Pre-Calculus Parametrics Worksheet #2
Show work on separate paper.
Name
1. Fill in the table and sketch the parametric equation for t [-2,6]
x=
t2 + 1
t
x
y
-2
y=2–t
-1
0
1
2
3
4
5
6
Problems 2 – 11: Eliminate the parameter to write the parametric equations as a
rectangular equation.
1
2. x =
3. x = 6 – t
4. x = ½t + 4
t-2
y = 4t + 5
y=
y = t3
3t - 4
5. x = 3 cos t
y = 3 sin t
6. x = 4 sin (2t)
y = 2 cos (2t)
7. x = cos t
y = 2 sin2t
8. x = 4 sec t
y = 3 tan t
9. x = 4 + 2 cos t
y = -1 + 4 sin t
10. x = -4 + 3tan t
y = 7 – 2 sec t
Problems 11 and 12: Write two new sets of parametric equations for the following
rectangular equations.
11. y = (x + 2)3 – 4
12. x =
y2 - 3
Problems 13 – 15: Write a new set of parametric equations with the following
transformations for x = t4 – 3 and y = 2t
13. Shift right 7
14. Shift up 7 and horizonatally
stretched by a factor of 4
15. Shift left 3 and
shift down 5
16. For the parametric equations x = t and y = t2
a)
b)
c)
d)
Sketch the graph and state an appropriate window.
Graph x = t – 1 and y = t2. How does this compare to the graph in part (a)?
Graph x = t and y = t2 – 3. How does this compare to the graph in part (a)?
Write parametric equations which will give the graph in part (a) a vertical stretch by
a factor of 2 and move the graph 5 units to the right. (Hint: Verify on your
calculator!)
17. For the parametric equations x = t and y = t
a) Sketch the graph and state an appropriate window.
b) Graph x = t + 2 and y = t - 1. How does this compare to the graph in part (a)?
c) Write a pair of parametric equations which will move the graph in part (a) 4 units to
the left and three units down.
d) Describe the how the numbers 1, ½, and 3 in the following parametric equations
transform the graph from part (a).
x = ½(t + 1) and y = = t + 3
18. For the curve (x + 5)2 + y2 = 4 complete the following:
a)
b)
c)
d)
Write a pair of parametric equations for the curve.
Give an appropriate window and sketch the graph.
How could you graph just the top half of the curve?
How could you graph just the left side of the curve?
(x + 3)2 (y - 4)2
19. For the curve
+
= 1 complete the following:
1
16
a) Write a pair of parametric equations for the curve.
b) Give an appropriate window and sketch the graph.
c) How could you graph just the right side of the curve?
d) How could you graph just the bottom half of the curve?
20. Graph the parametric equations using the window provided.
x1 = 700t
and x2 = 300
T [0, 0.6]
Tstep = 0.005
2
2
y1 = -4.9t
y2 = -4.9t
X [0, 310] X scl = 100
Y [-2, 2]
Yscl = 0.5
a) Do the paths intersect or just cross? If the paths intersect, where and when?
b) Change x1 to 900t and then to 500t. What do you observe?
21. Do the following sets of parametric equations cross or intersect? Justify your
answer.
a) x1 = 3 – t and x2 = t + 19
y1 = t2 – 60
y2 = t + 12
b) x1 = 3 – t and
y1 = 2t + 1
x2 = 3 – 2t
y2 = 2 + 3t
c) x1 = 4t
and
y1 = ½t + 5
x2 = 5t - 6
y2 = t + 2
22. Horizontal Motion of a Particle.
a) Graph the parametric equations with the provided window:
x = 4t3 – 16t2 + 15t
T [0, 5] step 0.05
X [-4, 6] scl 1 Y [-3, 6] scl 1
y=2
b) When and where does the particle reverse direction?
c) Graph the parametric equations simultaneously with the graph from part (a)
x = 4t3 – 16t2 + 15t
y=t
d) What do you observe from part (c)?
23. Two tankers leave Corpus Cristi at the same time traveling toward St. Petersburg,
which is 900 miles east of Corpus Cristi. Tanker A travels at 18mph and Tanker B
travels at 22mph.
a) Write parametric equations for the situation.
b) Graph and identify appropriate windows.
c) How long does it take the faster tanker to reach St. Petersburg?
d) Where is the slower tanker when the faster tanker reaches its destination?
e) When, during the trip, is the faster tanker exactly 82 miles in front of the slower
tanker?
f) During what part of the trip are the tankers less than 60 miles apart?
Problems 24 – 28: Use parametric equations to solve each situation.
24. Jack and Mark live 4 miles apart from each other on Baseline Road. At 6pm, their
moms agree to let them both leave and ride their bikes toward one another. If Jack
can pedal 12 mph and Mark can pedal 8 mph, where and when will the boys meet each
other?
25. Andrew’s dog Toby ran out of the house at 4pm and continued to run along Guadalupe
Road without stopping at a speed of 4 mph. When Andrew’s mom returned home from
the grocery store at 4:30pm, she and Andrew hopped in the minivan to retrieve Toby.
If Andrew’s mom drove at a speed of 20 mph, when and where will they find Toby?
26. A dart is thrown from a point of 5 feet above the ground with an initial velocity of 58
ft/sec and angle of elevation of 41. Assume the only force acting on the dart is
gravity.
a)
b)
c)
d)
Write a pair of parametric equations to simulate the motion of the dart.
Sketch a graph of the motion and give an appropriate window.
What is the maximum height reached by the dart?
When and where will the dart hit the ground?
27. Kelly and Mary are standing 78 feet apart. At the same time, they each throw a
softball toward each other. Mary throws her softball with an initial velocity of 45
ft/sec with an angle of elevation of 44. Kelly throws her softball with an initial
velocity of 41 ft/sec with an angle of elevation of 39°.
a)
b)
c)
d)
Write the sets of parametric equations to simulate the motion of the softballs.
Will the softballs collide? Justify your answer.
Which softball hits the ground first?
How far does each softball travel in the horizontal direction?
28. A plane leaves point A at a bearing of N 40 E traveling 600 mph. At the same time, a
second plane leaves point B, 1000 miles east of point A, at a bearing of N 52 W
traveling 550 mph.
a) Write a set of parametric equations for the situation.
b) Is there any danger of a collision? Justify your answer.
Suppose a wind is blowing at a bearing of S 10 W at 20 mph.
c) Write new parametric equations for the situation.
d) Will the planes collide with the wind affecting their paths?
29. Write your own real life application problem in which you would use parametric
equations to solve it. (Hint: look at the last two pages of questions!!!)
a) Write a description of the situation, giving any needed information to create the set
of equations.
b) Write at least 3 questions related to your description.
c) State your assumptions and write an explanation and answer for each of your
questions in part (b).
30. Find the values of t that generated the graph described by the parametric equations:
x = t - 1 and y = ½t + 2
t
x
y
-5
0
-3
1
-1
2
1
3
3
4
Describe your thought process in solving for t.