To find the rate of change and the initial value of the linear function from the table, we can follow these steps:
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Identify the points from the table:
Given the values:- (1, 4)
- (-3, 8)
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Calculate the rate of change (slope):
The slope \( m \) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's assign the points:
- \( (x_1, y_1) = (1, 4) \)
- \( (x_2, y_2) = (-3, 8) \)
Plugging in the values:
\[ m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1 \]
-
Identify the initial value (y-intercept):
The initial value \( b \) (y-intercept) can be found using one of the points and the slope in the slope-intercept form of a linear equation \( y = mx + b \).Using point \( (1, 4) \) and plugging into the equation:
\[ 4 = -1(1) + b \]
\[ 4 = -1 + b \rightarrow b = 4 + 1 \rightarrow b = 5 \]
Now we have the rate of change and the initial value:
- The rate of change \( m = -1 \)
- The initial value \( b = 5 \)
Final answer: The rate of change is \( m = -1 \), and the initial value is \( b = 5 \).