Mike Bireley

When is the average rate from A to B positive? When is it negative?  The average rate is positive when B is both higher and further to the right than A.  The point is negative when A is higher and further to the right than point B.

Does A need to be on a particular side of B in order to have a positive rate of change?  No, A has to be either higher than B if it is on the right, or it has to be lower than B if it is on the left.

Why do the average rate of change and the slope always match when you drag segment AB around?  The average rate of change and the slope always match because the average rate is the measure of a slope between two points.

Why are all the average rate measurements equal to each other? Does moving any points change that?  All of the average rates are equal because the rate of change between all of the points on the line is the same, because the line will infinitely have the same slope.  Moving the points on the line up or down does not change the average rate of any of the intervals.

Are all the average rates still equal to each other after the expression for f(x) is changed?  No, because there is a different rate of change between each of the points.

How can you determine from the function where the average rate of change between two points on that function will be positive, negative, and zero?  The average rate of change between two points is positive when the point to the right is higher than the point on the left.  The average rate of change between two points is negative when the point to the left is higher than the point on the right.  The point is higher when the two points have the same y value.

When you drag A or B around in the plane, when does it have a negative y-value? When is y positive or zero?  The step has a positive value when the slope of AB is positive, the step has a negative value when the slope of AB is negative, and the step has a value of zero when the slope of AB is zero.

When a=0, why does this new function plot of “steps” make a horizontal line?  This is a horizontal line because when a=0, the slope becomes constant, and the graph of the slope becomes the same for all x values, so the graph of the slopes becomes horizontal.

Experiment with the slider for c. What happens to the step function as a result?  When c is raised or lowers, the y intercept of the function is raised or lowered, but the step function stays the same.

Experiment with slider b, what happens to your Step function as a result?  When B is raised or lowered, the slope of the linear function is raised or lowered, and the step function is raised or lowered respectively.

In this function, the steps go up from left to right. Why? How can the function be adjusted to so that they go down?  The steps go up from left to right because the slope is negative to the left of the y axis, and the slopes are positive after the y axis, so they are positive.  If the value for A is made negative, then the parabola will become negative, and the steps will go down from left to right.

How can you determine from the function plot when the rate of change step function will be positive, negative, or zero for the cubic on page 3?  The step function for the cubic graph will look like a parabola, it will be positive until the graph turns negative where the steps will cross the x axis and equal zero at the relative maximum, become negative until the relative minimum where the steps will equal zero, and then the steps will be positive.

Further Exploration: Try this for different functions, can you predict what the trace of the average rate of change “step” will look like, given the plot of the function?  All step functions resemble the derivative of the original function.