Question

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=
.
Remaining Attempts : 3
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Answer 1

To determine the rate of change (m) and the initial value (b) from the provided value table, we can follow these steps:

From the table:
- When \( x = 1 \), \( y = 4 \)
- When \( x = 3 \), \( y = 8 \)

1. **Calculate the rate of change (m):**
The rate of change (slope) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (3, 8)\).

Plugging in the values:
\[
m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2
\]

2. **Find the initial value (b):**
The initial value is the \(y\)-intercept of the linear function, which is the \(y\) value when \(x = 0\).
We can use the point-slope form of a linear equation, \(y = mx + b\), to find \(b\):
- We already calculated \(m = 2\).
- We can use 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 rate of change is \(m = 2\), and the initial value is \(b = 2\).

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