Question

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

x
y
1
5
2
5
3
5
4
5
0
1
4
5

Answer 1

To determine the rate of change of the function represented by the table, we examine the values of \( y \) at different values of \( x \).

From your table, we have the following pairs:

- For \( x = 1 \), \( y = 5 \)
- For \( x = 2 \), \( y = 5 \)
- For \( x = 3 \), \( y = 5 \)
- For \( x = 4 \), \( y = 5 \)
- For \( x = 0 \), \( y = 1 \)
- For \( x = 4 \), \( y = 5 \) again

It appears that the values of \( y \) are constant at \( 5 \) for \( x = 1, 2, 3, 4 \). This means that within this interval, the change in \( y \) is \( 0 \), and therefore, the rate of change for these \( x \) values is:

\[
\text{Rate of change} = \frac{\Delta y}{\Delta x} = \frac{5 - 5}{x_2 - x_1} = 0
\]

For \( x = 0 \), \( y = 1 \). If we compare \( (0, 1) \) to \( (1, 5) \):

\[
\text{Rate of change from } (0, 1) \text{ to } (1, 5) = \frac{5 - 1}{1 - 0} = \frac{4}{1} = 4
\]

Finally, we can summarize:
- The rate of change is \( 0 \) when \( x \) is in \( (1, 2, 3, 4) \).
- The rate of change from \( (0, 1) \) to \( (1, 5) \) is \( 4 \).

Therefore, there are two different rates of change based on the provided data:
- From \( x = 0 \) to \( x = 1 \): rate of change is \( 4 \).
- From \( x = 1 \) to \( x = 4 \): rate of change is \( 0 \).