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Question 1204975: suppose the function y=f(x) is decreasing on the interval (-2,7). Over what interval is the graph of y=f(x+2) decreasing?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: (-4, 5)
This represents the interval -4 < x < 5
Explanation
Going from f(x) to f(x+2) will shift the curve 2 units to the left.
The decreasing interval will also shift this amount.
Subtract 2 from each endpoint
-2 becomes -2-2 = -4
7 becomes 7-2 = 5
The decreasing interval for y = f(x+2) is (-4, 5) which is interval notation for -4 < x < 5.
If you are wondering "why does the x+2 mean shift 2 units left instead of 2 units right?" the reason is that the replacement of x with x+2 will move the xy axis 2 units to the right.
Holding the curve in place gives the illusion it moves 2 units left.
Take a look at the graph of y = x^2 and y = (x+2)^2 for instance.
Desmos and GeoGebra are two graphing tools I recommend.
What's an example of a function that decreases on the interval -2 < x < 7?
There are infinitely many possible, but one example is f(x) = (200/243)(x-2.5)^3 - 50x
Replace every x with x+2 to end up with g(x) = (200/243)(x+2-2.5)^3 - 50(x+2) to find the decreasing interval has shifted 2 units leftward.
Desmos graph link
https://www.desmos.com/calculator/49glyjmlkw
If you were to type decreasing (200/243)(x-2.5)^3 - 50x into WolframAlpha, then it will tell you that the decreasing interval is -2 < x < 7, and also highlight the decreasing portion.
https://www.wolframalpha.com/input?i=decreasing+%28200%2F243%29%28x-2.5%29%5E3+-+50x
Then type in decreasing (200/243)(x+2-2.5)^3 - 50(x+2) to see how the interval shifts.
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