Question

find the rate of change and the initial value of the linear function given
x 1, y 4
x -3, y 8

Answer 1

To find the rate of change (slope) and the initial value (y-intercept) of the linear function given the points \((1, 4)\) and \((-3, 8)\), we can use the following steps.

1. **Calculate the rate of change (slope)**:
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

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

Now plug in the values:

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

The slope (rate of change) is \(-1\).

2. **Find the initial value (y-intercept)**:
We can use the slope-intercept form of a linear equation, which is:

\[
y = mx + b
\]

Here, \(m\) is the slope and \(b\) is the y-intercept (the initial value). We can plug in one of the points along with the slope to solve for \(b\). Using the point \((1, 4)\):

\[
4 = -1(1) + b
\]
\[
4 = -1 + b
\]
\[
b = 4 + 1 = 5
\]

Thus, the initial value (y-intercept) is \(5\).

### Summary:
- Rate of change (slope): \(-1\)
- Initial value (y-intercept): \(5\)

The linear function can thus be expressed as:

\[
y = -1x + 5
\] or equivalently, \(\boxed{y = -x + 5}\).