To find the initial value of a function given a point and a rate of change (slope), we can use the slope-intercept form of a linear equation, which is:
\[ y = mx + b \]
Where:
- \( m \) is the slope (rate of change)
- \( b \) is the y-intercept (initial value)
- \( (x, y) \) is a point on the line
In this case, you've provided the point (2,2) and a rate of change (slope) as \( \frac{1}{2} \).
We can plug the point and the slope into the equation to solve for \( b \):
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Start with the point (2,2): \[ 2 = \frac{1}{2}(2) + b \]
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Calculate \( \frac{1}{2}(2) \): \[ 2 = 1 + b \]
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Subtract 1 from both sides to find \( b \): \[ b = 2 - 1 \] \[ b = 1 \]
Therefore, the initial value of the function (y-intercept) is \( 1 \).
So the equation of the function is: \[ y = \frac{1}{2}x + 1 \]