Question

x=-1.5 y=-3
x=0.5 y=1

Determine the initial value and the rate of change of the linear function as given in the graph.

Answer 1

To determine the initial value and the rate of change (slope) of the linear function based on the two points provided, we can use the following points:

1. \((-1.5, -3)\)
2. \((0.5, 1)\)

### Step 1: Calculate the Rate of Change (Slope)
The slope \(m\) of the line passing through these two points can be calculated using the formula:

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

where \((x_1, y_1) = (-1.5, -3)\) and \((x_2, y_2) = (0.5, 1)\).

Plugging in the values:

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

### Step 2: Determine the Initial Value (y-intercept)
To find the equation of the line, we can use the point-slope form of a linear equation:

\[
y - y_1 = m(x - x_1)
\]

Using the slope we found (2) and one of the points, such as \((0.5, 1)\):

\[
y - 1 = 2(x - 0.5)
\]

Now, to convert this into slope-intercept form \(y = mx + b\):

\[
y - 1 = 2x - 1 \\
y = 2x + 0
\]

From this, we can see that the y-intercept (initial value) is \(b = 0\).

### Conclusion
- The **initial value** (y-intercept) of the linear function is \(0\).
- The **rate of change** (slope) of the linear function is \(2\).