Answer:
- rate of change: -2
- linear function: y = -2x -1
Explanation:
You want the rate of change and a linear function that will have a graph that passes through the points (-2, 3) and (-3, 5).
a. Rate of change
The rate of change can be found using the formula ...
m = (y2 -y1)/(x2 -x1)
m = (5 -3)/(-3 -(-2)) = 2/-1 = -2
The rate of change is -2.
b. Linear function
We can start with the point-slope form of the equation for a line.
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
Using m=-2, and (h, k) = (-2, 3), we find the equation of the line to be ...
y -3 = -2(x -(-2))
Solving for y gives ...
y = -2x -4 +3 . . . . . . eliminate parentheses, add 3
y = -2x -1 . . . . . . simplify
The linear function is y = -2x -1.
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Additional comment
We can't tell the remaining parts of the development you are being guided through. We have written the linear function in a way that seems appropriate for the given information.
If you are asked to find the y-intercept, you could use the slope-intercept form of the equation for a line to do that:
y = mx +b
y -mx = b . . . . . . . . subtract mx to find b
3 -(-2)(-2) = b . . . . . using our value for m, and (x, y) = (-2, 3)
b = 3 -4 = -1
Then the y=mx+b equation can be written
y = -2x -1
If you need "functional form", you can replace y with f(x):
f(x) = -2x -1
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