Parametric equations are equations in which y is a function of x, but both x and y are defined in terms of a third variable. The third variable is the parameter of the equations. Often, the variable t is used in this type of equation.
Here, we will learn about parametric equations with solved exercises. Also, we will look at some practice problems.
Summary of parametric equations
In parametric equations, $latex y$ is defined as a function of $latex x$ by expressing both $latex y$ and $latex x$ in terms of a third variable known as a parameter.
For example, the following equations are parametric equations in which the parameter is $latex t$.
$latex x=t+1~~~[1]$
$latex y=t^2~~~[2]$
These parametric equations define the parabola with the equation:
$latex y=x^2-2x+1$
We can obtain this by eliminating the parameter $latex t$ in equations [1] and [2]. If we solve equation [1] for $latex t$, we have $latex t=x-1$. Substituting this into equation [2], we have:
$latex y=(x-1)^2$
$latex y=x^2-2x+1$
Parametric equations – Examples with answers
EXAMPLE 1
What is the Cartesian equation of the following parametric equations?
$latex x=\sqrt{t}$
$latex y=3t^2-4$
Solution
A Cartesian equation is an equation of y expressed in terms of x. Therefore, we have to eliminate the parameter $latex t$.
We start by solving the first equation for $latex t$. For this, we square both sides:
$latex x=\sqrt{t}$
$latex x^2=t$
Substituting this expression into the second equation, we have:
$latex y=3t^2-4$
$latex y=3(x^2)^2-4$
$latex y=3x^4-4$
EXAMPLE 2
Find the Cartesian equation for the following parametric equations:
$latex x=2t-1$
$latex y=12t^2-5$
Solution
To get a Cartesian equation, we have to get an equation of y in terms of x.
Then, we start by solving the first equation for $latex t$:
$latex x=2t-1$
$$t=\frac{x+1}{2}$$
Now, we substitute this expression into the second equation:
$latex y=12t^2-5$
$$y=12\left(\frac{x+1}{2}\right)^2-5$$
$$y=12\left(\frac{x^2+2x+1}{4}\right)-5$$
$$y=3x^2+6x+3-5$$
$latex y=3x^2+6x-2$
EXAMPLE 3
Express y as an equation of x using the following parametric equations:
$latex x=2\sqrt{t}$
$latex y=8t^2+5$
Solution
To express $latex y$ as an equation of $latex x$, we have to start by solving the first equation for $latex t$:
$latex x=2\sqrt{t}$
$$\sqrt{t}=\frac{x}{2}$$
$latex t=\frac{x^2}{4}$
Now, we use this expression in the second equation:
$latex y=8t^2+5$
$$y=8\left(\frac{x^2}{4}\right)^2+5$$
$$y=8\left(\frac{x^4}{16}\right)+5$$
$$y=\frac{x^4}{2}+5$$
EXAMPLE 4
Find the Cartesian equation for the following parametric equations:
$$x=\frac{1}{t}$$
$latex y=3t-2$
Solution
Solving the first equation for $latex t$, we have:
$$x=\frac{1}{t}$$
$$t=\frac{1}{x}$$
Substituting this expression in the second equation, we have:
$latex y=3t-2$
$$y=3\left(\frac{1}{x}\right)-2$$
$$y=\frac{3}{x}-2$$
We can multiply the whole equation by $latex x$ to get $latex xy+2x=3$.
EXAMPLE 5
Find an equation for y in terms of x using the equations below:
$$x=\frac{2}{\sqrt{x}}$$
$$y=\frac{3}{1+3}$$
Solution
We start by finding an expression for $latex t$ in terms of $latex x$:
$$x=\frac{2}{\sqrt{t}}$$
$$x^2=\frac{4}{t}$$
$$t=\frac{4}{x^2}$$
Now, we use this expression in the second equation:
$$y=\frac{3}{1+t}$$
$$y=\frac{3}{1+\frac{4}{x^2}}$$
To simplify, we can multiply both the numerator and the denominator by $latex x^2$:
$$y=\frac{3x^2}{x^2+4}$$
EXAMPLE 6
Find a Cartesian equation using the following parametric equations:
$$x=\frac{1}{2-t}$$
$$y=\frac{3}{1+2t}$$
Solution
The $latex x$ equation can be solved for $latex t$ as follows:
$$x=\frac{1}{2-t}$$
$latex x(x-t)=1$
$latex 2x-xt=1$
$latex xt=2x-1$
$$t=\frac{2x-1}{x}$$
Substituting this expression into the $latex y$ equation, we have:
$$y=\frac{3}{1+2t}$$
$$y=\frac{3}{1+2\left(\frac{2x-1}{x}\right)}$$
$$y=\frac{3x}{x+2(2x-1)}$$
$$y=\frac{3x}{5x-2}$$
Parametric equations – Practice problems
The Cartesian equation of the equations $latex x=\frac{t}{1-3t}$, $latex y=\frac{t}{1+2t}$ is written as a fraction. What is the denominator?
Write the denominator in the input box.
See also
Interested in learning more about parametric equations and calculus? You can look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
