Question

Use the table to answer the question.
xxyy
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=m=, and the initial value is b=b=.



Use the table to answer the question.
xxyy
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=m=, and the initial value is b=b=.

Answer 1

To determine the rate of change (slope) and the initial value (y-intercept) from the given table of values, we can use the following steps:

From the table:
- When \( x = 2 \), \( y = 22 \)
- When \( x = 8 \), \( y = 25 \)

### Step 1: Calculate the Rate of Change (slope, \( m \))

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

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

Substituting the values from the table:

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

Calculating:

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

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

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

We can use the slope-intercept form of a linear equation, which is:

\[
y = mx + b
\]

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

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

Calculating:

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

Thus, the initial value is \( b = 21 \).

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