Parametric Equations
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    This lesson page will inform you about parametric equations. Here are the sections within this lesson page:

    Parametric equations are equations that define the location of points using another variable. Usually, the independent variable is represented as a t-value while the x and y-values are written as functions of t.

    Here is an example of parametric equations that define a set of points in a plane.

parametric equations line

    Notice that the equations have been written in function notation. Recall, this means x(t) is a function called 'x' that is being defined using a t-variable.

    In the next section, you will be shown how to use these equations to get the corresponding graph.

    To gain a graph from parametric equations, choose values for the t-values and then evaluate the x and y-values.

    Here is a table that can be used for this purpose.

table of t-values

    To calculate the x-values, plug in the t-values in the function one at a time. Here is one such calculation.

calculating an x-value

    Likewise, we can do the calculation for the corresponding y-value.

calculating an y-value

    So, here is the first row in the table filled in.

table of t-values

    If we continue this process for all the t-values, we get these results.

table of t-values

    If we wanted to graph this data, we need only look at the second two columns, which are the columns that reflect the x and the y-values.

table of x and y-values

    The graph of these points form a line.

linear graph parametric equations

    ideo: Graphing Parametric Equations
    ctivity: Parametric Equations 1
    uiz: Parametric Equations 1

    If given a set of parametric equations, it may be necessary to collapse them to a single equation without the parameter.

    The algorithm for doing this is easily explained. Here is that strategy.

algorithm converting parametric equations single equation

    Let us examine two examples, one simple and the other not so simple, to see how this algorithm is used.

Example 1: Convert these parametric equations to a single equation.

linear parametric equations

    We should start with the second equation because it is easier to solve that equation for 't,' like so.

solving for t first step parametric equations

    Now, we can use this equation to substitute for the t-value in the first equation.

algorithm second step parametric equations

    The next example will be a little bit more difficult.

Example 2: Convert these parametric equations to a single equation.

parametric equations parabola

    First, we need to solve one of the parametric equations for 't.' Picking the second equation would be the easier of the two equations.

first step algorithm parametric equations

    Now, we can substitute this expression for 't' into the first equation.

second step algorithm parametric equations

    One could leave the equation like so...

parabola equation long as we understand that this is a sideways parabola (opens to the right).

parabola parametric equations

    This instructional video will help you understand how to graph parametric equations.

    ideo: Graphing Parametric Equations

    Use these activities to learn about parametric equations.

    ctivity: Parametric Equations 1
    ctivity: Parametric Equations 2 and 3

    Use this interactive quiz to test your skills.

    uiz: Parametric Equations 1
    uiz: Parametric Equations 2
    uiz: Parametric Equations 3

    Review these related lessons.

    esson: Graphing Lines
    esson: Functions