Calculus Examples

$x = t^{2} + 3$ , $y = t^{2} - 2$

Step 1

Set up the parametric equation for $x (t)$ to solve the equation for $t$ .

$x = t^{2} + 3$

Step 2

Rewrite the equation as $t^{2} + 3 = x$ .

$t^{2} + 3 = x$

Step 3

Subtract $3$ from both sides of the equation.

$t^{2} = x - 3$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{x - 3}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \sqrt{x - 3}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \sqrt{x - 3}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \sqrt{x - 3}$

$t = - \sqrt{x - 3}$

$t = \sqrt{x - 3}$

$t = - \sqrt{x - 3}$

Step 6

Replace $t$ in the equation for $y$ to get the equation in terms of $x$ .

$y = {(\sqrt{x - 3}, - \sqrt{x - 3})}^{2} - 2$