Question

Determine that initial value and the rate of change of the linear functions of (0,0) and 2,3)

Answer 1

To determine the initial value and the rate of change (slope) of the linear function that passes through the points (0, 0) and (2, 3), we can follow these steps:

1. **Define the Points**:
- Point A: \((0, 0)\)
- Point B: \((2, 3)\)

2. **Calculate the Slope (Rate of Change)**:
The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

Substituting our points into the formula:

\[
m = \frac{3 - 0}{2 - 0} = \frac{3}{2}
\]

So, the rate of change (slope) of the linear function is \(\frac{3}{2}\).

3. **Determine the Initial Value**:
The initial value of the function corresponds to the y-intercept, which is the value of \(y\) when \(x = 0\). Since one of our points is \((0, 0)\), the y-intercept is:

\[
b = 0
\]

Therefore, the linear function can be expressed in slope-intercept form as:

\[
y = \frac{3}{2}x + 0 \quad \text{or simply} \quad y = \frac{3}{2}x
\]

### Summary:
- **Initial Value (Y-intercept)**: \(0\)
- **Rate of Change (Slope)**: \(\frac{3}{2}\)