Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

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

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.4

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

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

Step 1.3.1.5

Multiply $- x \cdot - 1$ .

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Step 1.3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.5.2

Multiply $x$ by $1$ .

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

Step 1.3.1.6

Multiply $- 1 (- x)$ .

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Step 1.3.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.6.2

Multiply $x$ by $1$ .

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

Step 1.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Add $x$ and $x$ .

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

Step 2

Add $1$ and $3$ .

$f (x) = x^{2} + 2 x + 4$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

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Step 4.1

Substitute in the values of $a$ and $b$ .

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

Step 4.2

Remove parentheses.

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

Step 4.3

Simplify $- \frac{2}{2 (1)}$ .

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

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

Step 4.3.1.2

Rewrite the expression.

$x = - 1 \cdot 1$

Step 4.3.2

Multiply $- 1$ by $1$ .

$x = - 1$

Step 5

Evaluate $f (- 1)$ .

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (- 1) = 1 + 2 (- 1) + 4$

Step 5.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = 1 - 2 + 4$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $2$ from $1$ .

$f (- 1) = - 1 + 4$

Step 5.2.2.2

Add $- 1$ and $4$ .

$f (- 1) = 3$

Step 5.2.3

The final answer is $3$ .

$3$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(- 1, 3)$

Step 7