To determine the rate of change (m) and the initial value (b) of the linear function given the two points from the table, we will use the following formulas:
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Rate of Change (slope) \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (2, 22)\) and \((x_2, y_2) = (8, 25)\).
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Initial Value (y-intercept) \( b \): For a linear function, if we know the slope \( m \) and one point, we can find \( b \) using the equation of the line \( y = mx + b \).
Step 1: Calculate the Rate of Change (m)
Using the values from the table:
- \( y_1 = 22 \), \( x_1 = 2 \)
- \( y_2 = 25 \), \( x_2 = 8 \)
Substituting these values into the formula for \( m \): \[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
Step 2: Calculate the Initial Value (b)
We can use one of the points (we'll use (2, 22)) to find \( b \). Now we have \( m = \frac{1}{2} \).
The equation of the line is: \[ y = mx + b \] Substituting in \( m \) and the values for one of the points: \[ 22 = \frac{1}{2}(2) + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Final Answers
The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
So, the answers are:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)
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