Question

This table has the x and y values that will contribute to determining the rate of change and the initial value of the linear function.
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 (slope) and the initial value (y-intercept) of the linear function from the given table of values, we will follow these steps:

1. **Calculate the rate of change (m)** using the formula for the slope of a line, which is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the values provided:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)

Plugging into the formula:
\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]

2. **Determine the initial value (b)**, which is the y-intercept. We can use the slope-intercept form of a linear equation, \( y = mx + b \). We can substitute one of the points to solve for \( b \).

Using the point \( (2, 22) \):
\[
22 = \frac{1}{2}(2) + b
\]
\[
22 = 1 + b
\]
\[
b = 22 - 1 = 21
\]

Thus, we have:
- The rate of change \( m = \frac{1}{2} \)
- The initial value \( b = 21 \)

In summary:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)