Calculus Examples Parametric Equations and Polar Coordinates

Eliminate the Parameter

$x = t + 7$ , $y = t^{2}$

Step 1

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

$x = t + 7$

Step 2

Rewrite the equation as $t + 7 = x$ .

$t + 7 = x$

Step 3

Subtract $7$ from both sides of the equation.

$t = x - 7$

Step 4

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

$y = {(x - 7)}^{2}$

Step 5

Simplify ${(x - 7)}^{2}$ .

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Step 5.1

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$y = (x - 7) (x - 7)$

Step 5.2

Expand $(x - 7) (x - 7)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$y = x (x - 7) - 7 (x - 7)$

Step 5.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 7 - 7 (x - 7)$

Step 5.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7$

Step 5.3.1.2

Move $- 7$ to the left of $x$ .

$y = x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7$

Step 5.3.1.3

Multiply $- 7$ by $- 7$ .

$y = x^{2} - 7 x - 7 x + 49$

Step 5.3.2

Subtract $7 x$ from $- 7 x$ .

$y = x^{2} - 14 x + 49$

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Parametric Equations and Polar Coordinates

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Eliminate the Parameter

Parametric Equations and Polar Coordinates

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