Precalculus Examples

$x = 2 t$ , $y = \frac{1}{2} t^{2} + 1$

Step 1

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

$x = 2 t$

Step 2

Rewrite the equation as $2 t = x$ .

$2 t = x$

Step 3

Divide each term in $2 t = x$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 t = x$ by $2$ .

$\frac{2 t}{2} = \frac{x}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{x}{2}$

Step 3.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{2}$

Step 4

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

$y = \frac{1}{2} \cdot {(\frac{x}{2})}^{2} + 1$

Step 5

Simplify each term.

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Step 5.1

Apply the product rule to $\frac{x}{2}$ .

$y = \frac{1}{2} \cdot \frac{x^{2}}{2^{2}} + 1$

Step 5.2

Combine.

$y = \frac{1 x^{2}}{2 \cdot 2^{2}} + 1$

Step 5.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 5.3.1

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

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Step 5.3.1.1

Raise $2$ to the power of $1$ .

$y = \frac{1 x^{2}}{2^{1} \cdot 2^{2}} + 1$

Step 5.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 x^{2}}{2^{1 + 2}} + 1$

Step 5.3.2

Add $1$ and $2$ .

$y = \frac{1 x^{2}}{2^{3}} + 1$

Step 5.4

Multiply $x^{2}$ by $1$ .

$y = \frac{x^{2}}{2^{3}} + 1$

Step 5.5

Raise $2$ to the power of $3$ .

$y = \frac{x^{2}}{8} + 1$