Parametric Equations: Graphing and Rectangular Conversion

A car is moving along a path described by the parametric equations x(t) = 3t and y(t) = 2t^2. Graph the path of the car for t in the interval [0, 3]. What shape does the graph represent?

• A straight line

• A circle

• The graph represents a parabola.

• An ellipse

A projectile is launched with the parametric equations x(t) = 5t and y(t) = -4.9t^2 + 20t + 1. Convert these equations to rectangular form and identify the vertex of the parabola.

• Vertex: (15, 25)

• Vertex: (0, 1)

• Vertex: (10.2041, 21.2041)

• Vertex: (5, 10)

A cyclist travels along a circular path described by the parametric equations x(t) = 4cos(t) and y(t) = 4sin(t). For t in the interval [0, 2π], graph the path of the cyclist. What is the radius of the circle?

• 6

• 2

• 8

• 4

A drone follows a path given by the parametric equations x(t) = 2t + 1 and y(t) = 3t^2 - 4. Convert these equations to rectangular form and determine the type of curve represented.

• The curve represented is an ellipse.

• The curve represented is a circle.

• The curve represented is a straight line.

• The curve represented is a parabola.

A ball is thrown with a velocity described by the parametric equations x(t) = 10t and y(t) = -5t^2 + 15t. Graph the trajectory of the ball for t in the interval [0, 3]. What is the maximum height reached?

• 11.25

• 12.0

• 9.75

• 10.5

A roller coaster's height is modeled by the parametric equations x(t) = t and y(t) = -0.5t^2 + 5t. Convert these equations to rectangular form and find the height at t = 4 seconds.

• 10

• 12

• 15

• 8

A boat moves in a straight line described by the parametric equations x(t) = 6t and y(t) = 2t + 3. Graph the path of the boat for t in the interval [0, 5]. What is the slope of the line?

• 1/2

• 2/3

• 3/4

• 1/3

A satellite orbits the Earth following the parametric equations x(t) = 10cos(t) and y(t) = 10sin(t). For t in the interval [0, 2π], graph the orbit. What is the distance from the center of the Earth to the satellite?

• 5

• 15

• 20

• 10

A runner's path is described by the parametric equations x(t) = 4t and y(t) = 2t^2 - 8t + 10. Convert these equations to rectangular form and find the coordinates of the vertex of the parabola.

• (8, -4)

• (4, -6)

• (10, 0)

• (6, -2)

A fish swims in a pond following the path given by the parametric equations x(t) = 3t^2 and y(t) = 2t + 1. Graph the path for t in the interval [0, 2]. What type of curve does the fish's path represent?

• The fish's path represents a parabola.

• A circle

• An ellipse

• A straight line