Question

determine inital value and rate of change of linear function
2,22
8,25

Answer 1

To determine the initial value and the rate of change of the linear function given the two points (2, 22) and (8, 25), we can follow these steps:

1. **Identify Points**: The points are \((x_1, y_1) = (2, 22)\) and \((x_2, y_2) = (8, 25)\).

2. **Calculate the Rate of Change (Slope)**: The rate of change (slope) \(m\) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values:
\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]

3. **Determine the Initial Value (y-intercept)**: The equation of a linear function can be written in the slope-intercept form:
\[
y = mx + b
\]
We know \(m = \frac{1}{2}\). To find \(b\) (the initial value or y-intercept), we can substitute one of the points into the equation. Using the point (2, 22):
\[
22 = \frac{1}{2}(2) + b
\]
\[
22 = 1 + b
\]
\[
b = 22 - 1 = 21
\]

Thus, the initial value (y-intercept) is \(21\) and the rate of change (slope) is \(\frac{1}{2}\).

### Summary
- Initial value (y-intercept) \(b = 21\)
- Rate of change (slope) \(m = \frac{1}{2}\)