Calculus Examples

$x = 4 t + 2$ , $y = t^{2}$

Step 1

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

$x = 4 t + 2$

Step 2

Rewrite the equation as $4 t + 2 = x$ .

$4 t + 2 = x$

Step 3

Subtract $2$ from both sides of the equation.

$4 t = x - 2$

Step 4

Divide each term in $4 t = x - 2$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 t = x - 2$ by $4$ .

$\frac{4 t}{4} = \frac{x}{4} + \frac{- 2}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 t}{4} = \frac{x}{4} + \frac{- 2}{4}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{4} + \frac{- 2}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $- 2$ and $4$ .

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Step 4.3.1.1.1

Factor $2$ out of $- 2$ .

$t = \frac{x}{4} + \frac{2 (- 1)}{4}$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor $2$ out of $4$ .

$t = \frac{x}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 4.3.1.1.2.2

Cancel the common factor.

$t = \frac{x}{4} + \frac{2 \cdot - 1}{2 \cdot 2}$

Step 4.3.1.1.2.3

Rewrite the expression.

$t = \frac{x}{4} + \frac{- 1}{2}$

Step 4.3.1.2

Move the negative in front of the fraction.

$t = \frac{x}{4} - \frac{1}{2}$

Step 5

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

$y = {(\frac{x}{4} - \frac{1}{2})}^{2}$

Step 6

Simplify ${(\frac{x}{4} - \frac{1}{2})}^{2}$ .

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Step 6.1

Rewrite ${(\frac{x}{4} - \frac{1}{2})}^{2}$ as $(\frac{x}{4} - \frac{1}{2}) (\frac{x}{4} - \frac{1}{2})$ .

$y = (\frac{x}{4} - \frac{1}{2}) (\frac{x}{4} - \frac{1}{2})$

Step 6.2

Expand $(\frac{x}{4} - \frac{1}{2}) (\frac{x}{4} - \frac{1}{2})$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

$y = \frac{x}{4} (\frac{x}{4} - \frac{1}{2}) - \frac{1}{2} (\frac{x}{4} - \frac{1}{2})$

Step 6.2.2

Apply the distributive property.

$y = \frac{x}{4} \cdot \frac{x}{4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} (\frac{x}{4} - \frac{1}{2})$

Step 6.2.3

Apply the distributive property.

$y = \frac{x}{4} \cdot \frac{x}{4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $\frac{x}{4} \cdot \frac{x}{4}$ .

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Step 6.3.1.1.1

Multiply $\frac{x}{4}$ by $\frac{x}{4}$ .

$y = \frac{x \cdot x}{4 \cdot 4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.1.2

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

$y = \frac{x^{1} x}{4 \cdot 4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.1.3

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

$y = \frac{x^{1} x^{1}}{4 \cdot 4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.1.4

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

$y = \frac{x^{1 + 1}}{4 \cdot 4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.1.5

Add $1$ and $1$ .

$y = \frac{x^{2}}{4 \cdot 4} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.1.6

Multiply $4$ by $4$ .

$y = \frac{x^{2}}{16} + \frac{x}{4} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.2

Multiply $\frac{x}{4} (- \frac{1}{2})$ .

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Step 6.3.1.2.1

Multiply $\frac{x}{4}$ by $\frac{1}{2}$ .

$y = \frac{x^{2}}{16} - \frac{x}{4 \cdot 2} - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.2.2

Multiply $4$ by $2$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{1}{2} \cdot \frac{x}{4} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.3

Multiply $- \frac{1}{2} \cdot \frac{x}{4}$ .

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Step 6.3.1.3.1

Multiply $\frac{x}{4}$ by $\frac{1}{2}$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{4 \cdot 2} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.3.2

Multiply $4$ by $2$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{8} - \frac{1}{2} (- \frac{1}{2})$

Step 6.3.1.4

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Step 6.3.1.4.1

Multiply $- 1$ by $- 1$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{8} + 1 (\frac{1}{2}) \frac{1}{2}$

Step 6.3.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{8} + \frac{1}{2} \cdot \frac{1}{2}$

Step 6.3.1.4.3

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{8} + \frac{1}{2 \cdot 2}$

Step 6.3.1.4.4

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{16} - \frac{x}{8} - \frac{x}{8} + \frac{1}{4}$

Step 6.3.2

Subtract $\frac{x}{8}$ from $- \frac{x}{8}$ .

$y = \frac{x^{2}}{16} - 2 \frac{x}{8} + \frac{1}{4}$

Step 6.4

Simplify each term.

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Step 6.4.1

Cancel the common factor of $2$ .

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Step 6.4.1.1

Factor $2$ out of $- 2$ .

$y = \frac{x^{2}}{16} + 2 (- 1) \frac{x}{8} + \frac{1}{4}$

Step 6.4.1.2

Factor $2$ out of $8$ .

$y = \frac{x^{2}}{16} + 2 \cdot - 1 \frac{x}{2 \cdot 4} + \frac{1}{4}$

Step 6.4.1.3

Cancel the common factor.

$y = \frac{x^{2}}{16} + 2 \cdot - 1 \frac{x}{2 \cdot 4} + \frac{1}{4}$

Step 6.4.1.4

Rewrite the expression.

$y = \frac{x^{2}}{16} - 1 \frac{x}{4} + \frac{1}{4}$

Step 6.4.2

Rewrite $- 1 \frac{x}{4}$ as $- \frac{x}{4}$ .

$y = \frac{x^{2}}{16} - \frac{x}{4} + \frac{1}{4}$