Question

Use the image to answer the question. (-4, -10/3) (-1,-4/3)
Determine the initial value and the rate of change of the linear function given in the graph. (1 point) The initial value is () , and the rate of change is ()

Answer 1

To find the initial value and the rate of change of a linear function given two points, we can use the two points you've provided: (-4, -10/3) and (-1, -4/3).

1. **Calculate the Rate of Change (Slope)**:
The formula for the slope (rate of change) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (-4, -10/3)\) and \((x_2, y_2) = (-1, -4/3)\).

Plugging in the values:
\[
m = \frac{-4/3 - (-10/3)}{-1 - (-4)} = \frac{-4/3 + 10/3}{-1 + 4} = \frac{6/3}{3} = \frac{2}{3}
\]

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

2. **Determine the Initial Value (y-intercept)**:
We can use the slope-intercept form of a linear equation, \(y = mx + b\), where \(b\) is the y-intercept (initial value). We can use one of the points to find \(b\). Let's use the point \((-4, -10/3)\):

\[
-\frac{10}{3} = \frac{2}{3}(-4) + b
\]

Which simplifies to:
\[
-\frac{10}{3} = -\frac{8}{3} + b
\]

Adding \(\frac{8}{3}\) to both sides gives:
\[
b = -\frac{10}{3} + \frac{8}{3} = -\frac{2}{3}
\]

So, the initial value is \(-\frac{2}{3}\) and the rate of change is \(\frac{2}{3}\).

**Final Answer**: The initial value is \(-\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).