SOLUTION: 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?

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.

RELATED QUESTIONS
Suppose that the function y=f(x) is decreasing on the interval (-2,7). Over what interval (answered by ikleyn,MathLover1)

y=f(x) is decreasing on the interval (-2,7). what interval is the graph of y=f(x+2)... (answered by MathLover1,ikleyn)

Suppose that a function y=f(x) is increasing on the interval (-5,2) a.)Over what... (answered by josgarithmetic)

Let f(x) = x^2 + 18x + 83. On what interval is the function... (answered by psbhowmick,ewatrrr)

Suppose that the function y = f(x) is increasing on the interval (4, 7). Over what... (answered by Theo)

Find an interval over which the function f(x)=(x+5)^3(x-4)^2 is decreasing. (answered by KMST)

f(x)=x^2+6x+4 1. Find the vertex 2. State the range of the function 3. On what... (answered by lwsshak3)

Sketch the graph of the following absolute value equation. Identify the vertex (ordered... (answered by ikleyn)

f(x) = x^3 − 2x^2 − 4x + 8 / x − 2 Find any hole(s) in the graph of f. (answered by greenestamps)