Precalculus Examples

$x (t) = 2 - t$ , $y (t) = 2 t^{2} - t^{3}$

Step 1

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

$x = 2 - t$

Step 2

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

$2 - t = x$

Step 3

Subtract $2$ from both sides of the equation.

$- t = x - 2$

Step 4

Divide each term in $- t = x - 2$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- t = x - 2$ by $- 1$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Divide $t$ by $1$ .

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

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

Move the negative one from the denominator of $\frac{x}{- 1}$ .

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

Step 4.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

Step 4.3.1.3

Divide $- 2$ by $- 1$ .

$t = - x + 2$

Step 5

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

$y = 2 {(- x + 2)}^{2} - {(- x + 2)}^{3}$

Step 6

Simplify $2 {(- x + 2)}^{2} - {(- x + 2)}^{3}$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$y = 2 ((- x + 2) (- x + 2)) - {(- x + 2)}^{3}$

Step 6.1.2

Expand $(- x + 2) (- x + 2)$ using the FOIL Method.

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Step 6.1.2.1

Apply the distributive property.

$y = 2 (- x (- x + 2) + 2 (- x + 2)) - {(- x + 2)}^{3}$

Step 6.1.2.2

Apply the distributive property.

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

Step 6.1.2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 6.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.3.1.2.1

Move $x$ .

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

Step 6.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 6.1.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 6.1.3.1.4

Multiply $x^{2}$ by $1$ .

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

Step 6.1.3.1.5

Multiply $2$ by $- 1$ .

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

Step 6.1.3.1.6

Multiply $- 1$ by $2$ .

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

Step 6.1.3.1.7

Multiply $2$ by $2$ .

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

Step 6.1.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 6.1.4

Apply the distributive property.

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

Step 6.1.5

Simplify.

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Step 6.1.5.1

Multiply $- 4$ by $2$ .

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

Step 6.1.5.2

Multiply $2$ by $4$ .

$y = 2 x^{2} - 8 x + 8 - {(- x + 2)}^{3}$

Step 6.1.6

Use the Binomial Theorem.

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

Step 6.1.7

Simplify each term.

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Step 6.1.7.1

Apply the product rule to $- x$ .

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

Step 6.1.7.2

Raise $- 1$ to the power of $3$ .

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

Step 6.1.7.3

Apply the product rule to $- x$ .

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

Step 6.1.7.4

Raise $- 1$ to the power of $2$ .

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

Step 6.1.7.5

Multiply $x^{2}$ by $1$ .

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

Step 6.1.7.6

Multiply $2$ by $3$ .

$y = 2 x^{2} - 8 x + 8 - (- x^{3} + 6 x^{2} + 3 (- x) \cdot 2^{2} + 2^{3})$

Step 6.1.7.7

Multiply $- 1$ by $3$ .

$y = 2 x^{2} - 8 x + 8 - (- x^{3} + 6 x^{2} - 3 x \cdot 2^{2} + 2^{3})$

Step 6.1.7.8

Raise $2$ to the power of $2$ .

$y = 2 x^{2} - 8 x + 8 - (- x^{3} + 6 x^{2} - 3 x \cdot 4 + 2^{3})$

Step 6.1.7.9

Multiply $4$ by $- 3$ .

$y = 2 x^{2} - 8 x + 8 - (- x^{3} + 6 x^{2} - 12 x + 2^{3})$

Step 6.1.7.10

Raise $2$ to the power of $3$ .

$y = 2 x^{2} - 8 x + 8 - (- x^{3} + 6 x^{2} - 12 x + 8)$

Step 6.1.8

Apply the distributive property.

$y = 2 x^{2} - 8 x + 8 - - x^{3} - (6 x^{2}) - (- 12 x) - 1 \cdot 8$

Step 6.1.9

Simplify.

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Step 6.1.9.1

Multiply $- - x^{3}$ .

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Step 6.1.9.1.1

Multiply $- 1$ by $- 1$ .

$y = 2 x^{2} - 8 x + 8 + 1 x^{3} - (6 x^{2}) - (- 12 x) - 1 \cdot 8$

Step 6.1.9.1.2

Multiply $x^{3}$ by $1$ .

$y = 2 x^{2} - 8 x + 8 + x^{3} - (6 x^{2}) - (- 12 x) - 1 \cdot 8$

Step 6.1.9.2

Multiply $6$ by $- 1$ .

$y = 2 x^{2} - 8 x + 8 + x^{3} - 6 x^{2} - (- 12 x) - 1 \cdot 8$

Step 6.1.9.3

Multiply $- 12$ by $- 1$ .

$y = 2 x^{2} - 8 x + 8 + x^{3} - 6 x^{2} + 12 x - 1 \cdot 8$

Step 6.1.9.4

Multiply $- 1$ by $8$ .

$y = 2 x^{2} - 8 x + 8 + x^{3} - 6 x^{2} + 12 x - 8$

Step 6.2

Simplify by adding terms.

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Step 6.2.1

Combine the opposite terms in $2 x^{2} - 8 x + 8 + x^{3} - 6 x^{2} + 12 x - 8$ .

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Step 6.2.1.1

Subtract $8$ from $8$ .

$y = 2 x^{2} - 8 x + x^{3} - 6 x^{2} + 12 x + 0$

Step 6.2.1.2

Add $2 x^{2} - 8 x + x^{3} - 6 x^{2} + 12 x$ and $0$ .

$y = 2 x^{2} - 8 x + x^{3} - 6 x^{2} + 12 x$

Step 6.2.2

Subtract $6 x^{2}$ from $2 x^{2}$ .

$y = - 4 x^{2} - 8 x + x^{3} + 12 x$

Step 6.2.3

Add $- 8 x$ and $12 x$ .

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

Step 6.2.4

Reorder $- 4 x^{2}$ and $x^{3}$ .

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