Understand that for quadratic functions, the change in output values over equal intervals of the domain grows linearly.
Explain why the rate of change of the average rates of change of a quadratic function is constant.
Connect the concavity of a parabola to whether the average rates of change of the quadratic function are increasing (concave up) or decreasing (concave down).
Quick Lesson Plan
Activity: How Fast Does a Penny Fall from the Empire State Building?
We culminate this unit with an exploration of a quadratic function’s rate of change. Students are given the equation for the height of a penny after it is dropped from the Empire State Building. Questions 1-4 review concepts from the first 4 lessons of the unit including evaluating a function, solving a function, finding the average rate of change, and being able to describe how two quantities vary with respect to one another (for every two second interval of time, the height of the penny…)
Note that question 2 is asking about the average rate of change in the penny’s height, not the penny’s average speed.
Questions 5-8 look at the average rates of change over each two second interval and then ask students to notice how they are changing. Students should see that the average rate of change is decreasing by 64 feet per second over each consecutive interval.
What is the penny’s average speed? Is this the same as your answer to question 2?
What was your first clue that the feet dropped over each two second interval would not be constant?
Is the rate of change of the average rates of change positive or negative? How do you know?
How can you tell from a quadratic equation whether its graph will be concave up or concave down?
When debriefing the concavity question, you may wish to remind students that rates of change are decreasing because they are getting more negative, however the penny is speeding up because speed does not take into account the sign of the rate of change. Speed is about the steepness of the graph. You can be going very fast but in a negative direction.
Again, though you can point out the constant second differences in the original table, the real emphasis should be about the constant change in the average rates of change. These are ultimately pointing to the same feature of a quadratic function, but the constant second difference VALUE is not the same as the rate of change of the rate of change value (-128 is the constant second difference of the penny’s heights whereas the penny’s average rate of change is decreasing by 64 feet per second over consecutive 2-second intervals.)
Students sometimes struggle to understand that a function that is sometimes decreasing and sometimes increasing, like a quadratic, could have a constant anything. The slopes are changing from negative to positive! Question 1 of the Check Your Understanding really helps students conceptualize the idea of a constant rate of change in the average rates of change. The average rate of change is always increasing by 2, which can be understood by thinking of the slopes graphed on a number line.