline 4 & -13 \\ \hline 5 & -22 \\ \hline 6 & -33 \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline Interval & $\begin{array}{c}\text { Average rate } \\ \text { of change }\end{array}$ \\ \hline 0 to 1 & -1 \\ \hline 1 to 2 & -3 \\ \hline 2 to 3 & -5 \\ \hline 3 to 4 & -7 \\ \hline 4 to 5 & -9 \\ \hline 5 to 6 & -11 \\ \hline \end{tabular} A. -46 B. -2 C. -61 D. -15

#### STEP 1

Assumptions1. The function is a quadratic function with a vertex at (0,3)
. The average rate of change for the interval from $x=7$ to $x=8$ is what we are looking for3. The average rate of change between two points $(x1, y1)$ and $(x, y)$ on a function is calculated as $\frac{y - y1}{x - x1}$

4. The average rate of change for the intervals from $x=0$ to $x=6$ are given in the second table

#### STEP 2

We can see from the second table that the average rate of change is decreasing by2 for each increase in $x$ by1. This is a linear pattern. We can use this pattern to predict the average rate of change for the interval from $x=7$ to $x=8$.

#### STEP 3

The average rate of change for the interval from $x=5$ to $x=6$ is -11. To find the average rate of change for the interval from $x=6$ to $x=7$, we subtract2 from -11.

$\text{Average rate of change from } x=6 \text{ to } x=7 = -11 -2$

#### STEP 4

Calculate the average rate of change from $x=6$ to $x=7$.

$\text{Average rate of change from } x=6 \text{ to } x=7 = -11 -2 = -13$

#### STEP 5

Now, to find the average rate of change for the interval from $x=7$ to $x=8$, we subtract2 from -13.

$\text{Average rate of change from } x=7 \text{ to } x=8 = -13 -2$

##### SOLUTION

Calculate the average rate of change from $x=$ to $x=8$.

$\text{Average rate of change from } x= \text{ to } x=8 = -13 -2 = -15$The average rate of change for the interval from $x=$ to $x=8$ is -15, so the answer is D. -15.