Question

Use the table to answer the question.
x y
2 22
8 25
Determine the rate of change and the initial value of the linear function given here as a table of values. (1 point) The rate of change is m= , and the initial value is b= .

Answer 1

To determine the rate of change (\(m\)) and the initial value (\(b\)) of the linear function from the given table, we can follow these steps.

1. **Calculate the Rate of Change (Slope \(m\))**:
The formula for the rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Using the points \((2, 22)\) and \((8, 25)\):
- \(x_1 = 2\), \(y_1 = 22\)
- \(x_2 = 8\), \(y_2 = 25\)

Substitute these values into the formula:

\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]

2. **Determine the Initial Value (\(b\))**:
The initial value (\(b\)) is the value of \(y\) when \(x = 0\). We can use the slope-intercept form of the equation of a line, which is \(y = mx + b\).

We can use one of the points to find \(b\). Let's use the point \((2, 22)\):

\[
22 = \left(\frac{1}{2}\right)(2) + b
\]

Simplifying this:

\[
22 = 1 + b
\]

Therefore, \(b = 22 - 1 = 21\).

So, from the calculations:
- The rate of change is \(m = \frac{1}{2}\).
- The initial value is \(b = 21\).

Thus, the final answer is:
- The rate of change is \(m = \frac{1}{2}\), and the initial value is \(b = 21\).