Calculus Examples

$x = 4 t - 1$ , $y = 2 t^{2} + 3 t - 4$

Step 1

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

$x = 4 t - 1$

Step 2

Rewrite the equation as $4 t - 1 = x$ .

$4 t - 1 = x$

Step 3

Add $1$ to both sides of the equation.

$4 t = x + 1$

Step 4

Divide each term in $4 t = x + 1$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 t = x + 1$ by $4$ .

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

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.

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

Step 4.2.1.2

Divide $t$ by $1$ .

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

Step 5

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

$y = 2 {(\frac{x}{4} + \frac{1}{4})}^{2} + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6

Simplify $2 {(\frac{x}{4} + \frac{1}{4})}^{2} + 3 (\frac{x}{4} + \frac{1}{4}) - 4$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$y = 2 ((\frac{x}{4} + \frac{1}{4}) (\frac{x}{4} + \frac{1}{4})) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.2

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

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Step 6.1.2.1

Apply the distributive property.

$y = 2 (\frac{x}{4} (\frac{x}{4} + \frac{1}{4}) + \frac{1}{4} (\frac{x}{4} + \frac{1}{4})) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.2.2

Apply the distributive property.

$y = 2 (\frac{x}{4} \cdot \frac{x}{4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} (\frac{x}{4} + \frac{1}{4})) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.2.3

Apply the distributive property.

$y = 2 (\frac{x}{4} \cdot \frac{x}{4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

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

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Step 6.1.3.1.1.1

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

$y = 2 (\frac{x \cdot x}{4 \cdot 4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.1.2

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

$y = 2 (\frac{x^{1} x}{4 \cdot 4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.1.3

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

$y = 2 (\frac{x^{1} x^{1}}{4 \cdot 4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.1.4

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

$y = 2 (\frac{x^{1 + 1}}{4 \cdot 4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.1.5

Add $1$ and $1$ .

$y = 2 (\frac{x^{2}}{4 \cdot 4} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.1.6

Multiply $4$ by $4$ .

$y = 2 (\frac{x^{2}}{16} + \frac{x}{4} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.2

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

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Step 6.1.3.1.2.1

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

$y = 2 (\frac{x^{2}}{16} + \frac{x}{4 \cdot 4} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.2.2

Multiply $4$ by $4$ .

$y = 2 (\frac{x^{2}}{16} + \frac{x}{16} + \frac{1}{4} \cdot \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.3

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

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Step 6.1.3.1.3.1

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

$y = 2 (\frac{x^{2}}{16} + \frac{x}{16} + \frac{x}{4 \cdot 4} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.3.2

Multiply $4$ by $4$ .

$y = 2 (\frac{x^{2}}{16} + \frac{x}{16} + \frac{x}{16} + \frac{1}{4} \cdot \frac{1}{4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.4

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

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Step 6.1.3.1.4.1

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

$y = 2 (\frac{x^{2}}{16} + \frac{x}{16} + \frac{x}{16} + \frac{1}{4 \cdot 4}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.1.4.2

Multiply $4$ by $4$ .

$y = 2 (\frac{x^{2}}{16} + \frac{x}{16} + \frac{x}{16} + \frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.3.2

Add $\frac{x}{16}$ and $\frac{x}{16}$ .

$y = 2 (\frac{x^{2}}{16} + 2 \frac{x}{16} + \frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.4

Cancel the common factor of $2$ .

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Step 6.1.4.1

Factor $2$ out of $16$ .

$y = 2 (\frac{x^{2}}{16} + 2 \frac{x}{2 (8)} + \frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.4.2

Cancel the common factor.

$y = 2 (\frac{x^{2}}{16} + 2 \frac{x}{2 \cdot 8} + \frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.4.3

Rewrite the expression.

$y = 2 (\frac{x^{2}}{16} + \frac{x}{8} + \frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.5

Apply the distributive property.

$y = 2 \frac{x^{2}}{16} + 2 \frac{x}{8} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6

Simplify.

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Step 6.1.6.1

Cancel the common factor of $2$ .

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Step 6.1.6.1.1

Factor $2$ out of $16$ .

$y = 2 \frac{x^{2}}{2 (8)} + 2 \frac{x}{8} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.1.2

Cancel the common factor.

$y = 2 \frac{x^{2}}{2 \cdot 8} + 2 \frac{x}{8} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.1.3

Rewrite the expression.

$y = \frac{x^{2}}{8} + 2 \frac{x}{8} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.2

Cancel the common factor of $2$ .

