Question

Is the rate of change of the function 5?

On a coordinate plane, a line with positive slope goes through points (negative 2, 0) and (1, 2).

Yes, because y changes by 5 every time x changes by 1.
Yes, because y changes by 1 every time x changes by 5.
No, because y does not change by 5 every time x changes by 1.
No, because y does not change by 1 every time x changes by 5.

Answer 1

To determine if the rate of change of the function is 5, we can calculate the slope of the line passing through the points \((-2, 0)\) and \((1, 2)\).

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the given points:
- \(x_1 = -2\), \(y_1 = 0\)
- \(x_2 = 1\), \(y_2 = 2\)

Plugging in the values:

\[
m = \frac{2 - 0}{1 - (-2)} = \frac{2}{1 + 2} = \frac{2}{3}
\]

The slope (rate of change) is \(\frac{2}{3}\).

Now evaluating each option:

1. **Yes, because y changes by 5 every time x changes by 1.**
Incorrect, because the change in \(y\) is not 5, it's 2 over and 3 up gives the slope \( \frac{2}{3} \).

2. **Yes, because y changes by 1 every time x changes by 5.**
Incorrect, as this does not reflect the calculated slope.

3. **No, because y does not change by 5 every time x changes by 1.**
Correct, the change in \(y\) is 2, not 5.

4. **No, because y does not change by 1 every time x changes by 5.**
This is also correct but does not reflect the calculated slope accurately.

Thus, the most accurate and directly relevant response is:

**No, because y does not change by 5 every time x changes by 1.**