Question

Determine the initial value and the rate of change of the linear function of points (-4, -10/3) and (-1, -4/3)

Answer 1

To find the initial value and the rate of change of the linear function passing through the points \((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\), we can start by calculating the slope (rate of change) of the line.

The slope \(m\) can be calculated using the formula:

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

Here, we can take:
- Point 1: \((-4, -\frac{10}{3})\) \(\Rightarrow (x_1, y_1)\)
- Point 2: \((-1, -\frac{4}{3})\) \(\Rightarrow (x_2, y_2)\)

Plugging in the coordinates:

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

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

Next, we need to find the initial value (y-intercept) of the linear function. The equation of a linear function can be expressed in slope-intercept form as:

\[
y = mx + b
\]

where \(m\) is the slope and \(b\) is the y-intercept. We already found that \(m = \frac{2}{3}\).

To find \(b\), we can use one of the points. Let's use the point \((-4, -\frac{10}{3})\):

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

Calculating the right side:

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

Now, add \(\frac{8}{3}\) to both sides:

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

Simplifying the left side:

\[
-\frac{2}{3} = b
\]

Thus, the y-intercept \(b\) is \(-\frac{2}{3}\).

The equation of the linear function is:

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

In summary:
- The initial value (y-intercept) is \(-\frac{2}{3}\).
- The rate of change (slope) is \(\frac{2}{3}\).