Calculus Examples

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

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

y (t) = t + 6

$y (t) = t + 6$

Step 1

Set up the parametric equation for

x (t)

$x (t)$ to solve the equation for

t

$t$ .

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

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

Step 2

Rewrite the equation as

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

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

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

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

Step 3

Add

\frac{1}{2}

$\frac{1}{2}$ to both sides of the equation.

t = x + \frac{1}{2}

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

Step 4

Replace

t

$t$ in the equation for

y

$y$ to get the equation in terms of

x

$x$ .

y = (x + \frac{1}{2}) + 6

$y = (x + \frac{1}{2}) + 6$

Step 5

Remove parentheses.

y = x + \frac{1}{2} + 6

$y = x + \frac{1}{2} + 6$

Step 6

Remove parentheses.

y = (x + \frac{1}{2}) + 6

$y = (x + \frac{1}{2}) + 6$

Step 7

Simplify

(x + \frac{1}{2}) + 6

$(x + \frac{1}{2}) + 6$ .

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Step 7.1

To write

6

$6$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

y = x + \frac{1}{2} + 6 \cdot \frac{2}{2}

$y = x + \frac{1}{2} + 6 \cdot \frac{2}{2}$

Step 7.2

Combine

6

$6$ and

\frac{2}{2}

$\frac{2}{2}$ .

y = x + \frac{1}{2} + \frac{6 \cdot 2}{2}

$y = x + \frac{1}{2} + \frac{6 \cdot 2}{2}$

Step 7.3

Combine the numerators over the common denominator.

y = x + \frac{1 + 6 \cdot 2}{2}

$y = x + \frac{1 + 6 \cdot 2}{2}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Multiply

6

$6$ by

2

$2$ .

y = x + \frac{1 + 12}{2}

$y = x + \frac{1 + 12}{2}$

Step 7.4.2

Add

1

$1$ and

12

$12$ .

y = x + \frac{13}{2}

$y = x + \frac{13}{2}$

y = x + \frac{13}{2}

$y = x + \frac{13}{2}$

y = x + \frac{13}{2}

$y = x + \frac{13}{2}$