Calculus Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$y = (- x - 1) (- x - 1) + 3$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$y = - x (- x - 1) - 1 (- x - 1) + 3$

Step 1.1.2.2

Apply the distributive property.

$y = - x (- x) - x \cdot - 1 - 1 (- x - 1) + 3$

Step 1.1.2.3

Apply the distributive property.

$y = - x (- x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1 + 3$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Rewrite using the commutative property of multiplication.

$y = - 1 \cdot - 1 x \cdot x - x \cdot - 1 - 1 (- x) - 1 \cdot - 1 + 3$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

Move $x$ .

$y = - 1 \cdot - 1 (x \cdot x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1 + 3$

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.1.3.1.4

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

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

Step 1.1.3.1.5

Multiply $- x \cdot - 1$ .

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Step 1.1.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.3.1.5.2

Multiply $x$ by $1$ .

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

Step 1.1.3.1.6

Multiply $- 1 (- x)$ .

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Step 1.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.3.1.6.2

Multiply $x$ by $1$ .

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

Step 1.1.3.1.7

Multiply $- 1$ by $- 1$ .

$y = x^{2} + x + x + 1 + 3$

Step 1.1.3.2

Add $x$ and $x$ .

$y = x^{2} + 2 x + 1 + 3$

Step 1.2

Add $1$ and $3$ .

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

Step 2

Graph the equation in order to determine the intervals over which it is increasing or decreasing.

Increasing on: $(- 1, \infty)$

Decreasing on: $(- \infty, - 1)$

Step 3