Question

find the rate of change and the initial value of the linear function given
(0,0) and (2,3)

Answer 1

To find the rate of change (slope) and the initial value (y-intercept) of the linear function defined by the points (0, 0) and (2, 3), we can use the following steps:

1. **Calculate the slope (rate of change)** using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (2, 3)\).

Plugging in the values:
\[
m = \frac{3 - 0}{2 - 0} = \frac{3}{2}
\]

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

2. **Determine the initial value (y-intercept)**: In this case, since one of the points is \((0, 0)\), the y-intercept is \(0\).

Now we can write the linear equation in slope-intercept form \(y = mx + b\):
- We have \(m = \frac{3}{2}\) and \(b = 0\).

Thus, the equation of the linear function is:
\[
y = \frac{3}{2}x
\]

**Summary**:
- Rate of change: \(\frac{3}{2}\)
- Initial value (y-intercept): \(0\)