To determine the rate of change (slope) and the initial value (y-intercept) from the given table of values, we can use the following steps:
From the table:
- When \( x = 2 \), \( y = 22 \)
- When \( x = 8 \), \( y = 25 \)
Step 1: Calculate the Rate of Change (slope, \( m \))
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from the table:
- Let \( (x_1, y_1) = (2, 22) \) and \( (x_2, y_2) = (8, 25) \)
Calculating:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
So, the rate of change is \( m = \frac{1}{2} \).
Step 2: Determine the Initial Value (y-intercept, \( b \))
We can use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
We can use one of the points to find \( b \). Using the point \((2, 22)\):
\[ 22 = \frac{1}{2}(2) + b \]
Calculating:
\[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Thus, the initial value is \( b = 21 \).
Final Answer
The rate of change is \( m = \frac{1}{2} \) and the initial value is \( b = 21 \).
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