Parametric Equations — Practice
Let a third variable drive both coordinates, and curves that were impossible become easy.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Radius — Centre to edge. Half the diameter.
- Slope — How much y changes for each 1 that x increases.
- Parabola — The U-shaped graph of a quadratic function.
- Roots — The x values where y = 0.
- Cosine — cos θ = adjacenthypotenuse.
- Pythagorean identity — sin2θ + cos2θ = 1.
- Continuous — A function you could draw without lifting the pen — no holes, jumps or blow-ups.
Problem 1
For x=t+1, y=2t, find x when t=3.
Worked answer
4
3+1 = 4
Problem 2
For the same equations, find y when t=3.
Worked answer
6
2(3) = 6
Problem 3
Eliminate t from x=t, y=t2. Give y in terms of x.
Worked answer
x^2
t = x, so y = x2
Problem 4
Eliminate t from x=t-2, y=t. Give y in terms of x.
Worked answer
x+2
t = x+2, so y = x+2
Problem 5
Why can a circle not be written as y=f(x)?
- It fails the vertical line test
- It has no equation
- It is not continuous
- It has no slope
Worked answer
It fails the vertical line test
Two, for most x on a circle
Problem 6
Eliminate t from x=2t, y=t2+1. Give y in terms of x.
Worked answer
x^2/4+1
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Problem 7
What curve is x=5cos t, y=5sin t?
- A circle of radius 5
- An ellipse
- A line
- A parabola
Worked answer
A circle of radius 5
x2+y2 = 25(cos2 t + sin2 t) = 25
Problem 8
For x=6cos t, y=6sin t, find x2+y2.
Worked answer
36
36(cos2 t+sin2 t) = 36
Problem 9
For x=3cos t, y=3sin t, find x when t=0.
Worked answer
3
3(1) = 3
Problem 10
A ball has y = 30t - 16t2. At what t does it return to the ground? Give the positive value.
Worked answer
1.875
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Problem 11
With x = 20t, how far has that ball travelled horizontally at t=1.5?
Worked answer
30
20(1.5) = 30
Problem 12
Eliminate t from x=t2, y=t4. Give y in terms of x.
Worked answer
x^2
y = (t2)2 = x2
Note x ≥ 0, since it is a square
Problem 13
For x=4cos t, y=3sin t, find the largest value x can take.
Worked answer
4
4(1) = 4
Problem 14
What curve is x=4cos t, y=3sin t?
- An ellipse
- A circle
- A hyperbola
- A line
Worked answer
An ellipse
x216+y29 = cos2 t + sin2 t = 1
Problem 15
For x=t2 with t any real number, what is the smallest value x can take?
Worked answer
0
t2 ≥ 0
Problem 16
A ball has y = 48t - 16t2. Find the time at which it reaches its highest point.
Worked answer
1.5
Roots at t=0 and t=3
Halfway is 1.5