Question

Use the image to answer the question. (0.5,1) (-1.5,-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 determine the initial value and the rate of change of the linear function given the points (0.5, 1) and (-1.5, -3), we can follow these steps:

1. **Initial Value**: The initial value of a linear function is typically the y-coordinate when x = 0. Since the point (0.5, 1) does not have x = 0, we would have to determine the y-intercept (where x = 0) using the points provided.

2. **Rate of Change (Slope)**: The slope (rate of change) of a line can be calculated using the formula:

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

Using the points (-1.5, -3) and (0.5, 1):

\[
m = \frac{1 - (-3)}{0.5 - (-1.5)} = \frac{1 + 3}{0.5 + 1.5} = \frac{4}{2} = 2
\]

So, the rate of change is **2**.

3. **Finding Initial Value (y-intercept)**: We can use the slope-intercept form of the line, which is:

\[
y = mx + b
\]

where \( m = 2 \) (slope) and \( b \) is the y-intercept. To find \( b \), we can use one of the points. Let's use the point (0.5, 1):

\[
1 = 2(0.5) + b
\]
\[
1 = 1 + b
\]
\[
b = 1 - 1 = 0
\]

Hence, the initial value is **0**.

Therefore, the final answers are:

- The initial value is **(0)**,
- The rate of change is **(2)**.