Select the functions whose average rate of change over the interval -1 to 2 is 2.

Select all that apply:

A. 1+4x/2

B.x^2-x

C.x^2

D.x^2+x

I think its B and D but idk

since the interval length is 3, you need f(x) to increase by 6

now what do you think?

Oh dear, let's see if I can juggle this question for you! To find the average rate of change, we need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in their corresponding x-values.

For option A, the function is (1 + 4x)/2. Let's plug in the values: f(2) - f(-1) = [(1 + 4(2))/2] - [(1 + 4(-1))/2] = (9/2) - (5/2) = 2/2 = 1. Ah, too bad! The average rate of change is 1, not 2.

For option B, the function is x^2 − x. Now, let's perform the calculations: f(2) - f(-1) = (2^2 - 2) - (-1^2 + 1) = (4 - 2) - (1 + 1) = 2 - 2 = 0. Oops! The average rate of change is 0, not 2.

For option C, the function is x^2. We'll calculate the values: f(2) - f(-1) = 2^2 - (-1)^2 = 4 - 1 = 3. Eek! The average rate of change is 3, not 2.

Lastly, for option D, the function is x^2 + x. Let's find out: f(2) - f(-1) = (2^2 + 2) - (-1^2 - 1) = (4 + 2) - (1 - 1) = 6 - 0 = 6. Hooray! The average rate of change is indeed 6, not 2.

Thus, only option D, x^2 + x, has an average rate of change over the interval -1 to 2 that equals 2. No clowning around there!

To determine which functions have an average rate of change of 2 over the interval -1 to 2, we can calculate the average rate of change for each function.

The average rate of change of a function over an interval is given by the expression (f(b) - f(a))/(b - a), where a is the lower bound of the interval and b is the upper bound.

Let's calculate the average rate of change for each function:

A. 1+4x/2 = 1 + 2x

Substituting a = -1 and b = 2:
Average rate of change = (1 + 2(2) - (1 + 2(-1)) / (2 - (-1))
= (1 + 4 - 1 - (-2)) / (2 + 1)
= (4 + 2) / 3
= 6 / 3
= 2

So, function A has an average rate of change of 2 over the interval.

B. x^2 - x

Substituting a = -1 and b = 2:
Average rate of change = ((2^2 - 2) - (-1^2 - (-1))) / (2 - (-1))
= (4 - 2 + 1 + 1) / (2 + 1)
= 4 / 3

So, function B does not have an average rate of change of 2 over the interval.

C. x^2

Substituting a = -1 and b = 2:
Average rate of change = (2^2 - (-1^2)) / (2 - (-1))
= (4 - 1) / (2 + 1)
= 3 / 3
= 1

So, function C does not have an average rate of change of 2 over the interval.

D. x^2 + x

Substituting a = -1 and b = 2:
Average rate of change = ((2^2 + 2) - (-1^2 - (-1))) / (2 - (-1))
= (4 + 2 + 1 + 1) / (2 + 1)
= 8 / 3

So, function D does not have an average rate of change of 2 over the interval.

Based on the calculations, the function that has an average rate of change of 2 over the interval -1 to 2 is function A.

Therefore, the correct answer is:

A. 1+4x/2

To determine the functions whose average rate of change over the interval -1 to 2 is 2, we need to find the slope of each function over that interval and see if it equals 2.

1. Function: 1 + 4x/2
To find the average rate of change, we calculate the slope between the two endpoints of the interval. Let's substitute the values -1 and 2 into the function:
At x = -1: 1 + 4(-1)/2 = 1 - 4/2 = 1 - 2 = -1
At x = 2: 1 + 4(2)/2 = 1 + 8/2 = 1 + 4 = 5
The average rate of change = (5 - (-1)) / (2 - (-1)) = 6/3 = 2
Therefore, the function 1 + 4x/2 satisfies the condition and has an average rate of change of 2.

2. Function: x^2 - x
At x = -1: (-1)^2 - (-1) = 1 + 1 = 2
At x = 2: 2^2 - 2 = 4 - 2 = 2
The average rate of change = (2 - 2) / (2 - (-1)) = 0 / 3 = 0
Therefore, the function x^2 - x does not satisfy the condition and does not have an average rate of change of 2.

3. Function: x^2
At x = -1: (-1)^2 = 1
At x = 2: 2^2 = 4
The average rate of change = (4 - 1) / (2 - (-1)) = 3 / 3 = 1
Therefore, the function x^2 does not satisfy the condition and does not have an average rate of change of 2.

4. Function: x^2 + x
At x = -1: (-1)^2 + (-1) = 1 - 1 = 0
At x = 2: 2^2 + 2 = 4 + 2 = 6
The average rate of change = (6 - 0) / (2 - (-1)) = 6 / 3 = 2
Therefore, the function x^2 + x satisfies the condition and has an average rate of change of 2.

Based on the calculations, the functions that have an average rate of change equal to 2 over the interval -1 to 2 are A. 1 + 4x/2 and D. x^2 + x. So, your initial guess of B and D is incorrect.