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=

Answers

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Answered by GPT-4o mini
To determine the rate of change (m) and the initial value (b) of the linear function given the table of values, we use the following information from the table:

- Point 1: (2, 22)
- Point 2: (8, 25)

**Step 1: Calculate the rate of change (m)**

The rate of change (slope) \( m \) is calculated using the formula:

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

Substituting the values from the points:

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

So, the rate of change \( m = \frac{1}{2} \).

**Step 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 linear equation:

\[
y = mx + b
\]

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

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

Calculating this gives:

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

So, the initial value \( b = 21 \).

**Final Answer:**

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

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