Calculus Examples

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

Step 1

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

$x = t + 1$

Step 2

Rewrite the equation as $t + 1 = x$ .

$t + 1 = x$

Step 3

Subtract $1$ from both sides of the equation.

$t = x - 1$

Step 4

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

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

Step 5

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

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$y = (x - 1) (x - 1) + 4 (x - 1)$

Step 5.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 5.1.2.1

Apply the distributive property.

$y = x (x - 1) - 1 (x - 1) + 4 (x - 1)$

Step 5.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 1 - 1 (x - 1) + 4 (x - 1)$

Step 5.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 4 (x - 1)$

Step 5.1.3

Simplify and combine like terms.

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Step 5.1.3.1

Simplify each term.

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Step 5.1.3.1.1

Multiply $x$ by $x$ .

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

Step 5.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 5.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 5.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.1.3.2

Subtract $x$ from $- x$ .

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

Step 5.1.4

Apply the distributive property.

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

Step 5.1.5

Multiply $4$ by $- 1$ .

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

Step 5.2

Simplify by adding terms.

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Step 5.2.1

Add $- 2 x$ and $4 x$ .

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

Step 5.2.2

Subtract $4$ from $1$ .

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