Question

Functions Unit Test
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Question
Use the table to answer the 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=
.
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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 can use the following information from the table:

- When \( x = 2 \), \( y = 22 \)
- When \( x = 8 \), \( y = 25 \)

### Step 1: Calculate the Rate of Change (Slope)

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

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

Substituting the values:

- Let \( (x_1, y_1) = (2, 22) \)
- Let \( (x_2, y_2) = (8, 25) \)

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

### Step 2: Determine the Initial Value (y-intercept)

To write the equation of the linear function in the form \( y = mx + b \), we can substitute one of the points and the slope into the equation to find \( b \).

Using the slope \( m = \frac{1}{2} \) and the point \( (2, 22) \):

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

Calculating \( b \):

\[
22 = 1 + b \\
b = 22 - 1 \\
b = 21
\]

### Final Values

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

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