Question

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Slope and y-intercepts Practice
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Question
Use the table to answer the question.

Determining the starting value and rate of change from a given value table.
x y
1 4
3 8
Find the rate of change and the initial value of the linear function given as a table of values.

(1 point)
The rate of change is m=
, and the initial value is b=

Answer 1

To find the rate of change (\(m\)) and the initial value (\(b\)) based on the given values from the table, we can follow these steps:

From the given points:
- When \(x = 1\), \(y = 4\) (point 1: (1, 4))
- When \(x = 3\), \(y = 8\) (point 2: (3, 8))

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

The formula for the rate of change (slope) is given by:

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

Using the coordinates of the points:
- \(y_1 = 4\), \(y_2 = 8\)
- \(x_1 = 1\), \(x_2 = 3\)

Substituting the values into the formula:

\[
m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2
\]

So, the rate of change is \(m = 2\).

**Step 2: Determine the initial value (\(b\))**

The initial value or y-intercept (\(b\)) in a linear function can be found using the slope-intercept form of the equation, \(y = mx + b\).

Using one of the points to solve for \(b\). Let's use the point (1, 4):

\[
4 = 2(1) + b
\]
\[
4 = 2 + b
\]
\[
b = 4 - 2 = 2
\]

So, the initial value is \(b = 2\).

**Final Answer:**
- The rate of change is \(m = 2\), and the initial value is \(b = 2\).