Question

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. An upward slanting line is plotted on the coordinate plane. The upward slanting line with arrows at both ends passes through the following closed points: left parenthesis negative 4 comma start fraction negative 10 over 3 end fraction right parenthesis, and left parenthesis negative 1 comma start fraction negative 4 over 3 end fraction right parenthesis.
Determine the initial value and the rate of change of the linear function as given in the graph.

(1 point)
Responses

The initial value is −2/3, and the rate of change is 2/3.

The initial value is 2/3, and the rate of change is −2/3.

The initial value is 1/6, and the rate of change is 3/2.

The initial value is −2, and the rate of change is 2/3.

Answer 1

To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the line on the coordinate plane, we will perform the following steps:

1. **Calculate the rate of change (slope)**:
The slope \( m \) can be calculated using the formula:

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

Using the two points provided:
\( (-4, -\frac{10}{3}) \) and \( (-1, -\frac{4}{3}) \):

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

Substitute into the slope formula:

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

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

2. **Calculate the y-intercept (initial value)**:
We can use the slope-intercept form of the line \( y = mx + b \), where \( b \) is the y-intercept.

We can take one of the points, let's use \( (-1, -\frac{4}{3}) \), and substitute into the equation:

\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]

Simplifying this:

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

Adding \( \frac{2}{3} \) to both sides:

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

So, summarizing the findings:
- The rate of change (slope) is \( \frac{2}{3} \).
- The initial value (y-intercept) is \( -\frac{2}{3} \).

Thus, the correct response is:

**The initial value is −2/3, and the rate of change is 2/3.**