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 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 .

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 −23
, and the rate of change is 23
.

Answer 1

To determine the initial value and the rate of change of the linear function represented by the line in the coordinate plane, we will follow these steps:

1. **Calculate the slope (rate of change)** using the two points provided:
\((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\).

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

Here,
\[
x_1 = -4, \quad y_1 = -\frac{10}{3}, \quad x_2 = -1, \quad y_2 = -\frac{4}{3}
\]
Substituting these values into the slope formula gives us:
\[
m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]

Therefore, the **rate of change** is \(\frac{2}{3}\).

2. **Find the initial value**. The initial value is the y-intercept of the line (the value of \(y\) when \(x = 0\)). We can use the equation of the line in slope-intercept form, \(y = mx + b\), or we can substitute one of the points to solve for \(b\).

Using the slope \(m = \frac{2}{3}\) and the point \((-1, -\frac{4}{3})\):
\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]
\[
-\frac{4}{3} = -\frac{2}{3} + b
\]
\[
b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]

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

3. **Summarizing the results**:
- The initial value is \(-\frac{2}{3}\).
- The rate of change is \(\frac{2}{3}\).

Given this analysis, the correct response choice based on the options provided is:

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