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:
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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 \( \frac{2}{3} \).
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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.
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