Question

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 −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 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 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 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 .
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Answer 1

To determine the initial value and rate of change of the linear function represented in the graph, we need to analyze the points provided and the characteristics of a linear equation.

1. **Identify Points**:
- The points provided on the line are:
- \( A(-4, -\frac{10}{3}) \)
- \( B(-1, -\frac{4}{3}) \)

2. **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 points \( A(-4, -\frac{10}{3}) \) and \( B(-1, -\frac{4}{3}) \):
\[
m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{3} = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]
Thus, the rate of change (slope) is \(\frac{2}{3}\).

3. **Determine the Initial Value**:
The initial value (y-intercept) of a linear function can be found using the slope-intercept form of the equation \( y = mx + b \). To find \( b \), we can substitute one of the points into the line equation.

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

So, the initial value is \(-\frac{2}{3}\).

4. **Conclusion**:
Based on the calculations:
- The initial value is \(-\frac{2}{3}\).
- The rate of change is \(\frac{2}{3}\).

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