What is Parametric Equations?

Understanding parametric equations and how to convert between parametric and rectangular forms

How can I study Parametric Equations effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Parametric Equations study guide free?

Yes — all study notes, flashcards, and practice problems for Parametric Equations on Study Mondo are free to access. No account is needed.

What course covers Parametric Equations?

Parametric Equations is part of the AP Precalculus course on Study Mondo, specifically in the Functions Involving Parameters, Vectors, and Matrices section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Parametric Equations?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Parametric Equations

What Are Parametric Equations?

$t$	$x = 2t + 1$	$y = 3t - 2$	Point $(x, y)$
$0$	$1$	$-2$	$(1, -2)$

❓ Question:

Eliminate the parameter to find the rectangular equation: $x = t + 3$ , $y = t^2 - 1$

Solution:

Step 1: Solve one equation for $t$ .

From $x = t + 3$ : $t = x - 3$

❓ Question:

Consider the parametric equations: $x = 2t - 1$ and $y = t^2 + 3$

❓ Question:

Find parametric equations for a circle with center $(2, -3)$ and radius $5$ .

Solution:

Step 1: Recall the parametric form for a circle centered at origin.

For a circle centered at with radius :

❓ Question:

A projectile is launched with parametric equations: $x = 40t$ $y = -16t^2 + 60t + 5$

❓ Question:

Given $x = 2t$ and $y = t^2 + 1$ , find at .

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

Solution:

Part (a): From the first equation: $x = 2t - 1$

Solve for $t$ : $t = \frac{x + 1}{2}$

Substitute into the second equation:

$y = t^2 + 3 = \left(\frac{x + 1}{2}\right)^2 + 3$

$y = \frac{(x + 1)^2}{4} + 3$

$y = \frac{x^2 + 2x + 1}{4} + 3$

$y = \frac{x^2 + 2x + 1 + 12}{4} = \frac{x^2 + 2x + 13}{4}$

Or: $4y = x^2 + 2x + 13$

Part (b): When $t = 2$ :

$x = 2(2) - 1 = 3$ $y = 2^2 + 3 = 7$

Point: $(3, 7)$

Part (c): As $t$ increases from negative to positive:

When $t = -1$ : $x = -3$ , $y = 4$
When $t = 0$ : ,

The curve is a parabola opening upward, and motion is from left to right as $t$ increases.

Solution:

Part (a): Maximum height occurs at the vertex of the parabola $y = -16t^2 + 60t + 5$ .

For $y = at^2 + bt + c$ , vertex is at $t = -\frac{b}{2a}$

$t = -\frac{60}{2(-16)} = \frac{60}{32} = 1.875$ seconds

Maximum height:

$y = -16(1.875)^2 + 60(1.875) + 5$ $y = -16(3.516) + 112.5 + 5$ feet

Part (b): The projectile hits the ground when $y = 0$ :

$-16t^2 + 60t + 5 = 0$

Using the quadratic formula: $t = \frac{-60 \pm \sqrt{60^2 - 4(-16)(5)}}{2(-16)}$

$t = \frac{-60 \pm \sqrt{3600 + 320}}{-32} = \frac{-60 \pm \sqrt{3920}}{-32}$

$t = \frac{-60 \pm 62.61}{-32}$

$t = \frac{-60 + 62.61}{-32} = -0.082$ (reject, negative time)

$t = \frac{-60 - 62.61}{-32} = \frac{-122.61}{-32} = 3.832$ seconds

Horizontal distance: $x = 40(3.832) = 153.3$ feet

The projectile hits the ground after 3.83 seconds at a distance of 153.3 feet.

Solution:

Step 1: Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$ .

$\frac{dx}{dt} = 2$

$\frac{dy}{dt} = 2t$

Step 2: Use the formula for the derivative.

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2t}{2} = t$

Step 3: Evaluate at $t = 3$ .

$\frac{dy}{dx}\bigg|_{t=3} = 3$

Answer: $\frac{dy}{dx} = 3$ at $t = 3$

This means the slope of the tangent line at the point where $t = 3$ is $3$ .

Parametric Equations

Try the Interactive Version!

Parametric Equations

What Are Parametric Equations?

Why Use Parametric Equations?

Example: Linear Motion

Converting to Rectangular Form

Example