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Bot GPT 3.5 To find the rate of change and the initial value of the linear function, we need to determine the equation of the line that passes through the given points.
Using the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1)
For the points (x1, y1) = (x, y) = (4, -3), and (x2, y2) = (x, y) = (-3, 8):
m = (8 - (-3)) / (-3 - 4)
m = 11 / -7
m = -11/7
The rate of change, or the slope, of the linear function is -11/7.
To determine the y-intercept or the initial value of the linear function, we can use the slope-intercept form of a linear equation: y = mx + b, where b is the y-intercept.
Substituting the values of the slope (m = -11/7), and one of the points (x, y) = (4, -3):
-3 = (-11/7)(4) + b
Multiplying -11/7 by 4:
-3 = (-44/7) + b
Find the common denominator of 7:
-3 = (-44 + 7b) / 7
Multiplying both sides of the equation by 7 to eliminate the fraction:
-21 = -44 + 7b
Adding 44 to both sides to solve for b:
23 = 7b
Dividing both sides by 7:
b = 23/7
The y-intercept, or the initial value of the linear function, is 23/7.
The equation of the linear function is y = (-11/7)x + 23/7.
Therefore, the rate of change of the linear function is -11/7 and the initial value is 23/7.