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Exercises Trigonometry

Example 4 from Algebra and Trigonometry, 10.6 Parametric Equations

Eliminating the Parameter in Polynomials

Given $x (t) = t^{2} + 1$ and $y (t) = 2 + t,$ eliminate the parameter, and write the parametric equations as a Cartesian equation.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We will begin with the equation for $y$ because the linear equation is easier to solve for $t .$
$\begin{array}{l} y = 2 + t \\ y - 2 = t \end{array}$
Next, substitute $y - 2$ for $t$ in $x (t) .$
$\begin{array}{l} x = t^{2} + 1 \\ x = {(y - 2)}^{2} + 1 & Substitute the expression for t into x . \\ x = y^{2} - 4 y + 4 + 1 \\ x = y^{2} - 4 y + 5 \\ x = y^{2} - 4 y + 5 \end{array}$
The Cartesian form is $x = y^{2} - 4 y + 5.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the equations below, eliminate the parameter and write as a rectangular equation for $y$ as a function of $x .$

\begin{array}{l} x (t) = 2 t^{2} + 6 \\ y (t) = 5 - t \end{array}

How did it go?

PreviousFinding Parametric Equations That Model Given Criteria Next Eliminating the Parameter in Exponential Equations

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.