Understand that for parametric functions, the expression representing the slope of the curve, dy/dx, is generated from expressions for dy/dt and dx/dt.
Find values of the parameter at which the line tangent to a parametric curve is horizontal or vertical.
Quick Lesson Plan
The goal of this lesson is to allow students to explore the usefulness of parametric functions in the context of analyzing the motion of a baseball. Although these functions are usually studied in Pre-calculus, this lesson is approachable even for students who have not encountered this topic previously. The ordered pairs that determine the path of a baseball are first presented without the “hidden” t-values that generate them. When students are graphing the position of the ball, continue to emphasize what each point represents. Make sure students can articulate that a given x-coordinate represents the horizontal distance from the player and the y-coordinate represents the height of the ball at some instant in the ball’s path. This sets the stage for students to realize that they don’t know how long it takes for the ball to move from one position to the next. There is no information given about when the ball is at a particular position. After groups have had this discussion, the teacher will pause the class and provide the t-values that students will add to their table.
In question 2, the students will then work to discover patterns in how the t-value generates the x- and y-coordinate. In questions 3-7, students use their newfound equations to explore the vertical and horizontal velocity of the ball. Some students will be able to make these connections quickly while others may need suggestions (see Monitoring Questions below). Although initially students may refer to the rate of change of x or y as x’(t) or y’(t) respectively, it will be important to emphasize the dx/dt and dy/dt notation in the debrief, which affirms how dy/dx will be determined.
What does this point on your graph represent?
What kind of equation do you expect to have for projectile motion?
What information do you already know about this motion?
How can you use the ball’s highest point (i.e. vertex) to create an equation for the y-values?
What does it mean for a ball to be at rest?
If one variable’s rate of change is zero, does that mean the ball is stopped? Why?
Before discussing the slope of a parametric function, it should be stressed that dx/dt and dy/dt have meaning as individual rates of change, but neither represents the slope of a curve. By constructing the ratio of these rates of change, though, the actual slope, dy/dx, is found, and the algebraic simplification should be obvious. A clear articulation of what dy/dx values result in a horizontal or vertical tangent line and how to find the t-values that cause them will prepare students for upcoming slope problems that may arise when studying vector functions.
It is important for students to be clear when labeling final answers in calculus, but especially when dealing with parametric functions. If a problem were to ask for the rate of change of x(t) at t=2 and the student responds, “x’(t)=___” instead of “x’(2)=_____”, the message being communicated is that x’(t) equals that value for ALL t, not just at t=2. This poor communication should be addressed and remedied consistently.