Question

Points given:

(-4, -10/3)
(-1, -4/3)

Determine the intial value and rate of change of the linear function.

Answer 1

To determine the initial value and rate of change (slope) of the linear function defined by the given points \((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\), let's use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

### Step 1: Substitute the values into the formula

Let \((x_1, y_1) = (-4, -\frac{10}{3})\) and \((x_2, y_2) = (-1, -\frac{4}{3})\).

Continuing with the calculation, we have:

- \(y_2 = -\frac{4}{3}\)
- \(y_1 = -\frac{10}{3}\)
- \(x_2 = -1\)
- \(x_1 = -4\)

### Step 2: Calculate the slope (rate of change)

\[
m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)}
\]

This simplifies to:

\[
m = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]

### Step 3: Identify the initial value

To express the linear function in the slope-intercept form \(y = mx + b\), we need to find the y-intercept \(b\) (initial value).

We can use one of the points and the slope to solve for \(b\). Using the point \((-4, -\frac{10}{3})\):

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

Calculating \( \frac{2}{3}(-4) \):

\[
\frac{2}{3} \cdot -4 = -\frac{8}{3}
\]

So the equation becomes:

\[
-\frac{10}{3} = -\frac{8}{3} + b
\]

Adding \(\frac{8}{3}\) to both sides:

\[
b = -\frac{10}{3} + \frac{8}{3} = -\frac{2}{3}
\]

### Summary of Results

- **Rate of Change (Slope)**: \( \frac{2}{3} \)
- **Initial Value (y-intercept)**: \( -\frac{2}{3} \)

Thus, the linear function can be expressed as:

\[
y = \frac{2}{3}x - \frac{2}{3}
\]