Parametric Equations Worksheet
Explore and analyze parametric equations, converting between parametric and Cartesian forms, and understanding their graphs.
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Parametric Equations Exploration
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Read each question carefully and provide clear, concise answers. Show all your work for full credit.
1. Consider the parametric equations x(t) = 2t - 1 and y(t) = t^2 + 3.
a) Complete the table of values for t, x, and y.
t
x(t)
y(t)
-2
-1
0
1
2
b) Plot the points (x, y) from the table on the coordinate plane below. Indicate the direction of increasing t.
2. Convert the following parametric equations into Cartesian form (y = f(x)).
a) x(t) = t + 3, y(t) = 2t - 1
Cartesian Equation:
b) x(t) = 3cos(t), y(t) = 3sin(t)
Cartesian Equation:
3. Which of the following parametric equations represents a parabola opening upwards?
x(t) = t, y(t) = -t^2
x(t) = t^2, y(t) = t
x(t) = t, y(t) = t^2 + 1
x(t) = sin(t), y(t) = cos(t)
4. A projectile is launched with parametric equations x(t) = 20t and y(t) = 30t - 5t^2, where x and y are in meters and t is in seconds.
a) What is the horizontal distance traveled by the projectile after 3 seconds?
b) What is the height of the projectile after 3 seconds?
c) At what time does the projectile hit the ground (when y = 0)?
5. Determine whether the following statements are True or False.
a) Parametric equations always represent a function y = f(x).
True
False
b) The parameter 't' often represents time in real-world applications.
True
False