Question

What is the rate of change of the function represented by the table?

x
y
1
–8.5
2
–6
3
–3.5
4
–1
–2.5
–1
1
2.5

Answer 1

To determine the rate of change of the function represented by the table, we can calculate the change in \( y \) divided by the change in \( x \) for consecutive points.

The table provided is:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & -8.5 \\
2 & -6 \\
3 & -3.5 \\
4 & -1 \\
5 & -2.5 \\
6 & -1 \\
7 & 1 \\
8 & 2.5 \\
\hline
\end{array}
\]

Let's calculate the rate of change between each consecutive pair of \( x \) and \( y \) values:

1. Between \( (1, -8.5) \) and \( (2, -6) \):
\[
\text{Rate of change} = \frac{-6 - (-8.5)}{2 - 1} = \frac{2.5}{1} = 2.5
\]

2. Between \( (2, -6) \) and \( (3, -3.5) \):
\[
\text{Rate of change} = \frac{-3.5 - (-6)}{3 - 2} = \frac{2.5}{1} = 2.5
\]

3. Between \( (3, -3.5) \) and \( (4, -1) \):
\[
\text{Rate of change} = \frac{-1 - (-3.5)}{4 - 3} = \frac{2.5}{1} = 2.5
\]

4. Between \( (4, -1) \) and \( (5, -2.5) \):
\[
\text{Rate of change} = \frac{-2.5 - (-1)}{5 - 4} = \frac{-1.5}{1} = -1.5
\]

5. Between \( (5, -2.5) \) and \( (6, -1) \):
\[
\text{Rate of change} = \frac{-1 - (-2.5)}{6 - 5} = \frac{1.5}{1} = 1.5
\]

6. Between \( (6, -1) \) and \( (7, 1) \):
\[
\text{Rate of change} = \frac{1 - (-1)}{7 - 6} = \frac{2}{1} = 2
\]

7. Between \( (7, 1) \) and \( (8, 2.5) \):
\[
\text{Rate of change} = \frac{2.5 - 1}{8 - 7} = \frac{1.5}{1} = 1.5
\]

To summarize:

- From \( x = 1 \) to \( x = 2 \): \( 2.5 \)
- From \( x = 2 \) to \( x = 3 \): \( 2.5 \)
- From \( x = 3 \) to \( x = 4 \): \( 2.5 \)
- From \( x = 4 \) to \( x = 5 \): \( -1.5 \)
- From \( x = 5 \) to \( x = 6 \): \( 1.5 \)
- From \( x = 6 \) to \( x = 7 \): \( 2 \)
- From \( x = 7 \) to \( x = 8 \): \( 1.5 \)

Thus, the rates of change vary between these points, showing an increasing trend followed by fluctuations.