Two students in your class, Wilson and Alexis, are disputing a function. Wilson says that for the function, between x = –1 and x
= 1, the average rate of change is 0. Alexis says that for the function, between x = –1 and x = 1, the graph goes up through a turning point, and then back down. Explain how Wilson and Alexis can both be correct, using complete sentences. What is the equation of the parabola? (I think it's a parabola)
Wilson can be correct because the functions x=-1 and x=1 are linear functions with no slope (no rate of change). The average y value of these functions is zero. Another function that has an average y value of 0 and no rate of change would be the function x=0. Since all of the y values of this function are 0, there is no rate of change and the average y value is zero itself is zero. Therefore, Wilson is correct. The function that Alexis described is a sinusoidal function. The equation for the function that Alexis described is f(x)=sin(x). The natural sinusoidal function oscillates around the average y=0 with maximums at y=-1 and y=1. A sinusoidal function goes up through a turning point and comes back down, and given this, Alexis is also correct.
