Question

which of the following functions have an average rate of change equal to 0 on the interval from x = -1 to x = 1? select all that apply. f(x) = 4 f(x) = -4x + 1 f(x) = 4 - x² f(x) = 4x² - 1 work it out not feeling ready yet? these can help: find the slope from two points characteristics of quadratic functions: equations match quadratic functions and graphs domain and range of quadratic functions: graphs domain and range of quadratic functions: equations write a quadratic function in vertex form from its vertex and a point. identify parts of quadratic expressions: word problems write a quadratic function from its zeros

Explanation:

Step1: Recall Average Rate of Change Formula

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \( \frac{f(b) - f(a)}{b - a} \). Here, \( a=-1 \) and \( b = 1 \), so the formula becomes \( \frac{f(1)-f(-1)}{1-(-1)}=\frac{f(1)-f(-1)}{2} \). We need this to equal 0, so \( f(1)-f(-1)=0 \) or \( f(1)=f(-1) \).

Step2: Analyze \( f(x)=4 \)

For \( f(x)=4 \), \( f(1)=4 \) and \( f(-1)=4 \). Then \( \frac{4 - 4}{2}=0 \). So this function has an average rate of change of 0.

Step3: Analyze \( f(x)=-4x + 1 \)

\( f(1)=-4(1)+1=-3 \), \( f(-1)=-4(-1)+1 = 5 \). Then \( \frac{-3 - 5}{2}=\frac{-8}{2}=-4
eq0 \). So this function does not have an average rate of change of 0.

Step4: Analyze \( f(x)=4 - x^{2} \)

\( f(1)=4-(1)^{2}=3 \), \( f(-1)=4-(-1)^{2}=3 \). Then \( \frac{3 - 3}{2}=0 \). So this function has an average rate of change of 0.

Step5: Analyze \( f(x)=4x^{2}-1 \)

\( f(1)=4(1)^{2}-1 = 3 \), \( f(-1)=4(-1)^{2}-1 = 3 \). Then \( \frac{3 - 3}{2}=0 \). Wait, no, wait: Wait, \( f(1)=4(1)^2 - 1=3 \), \( f(-1)=4(-1)^2 - 1 = 3 \), so \( \frac{3 - 3}{2}=0 \)? Wait, no, earlier mistake? Wait, no, let's recalculate. Wait, \( f(x)=4x^2 - 1 \): \( f(1)=4(1)-1 = 3 \), \( f(-1)=4(1)-1 = 3 \), so \( f(1)-f(-1)=0 \), so average rate of change is 0? Wait, but wait, the original selection in the image had this checked, but let's recheck. Wait, no, wait, \( f(x)=4x^2 - 1 \): \( f(1)=3 \), \( f(-1)=3 \), so \( \frac{3 - 3}{2}=0 \). But wait, earlier for \( f(x)=-4x + 1 \), it's not zero. Wait, but let's confirm again.

Wait, no, for \( f(x)=4x^2 - 1 \), \( f(1)=4(1)^2 -1=3 \), \( f(-1)=4(-1)^2 -1=3 \), so the difference is 0, so average rate of change is 0. But wait, the function \( f(x)=-4x + 1 \): \( f(1)=-4 +1=-3 \), \( f(-1)=4 +1=5 \), difference is -8, average rate of change -4. So the correct functions are \( f(x)=4 \), \( f(x)=4 - x^2 \), and \( f(x)=4x^2 - 1 \)? Wait, no, wait \( f(x)=4x^2 -1 \): let's compute \( f(1) - f(-1) \): \( (4(1)^2 -1)-(4(-1)^2 -1)= (4 -1)-(4 -1)=3 - 3=0 \), so yes. Wait, but the initial check in the image had \( f(x)=-4x +1 \) checked, which is wrong. So the correct functions are \( f(x)=4 \), \( f(x)=4 - x^2 \), and \( f(x)=4x^2 - 1 \)? Wait, no, wait \( f(x)=4x^2 -1 \): is that symmetric about the y-axis? Yes, because it's an even function (since \( f(-x)=f(x) \)). So for even functions, \( f(1)=f(-1) \), so average rate of change on \([-1,1]\) is 0. \( f(x)=4 \) is a constant function, so \( f(1)=f(-1) \). \( f(x)=4 - x^2 \) is also even, so \( f(1)=f(-1) \). \( f(x)=-4x +1 \) is linear with slope -4, so it's not even, and \( f(1)
eq f(-1) \). So the correct functions are \( f(x)=4 \), \( f(x)=4 - x^2 \), and \( f(x)=4x^2 - 1 \)? Wait, but let's recheck \( f(x)=4x^2 -1 \):

\( f(1)=4(1)^2 -1=3 \), \( f(-1)=4(-1)^2 -1=3 \), so \( \frac{3 - 3}{2}=0 \). Correct. \( f(x)=4 - x^2 \): \( f(1)=3 \), \( f(-1)=3 \), so \( \frac{3 - 3}{2}=0 \). Correct. \( f(x)=4 \): \( f(1)=4 \), \( f(-1)=4 \), so \( \frac{4 - 4}{2}=0 \). Correct. \( f(x)=-4x +1 \): \( f(1)=-3 \), \( f(-1)=5 \), \( \frac{-3 - 5}{2}=-4
eq 0 \). So the correct functions are \( f(x)=4 \), \( f(x)=4 - x^2 \), and \( f(x)=4x^2 - 1 \). But in the image, \( f(x)=-4x +1 \) was checked, which is incorrect.

Answer:

The functions with an average rate of change equal to 0 on \([-1, 1]\) are \( \boldsymbol{f(x) = 4} \), \( \boldsymbol{f(x) = 4 - x^2} \), and \( \boldsymbol{f(x) = 4x^2 - 1} \) (note: \( f(x) = -4x + 1 \) is incorrect as its average rate of change is \(-4\)).