Question

Determine the initial value and rate of change of the linear function (-3,4) and (0,3)

Answer 1

To determine the initial value and rate of change of the linear function defined by the two points \((-3,4)\) and \((0,3)\), we first need to find the slope (rate of change) of the line and then the y-intercept (initial value).

1. **Finding the Slope (Rate of Change)**:
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-3, 4)\) and \((0, 3)\):
\[
(x_1, y_1) = (-3, 4) \quad \text{and} \quad (x_2, y_2) = (0, 3)
\]
Substituting these values into the slope formula:
\[
m = \frac{3 - 4}{0 - (-3)} = \frac{-1}{3} = -\frac{1}{3}
\]

2. **Finding the Initial Value (y-intercept)**:
The y-intercept occurs when \(x = 0\). From the points provided, we see that when \(x = 0\), \(y = 3\). Thus, the initial value (y-intercept) is:
\[
b = 3
\]

3. **Writing the Equation of the Line**:
Now we can use the point-slope form of the linear equation \(y = mx + b\) to write the equation based on our slope and y-intercept:
\[
y = -\frac{1}{3}x + 3
\]

### Summary:
- **Initial Value** (y-intercept): \(3\)
- **Rate of Change** (slope): \(-\frac{1}{3}\)