Verify if question a is correct I calcuated twice now and I am getting 0 as an answer.

I don't understand question b

a) Examine the function f9x) =2(0.5x-1)^2(x+2)^2. Determine the function's average of change from -2<x<2.

I got answer:

=0-0/-2-2
=0

Avg rate of change: 0

b) Explain how why you could have predicted what the average rate of change would have been without doing any computations

the equation was f(x) = 2(.5x-1)^2 (x+2)^2

Quick observation is that for x=-2 the first factor is zero and for x = +2 the second factor is zero.
In either case f(x) = 0 or y = 0 for those two end values.
"Without any calculation" ?, I had to calculate the value inside the brackets.

So, question a is correct right? The answer is 0?

As for question b, the question is wrong? there is no way to predict the average rate of change unless you calculate inside the bracketS?

yes, the average rate of change is 0

as for b), I don't say the "question is wrong", all I meant was that you had to do a bit of "calculation" to realize that the brackets were zero.

To verify if question a is correct and to determine the average rate of change of the given function, you can follow these steps:

1. Start by calculating the value of the function at the endpoints of the given interval. In this case, the interval is -2 < x < 2.

When x = -2:
f(-2) = 2(0.5(-2) - 1)^2(-2+2)^2
= 2(-1 - 1)^2(0)^2
= 2(-2)^2(0)
= 2(4)(0)
= 0

When x = 2:
f(2) = 2(0.5(2) - 1)^2(2+2)^2
= 2(1 - 1)^2(4)^2
= 2(0)^2(16)
= 2(0)(16)
= 0

2. Find the difference in the function values at the endpoints:
Δf = f(2) - f(-2)
= 0 - 0
= 0

3. Determine the average rate of change by dividing the difference in function values by the difference in x-values:
Average rate of change = Δf / Δx
= 0 / (2 - (-2))
= 0 / 4
= 0

Therefore, you have correctly calculated the average rate of change of the given function to be 0.

Moving on to question b, "Explain how you could have predicted what the average rate of change would have been without doing any computations."

To predict the average rate of change without performing any calculations, you can analyze the given function and make some observations:

1. Look for any symmetry in the function. If the function is symmetrical with respect to the y-axis, then the average rate of change will be zero. This is because any positive change on one side of the y-axis will be mirrored by an equal negative change on the other side, resulting in a net change of zero.

2. Examine the power of the highest-degree term in the function. In this case, the highest-degree term is (x+2)^2 which has a degree of 2. If the degree of the highest-degree term is even, then the average rate of change will be zero. This is because the function will have the same values on both sides of the highest-degree term, resulting in a net change of zero.

By observing the function f(x) = 2(0.5x - 1)^2(x + 2)^2, we can see that it is symmetrical with respect to the y-axis (due to the square terms) and the highest-degree term has an even degree of 2. Therefore, we can predict that the average rate of change of the function will be zero without performing any computations.