Pre-Calculus Parametrics Worksheet #2



Parametrics Review Worksheet Name

Show work on separate paper.

1. Fill in the table and sketch the parametric equation for t [-2,6]

|t |x |y |

|-2 |  |  |

|-1 |  |  |

|0 |  |  |

|1 |  |  |

|2 |  |  |

|3 |  |  |

|4 |  |  |

|5 |  |  |

|6 |  |  |

x =

y = 2 – t

Write equations to rotate this graph 30° CW and sketch the graph.

Problems 2 – 11: Eliminate the parameter to write the parametric equations as a rectangular equation.

2. x = 3. x = 6 – t 4. x = ½t + 4

y = 4t + 5 y = y = t3

5. x = 3 log t 6. x = 4 sin (2t) 7. x = cos t

y = 3 log t2 y = 2 cos (2t) y = 2 sin2t

8. x = 4 et 9. x = 4 + 2 cos t 10. x = -4 + 3tan t

y = 3 e-2t y = -1 + 4 sin 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 =

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 15. Shift left 3 and

stretched by a factor of 4 shift down 5

16. For the parametric equations x = t and y = t2

a) Sketch the graph and state an appropriate window.

b) Graph x = t – 1 and y = t2- 3. How does this compare to the graph in part (a)?

c) 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!)

d) Write parametric equations that will rotate the original graph 60° CCW.

17. For the curve (x + 5)2 + y2 = 4 complete the following:

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 top half of the curve?

d) How could you graph just the left side of the curve?

18. For the curve + = 1 complete the following:

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?

19. For the curve [pic] - [pic] = 1 complete the following:

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

y1 = -4.9t2 y2 = -4.9t2 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. If they don’t intersect, determine how close they get.

a) x1 = 3 – t and x2 = t + 19

y1 = t2 – 60 y2 = t + 12

b) x1 = 3 – t and x2 = 3 – 2t

y1 = 2t + 1 y2 = 2 + 3t

c) x1 = 4t and x2 = 5t - 6

y1 = ½t + 5 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. At 4 PM Andrew programmed a drone to fly above I-75 without stopping at a speed of 4 mph. When Andrew’s mom found Andrew at 4:30pm, she and Andrew hopped in the minivan to retrieve the drone. If Andrew’s mom drove at a speed of 20 mph, when and where will they catch up with the drone?

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) Write a pair of parametric equations to simulate the motion of the dart.

b) Sketch a graph of the motion and give an appropriate window.

c) What is the maximum height reached by the dart?

d) When and where will the dart hit the ground?

27. Kelly and Mary are standing 80 feet apart. At the same time, they each throw a softball toward each other. Mary throws her softball with an initial velocity of 50 ft/sec with an angle of elevation of 45(. Kelly throws her softball with an initial velocity of 45 ft/sec with an angle of elevation of 50°.

a) Write the sets of parametric equations to simulate the motion of the softballs.

b) Will the softballs collide? If not, how close do they get? Justify your answer.

c) Which softball hits the ground first?

d) 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. Amy hits a volleyball from a height of 5’ off the ground. Two seconds later the ball hits the ground, 50’ from Amy. Find a possible initial velocity and angle for Amy’s ball.

30. The Texas State Fair has a Ferris wheel that turns CW (rom the gate) taking riders to a height of 212 feet. The wheel has a diameter of about 203 feet and a maximum speed of 1.5 rotations per minute. A “ride” consists of 6 complete revolutions of the wheel.

a) How high is the center of the wheel?

b) If a person gets on the Ferris wheel at the bottom of the wheel, write parametric equations of his path.

c) What must the t values be to model a complete ride?

d) How high will the person in part b be at 2.5 minutes into the ride?

31. A zookeeper needs to tranquilize a small monkey perched in a tree. He estimates that the monkey is 30’ in front of, but 20’ above him. Suppose he aims exactly at the monkey and the tranquilizing dart leaves the muzzle at 200 ft/sec,

a) Write parametric equations of the dart.

b) If the monkey stays in the tree, how close will the dart come to him?

c) If the monkey hears the shot and immediately falls off the perch when the gun is shot, how close will the dart come to him?

32. 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).

33. 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.

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