Question

Use the image to answer the 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 −23
, and the rate of change is 23
.
The initial value is negative Start Fraction 2 over 3 End Fraction , and the rate of change is Start Fraction 2 over 3 End Fraction .

The initial value is 23
, and the rate of change is −23
.
The initial value is Start Fraction 2 over 3 End Fraction , and the rate of change is negative Start Fraction 2 over 3 End Fraction .

The initial value is 16
, and the rate of change is 32
.
The initial value is Start Fraction 1 over 6 End Fraction , and the rate of change is Start Fraction 3 over 2 End Fraction .

The initial value is −2
, and the rate of change is 23
.
The initial value is negative 2 , and the rate of change is Start Fraction 2 over 3 End Fraction .
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Answer 1

To determine the initial value and the rate of change of the linear function described in the problem, we’ll do the following:

1. **Find the slope (rate of change)** using the two points given:
- Point 1: \((-4, -\frac{10}{3})\)
- Point 2: \((-1, -\frac{4}{3})\)

The formula for slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

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

2. **Determine the initial value (y-intercept)** using the slope-intercept form \(y = mx + b\):
We have calculated \(m = \frac{2}{3}\). We can use one of the points to solve for \(b\).

Let's use the point \((-1, -\frac{4}{3})\):
\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]
\[
-\frac{4}{3} = -\frac{2}{3} + b
\]
Adding \(\frac{2}{3}\) to both sides:
\[
b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]

Thus, the initial value (y-intercept) is \(-\frac{2}{3}\) and the rate of change (slope) is \(\frac{2}{3}\).

The correct response is:

**The initial value is negative \(\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).**