Parametric Equations of Lines

by

Mary Eager

Objectives : Students will write parametric equations of lines.
Students will graph parametric equations of lines.

Materials:
Textbook, Merril Advanced Mathematical Concept , by Gordon, et. al., Glencoe Publishing, 1994.
Graphing calculator, or graphing utility, with simultaneous, parametric graphing capabilities

Lesson Description : Building on the information presented by students, Jeremy Grizzle and Jeff Roberts, and the definition of parametric equations provided by Jess Bloomingdale, LCHS senior, the teacher provides closure for the process of writing parametric equations for lines. Problems are given such as, "Find the parametric equations for a line parallel to vector (-2,5) and passing through (3,-5). Then make a table of values and graph the line". From previous study of vectors, students realize the vector (-2,5) is equivalent to having a slope of (-5/2). Jess' definition for parametric equations states two equations are needed...one for x and one for y . Because of the graphing exploration from the two days prior to this lesson, students readily accept the equations necessary for the line are

x(t)=3+(-2)t and y(t)=-5+5t

. The question arises, "What's t ?" The table of values helps students see that t is any number. More problems are discussed and assigned. Given information is changed in order to vary the practice and solidify the concepts. For example, the problem reads, "Write the parametric equations for the line y=2x-3 " or, "Find the parametric equations for the line through (-4,-3) and (5,2)". Students also reverse the process and write an equation in slope-intercept form when given the line's parametric equations.

Graphing Calculator Activity: Using the parametric mode on the TI-82 and graphing two sets of parametric equations simulanteously, students simulate the linear relationship of a car race. This proves to be quite entertaining. The race is 500 miles. One car averages 105 mph and the other averages 120 mph but is delayed 30 minutes at the start of the race due to electrical problems. Which of the two cars will finish before the other, assuming they both finish the race?

First, students write a set of parametric equations to represent each car's position after t hours. Car 1 is on track one so its equations are

x=105t and y=1.

Car 2, on track two, is represented by

x=120(t-.5) and y=2.

Discussion ensues for establishing the values of t-min and t-max, x-min, x-max, y-min and y-max. What does t represent in this activity? How much time is enough for this simulation? What does x represent? Having answered these questions, students enter the values t-min=0, t-max=5, x-min=0, x-max=500, y-min=0, y-max=5....and its off to the races....graphically. The results of the visualization of the race can be confirmed by finding the time that each car finished the race. John Woody, a junior, says, "I want to do this algebraically", and so he did. By comparing the paths of each car, students could tell when car 2 passed car 1...and then find that time exactly using algebra. Excitement ran high for this activity....Mathematics can be fun!

Assessment: In order to evaluate how well students met the objectives of this lesson the following problem is assigned.

"Two semi-trucks are driving loads for Chicago to Denver, a distance of 1125 miles. The first truck leaves at 8:00am and averages 50 mph. The second truck leaves at 9:00am. Since it has a lighter load, the second truck averages 54 mph. Set up two sets of parametric equations to model this situation and use a graphing calculator to analyze the model.

a. How long is it until the second truck overtakes the first truck?

b. How far are they from Chicago when the second truck overtakes the first truck?

c. If each of the drivers stops for meals for a total of 3 hours, what time will it be in Chicago when each truck reaches Denver?

d. How much would the driver of the first truck have to increase her speed in order to arrive in Denver first?" p447, Merril Advanced Mathematical Concepts