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Step 6.1.6.2.1

Factor $2$ out of $8$ .

$y = \frac{x^{2}}{8} + 2 \frac{x}{2 (4)} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.2.2

Cancel the common factor.

$y = \frac{x^{2}}{8} + 2 \frac{x}{2 \cdot 4} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.2.3

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{x}{4} + 2 (\frac{1}{16}) + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.3

Cancel the common factor of $2$ .

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Step 6.1.6.3.1

Factor $2$ out of $16$ .

$y = \frac{x^{2}}{8} + \frac{x}{4} + 2 \frac{1}{2 (8)} + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.3.2

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{x}{4} + 2 \frac{1}{2 \cdot 8} + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.6.3.3

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{x}{4} + \frac{1}{8} + 3 (\frac{x}{4} + \frac{1}{4}) - 4$

Step 6.1.7

Apply the distributive property.

$y = \frac{x^{2}}{8} + \frac{x}{4} + \frac{1}{8} + 3 \frac{x}{4} + 3 (\frac{1}{4}) - 4$

Step 6.1.8

Combine $3$ and $\frac{x}{4}$ .

$y = \frac{x^{2}}{8} + \frac{x}{4} + \frac{1}{8} + \frac{3 x}{4} + 3 (\frac{1}{4}) - 4$

Step 6.1.9

Combine $3$ and $\frac{1}{4}$ .

$y = \frac{x^{2}}{8} + \frac{x}{4} + \frac{1}{8} + \frac{3 x}{4} + \frac{3}{4} - 4$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine the numerators over the common denominator.

$y = - 4 + \frac{x^{2}}{8} + \frac{1}{8} + \frac{x + 3 x + 3}{4}$

Step 6.2.2

Add $x$ and $3 x$ .

$y = - 4 + \frac{x^{2}}{8} + \frac{1}{8} + \frac{4 x + 3}{4}$

Step 6.3

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{x^{2}}{8} - 4 \cdot \frac{8}{8} + \frac{1}{8} + \frac{4 x + 3}{4}$

Step 6.4

Combine $- 4$ and $\frac{8}{8}$ .

$y = \frac{x^{2}}{8} + \frac{- 4 \cdot 8}{8} + \frac{1}{8} + \frac{4 x + 3}{4}$

Step 6.5

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{8} + \frac{- 4 \cdot 8 + 1}{8} + \frac{4 x + 3}{4}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $- 4$ by $8$ .

$y = \frac{x^{2}}{8} + \frac{- 32 + 1}{8} + \frac{4 x + 3}{4}$

Step 6.6.2

Add $- 32$ and $1$ .

$y = \frac{x^{2}}{8} + \frac{- 31}{8} + \frac{4 x + 3}{4}$

Step 6.7

Move the negative in front of the fraction.

$y = \frac{x^{2}}{8} - \frac{31}{8} + \frac{4 x + 3}{4}$

Step 6.8

To write $\frac{4 x + 3}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x^{2}}{8} + \frac{4 x + 3}{4} \cdot \frac{2}{2} - \frac{31}{8}$

Step 6.9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.9.1

Multiply $\frac{4 x + 3}{4}$ by $\frac{2}{2}$ .

$y = \frac{x^{2}}{8} + \frac{(4 x + 3) \cdot 2}{4 \cdot 2} - \frac{31}{8}$

Step 6.9.2

Multiply $4$ by $2$ .

$y = \frac{x^{2}}{8} + \frac{(4 x + 3) \cdot 2}{8} - \frac{31}{8}$

Step 6.10

Simplify terms.

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Step 6.10.1

Combine the numerators over the common denominator.

$y = \frac{x^{2} + (4 x + 3) \cdot 2}{8} - \frac{31}{8}$

Step 6.10.2

Combine the numerators over the common denominator.

$y = \frac{x^{2} + (4 x + 3) \cdot 2 - 31}{8}$

Step 6.11

Simplify each term.

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Step 6.11.1

Apply the distributive property.

$y = \frac{x^{2} + 4 x \cdot 2 + 3 \cdot 2 - 31}{8}$

Step 6.11.2

Multiply $2$ by $4$ .

$y = \frac{x^{2} + 8 x + 3 \cdot 2 - 31}{8}$

Step 6.11.3

Multiply $3$ by $2$ .

$y = \frac{x^{2} + 8 x + 6 - 31}{8}$

Step 6.12

Subtract $31$ from $6$ .

$y = \frac{x^{2} + 8 x - 25}{8}$