To determine the rate of change (slope) and the initial value (y-intercept) from the given x and y values, we can follow these steps:
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Calculate the Rate of Change (m): The rate of change (slope) \( m \) of a linear function can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points provided:
- Point 1: \( (x_1, y_1) = (2, 22) \)
- Point 2: \( (x_2, y_2) = (8, 25) \)
Plugging in the values:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
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Determine the Initial Value (b): The initial value (y-intercept) can be found using the linear equation \( y = mx + b \). We can rearrange this to solve for \( b \):
\[ b = y - mx \]
We can use either of the points. Let's use the first point \( (2, 22) \):
\[ b = 22 - \left(\frac{1}{2} \cdot 2\right) = 22 - 1 = 21 \]
Now we have the values:
- Rate of Change \( m = \frac{1}{2} \)
- Initial Value \( b = 21 \)
Final Answer: The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).