Calculus Examples

$x = \sqrt{t}$ , $y = 2 t + 5$

Step 1

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

$x = \sqrt{t}$

Step 2

Rewrite the equation as $\sqrt{t} = x$ .

$\sqrt{t} = x$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{t}}^{2} = x^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

${(t^{\frac{1}{2}})}^{2} = x^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(t^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = x^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$t^{1} = x^{2}$

Step 4.2.1.2

Simplify.

$t = x^{2}$

Step 5

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

$y = 2 (x^{2}) + 5$

Step 6

Multiply $2$ by $x^{2}$ .

$y = 2 x^{2} + 5$