Precalculus Examples

$f (x) = - 2 x^{2} - 8 x - 6$

Step 1

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

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

Step 2

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

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Step 2.1

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

$x = - \frac{- 8}{2 (- 2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 8}{2 (- 2)}$

Step 2.3

Simplify $- \frac{- 8}{2 (- 2)}$ .

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Step 2.3.1

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 8$ .

$x = - \frac{2 \cdot - 4}{2 \cdot - 2}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 2$ .

$x = - \frac{2 \cdot - 4}{2 (- 2)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 4}{2 \cdot - 2}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 4}{- 2}$

Step 2.3.2

Cancel the common factor of $- 4$ and $- 2$ .

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Step 2.3.2.1

Factor $- 2$ out of $- 4$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $- 2$ out of $- 2$ .

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $2$ by $1$ .

$x = - 1 \cdot 2$

Step 2.3.3

Multiply $- 1$ by $2$ .

$x = - 2$

Step 3

Evaluate $f (- 2)$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - 2 {(- 2)}^{2} - 8 \cdot - 2 - 6$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

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Step 3.2.1.1.1

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 3.2.1.1.1.1

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

$f (- 2) = (- 2) {(- 2)}^{2} - 8 \cdot - 2 - 6$

Step 3.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 2) = {(- 2)}^{1 + 2} - 8 \cdot - 2 - 6$

Step 3.2.1.1.2

Add $1$ and $2$ .

$f (- 2) = {(- 2)}^{3} - 8 \cdot - 2 - 6$

Step 3.2.1.2

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

$f (- 2) = - 8 - 8 \cdot - 2 - 6$

Step 3.2.1.3

Multiply $- 8$ by $- 2$ .

$f (- 2) = - 8 + 16 - 6$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $- 8$ and $16$ .

$f (- 2) = 8 - 6$

Step 3.2.2.2

Subtract $6$ from $8$ .

$f (- 2) = 2$

Step 3.2.3

The final answer is $2$ .

$2$

Step 4

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

$(- 2, 2)$

Step 5