Calculus Examples

$x = 2 t - 5$ , $y = t^{2} + 4$

Step 1

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

$x = 2 t - 5$

Step 2

Rewrite the equation as $2 t - 5 = x$ .

$2 t - 5 = x$

Step 3

Add $5$ to both sides of the equation.

$2 t = x + 5$

Step 4

Divide each term in $2 t = x + 5$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 t = x + 5$ by $2$ .

$\frac{2 t}{2} = \frac{x}{2} + \frac{5}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{x}{2} + \frac{5}{2}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{x}{2} + \frac{5}{2}$

Step 5

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

$y = {(\frac{x}{2} + \frac{5}{2})}^{2} + 4$

Step 6

Simplify ${(\frac{x}{2} + \frac{5}{2})}^{2} + 4$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite ${(\frac{x}{2} + \frac{5}{2})}^{2}$ as $(\frac{x}{2} + \frac{5}{2}) (\frac{x}{2} + \frac{5}{2})$ .

$y = (\frac{x}{2} + \frac{5}{2}) (\frac{x}{2} + \frac{5}{2}) + 4$

Step 6.1.2

Expand $(\frac{x}{2} + \frac{5}{2}) (\frac{x}{2} + \frac{5}{2})$ using the FOIL Method.

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Step 6.1.2.1

Apply the distributive property.

$y = \frac{x}{2} (\frac{x}{2} + \frac{5}{2}) + \frac{5}{2} (\frac{x}{2} + \frac{5}{2}) + 4$

Step 6.1.2.2

Apply the distributive property.

$y = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} (\frac{x}{2} + \frac{5}{2}) + 4$

Step 6.1.2.3

Apply the distributive property.

$y = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3

Simplify and combine like terms.

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Step 6.1.3.1

Simplify each term.

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Step 6.1.3.1.1

Multiply $\frac{x}{2} \cdot \frac{x}{2}$ .

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Step 6.1.3.1.1.1

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

$y = \frac{x \cdot x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.1.2

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

$y = \frac{x^{1} x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.1.3

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

$y = \frac{x^{1} x^{1}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.1.4

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

$y = \frac{x^{1 + 1}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.1.5

Add $1$ and $1$ .

$y = \frac{x^{2}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.1.6

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x}{2} \cdot \frac{5}{2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.2

Multiply $\frac{x}{2} \cdot \frac{5}{2}$ .

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Step 6.1.3.1.2.1

Multiply $\frac{x}{2}$ by $\frac{5}{2}$ .

$y = \frac{x^{2}}{4} + \frac{x \cdot 5}{2 \cdot 2} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.2.2

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{x \cdot 5}{4} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.3

Move $5$ to the left of $x$ .

$y = \frac{x^{2}}{4} + \frac{5 \cdot x}{4} + \frac{5}{2} \cdot \frac{x}{2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.4

Multiply $\frac{5}{2} \cdot \frac{x}{2}$ .

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Step 6.1.3.1.4.1

Multiply $\frac{5}{2}$ by $\frac{x}{2}$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{4} + \frac{5 x}{2 \cdot 2} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.4.2

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{4} + \frac{5 x}{4} + \frac{5}{2} \cdot \frac{5}{2} + 4$

Step 6.1.3.1.5

Multiply $\frac{5}{2} \cdot \frac{5}{2}$ .

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Step 6.1.3.1.5.1

Multiply $\frac{5}{2}$ by $\frac{5}{2}$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{4} + \frac{5 x}{4} + \frac{5 \cdot 5}{2 \cdot 2} + 4$

Step 6.1.3.1.5.2

Multiply $5$ by $5$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{4} + \frac{5 x}{4} + \frac{25}{2 \cdot 2} + 4$

Step 6.1.3.1.5.3

Multiply $2$ by $2$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{4} + \frac{5 x}{4} + \frac{25}{4} + 4$

Step 6.1.3.2

Add $\frac{5 x}{4}$ and $\frac{5 x}{4}$ .

$y = \frac{x^{2}}{4} + 2 \frac{5 x}{4} + \frac{25}{4} + 4$

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Factor $2$ out of $4$ .

$y = \frac{x^{2}}{4} + 2 \frac{5 x}{2 (2)} + \frac{25}{4} + 4$

Step 6.1.4.2

Cancel the common factor.

$y = \frac{x^{2}}{4} + 2 \frac{5 x}{2 \cdot 2} + \frac{25}{4} + 4$

Step 6.1.4.3

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{25}{4} + 4$

Step 6.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{25}{4} + 4 \cdot \frac{4}{4}$

Step 6.3

Combine $4$ and $\frac{4}{4}$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{25}{4} + \frac{4 \cdot 4}{4}$

Step 6.4

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{25 + 4 \cdot 4}{4}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $4$ by $4$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{25 + 16}{4}$

Step 6.5.2

Add $25$ and $16$ .

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{41}{4}$