Precalculus Examples

$92 t - 16 t^{2}$

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 $f (- \frac{b}{2 a})$ .

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

Step 2

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

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Step 2.1

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

$t = - \frac{92}{2 (- 16)}$

Step 2.2

Remove parentheses.

$t = - \frac{92}{2 (- 16)}$

Step 2.3

Simplify $- \frac{92}{2 (- 16)}$ .

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Step 2.3.1

Cancel the common factor of $92$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $92$ .

$t = - \frac{2 \cdot 46}{2 \cdot - 16}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot - 16$ .

$t = - \frac{2 \cdot 46}{2 (- 16)}$

Step 2.3.1.2.2

Cancel the common factor.

$t = - \frac{2 \cdot 46}{2 \cdot - 16}$

Step 2.3.1.2.3

Rewrite the expression.

$t = - \frac{46}{- 16}$

Step 2.3.2

Cancel the common factor of $46$ and $- 16$ .

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Step 2.3.2.1

Factor $2$ out of $46$ .

$t = - \frac{2 (23)}{- 16}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $2$ out of $- 16$ .

$t = - \frac{2 \cdot 23}{2 \cdot - 8}$

Step 2.3.2.2.2

Cancel the common factor.

$t = - \frac{2 \cdot 23}{2 \cdot - 8}$

Step 2.3.2.2.3

Rewrite the expression.

$t = - \frac{23}{- 8}$

Step 2.3.3

Move the negative in front of the fraction.

$t = - - \frac{23}{8}$

Step 2.3.4

Multiply $- - \frac{23}{8}$ .

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{23}{8})$

Step 2.3.4.2

Multiply $\frac{23}{8}$ by $1$ .

$t = \frac{23}{8}$

Step 3

Evaluate $f (\frac{23}{8})$ .

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Step 3.1

Replace the variable $t$ with $\frac{23}{8}$ in the expression.

$f (\frac{23}{8}) = 92 (\frac{23}{8}) - 16 {(\frac{23}{8})}^{2}$

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

Cancel the common factor of $4$ .

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Step 3.2.1.1.1

Factor $4$ out of $92$ .

$f (\frac{23}{8}) = 4 (23) (\frac{23}{8}) - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.1.2

Factor $4$ out of $8$ .

$f (\frac{23}{8}) = 4 \cdot (23 (\frac{23}{4 \cdot 2})) - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.1.3

Cancel the common factor.

$f (\frac{23}{8}) = 4 \cdot (23 (\frac{23}{4 \cdot 2})) - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.1.4

Rewrite the expression.

$f (\frac{23}{8}) = 23 (\frac{23}{2}) - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.2

Combine $23$ and $\frac{23}{2}$ .

$f (\frac{23}{8}) = \frac{23 \cdot 23}{2} - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.3

Multiply $23$ by $23$ .

$f (\frac{23}{8}) = \frac{529}{2} - 16 {(\frac{23}{8})}^{2}$

Step 3.2.1.4

Apply the product rule to $\frac{23}{8}$ .

$f (\frac{23}{8}) = \frac{529}{2} - 16 \frac{23^{2}}{8^{2}}$

Step 3.2.1.5

Raise $23$ to the power of $2$ .

$f (\frac{23}{8}) = \frac{529}{2} - 16 \frac{529}{8^{2}}$

Step 3.2.1.6

Raise $8$ to the power of $2$ .

$f (\frac{23}{8}) = \frac{529}{2} - 16 (\frac{529}{64})$

Step 3.2.1.7

Cancel the common factor of $16$ .

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Step 3.2.1.7.1

Factor $16$ out of $- 16$ .

$f (\frac{23}{8}) = \frac{529}{2} + 16 (- 1) (\frac{529}{64})$

Step 3.2.1.7.2

Factor $16$ out of $64$ .

$f (\frac{23}{8}) = \frac{529}{2} + 16 \cdot (- 1 \frac{529}{16 \cdot 4})$

Step 3.2.1.7.3

Cancel the common factor.

$f (\frac{23}{8}) = \frac{529}{2} + 16 \cdot (- 1 \frac{529}{16 \cdot 4})$

Step 3.2.1.7.4

Rewrite the expression.

$f (\frac{23}{8}) = \frac{529}{2} - 1 (\frac{529}{4})$

Step 3.2.1.8

Rewrite $- 1 (\frac{529}{4})$ as $- (\frac{529}{4})$ .

$f (\frac{23}{8}) = \frac{529}{2} - \frac{529}{4}$

Step 3.2.2

To write $\frac{529}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{23}{8}) = \frac{529}{2} \cdot \frac{2}{2} - \frac{529}{4}$

Step 3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{529}{2}$ by $\frac{2}{2}$ .

$f (\frac{23}{8}) = \frac{529 \cdot 2}{2 \cdot 2} - \frac{529}{4}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$f (\frac{23}{8}) = \frac{529 \cdot 2}{4} - \frac{529}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{23}{8}) = \frac{529 \cdot 2 - 529}{4}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $529$ by $2$ .

$f (\frac{23}{8}) = \frac{1058 - 529}{4}$

Step 3.2.5.2

Subtract $529$ from $1058$ .

$f (\frac{23}{8}) = \frac{529}{4}$

Step 3.2.6

The final answer is $\frac{529}{4}$ .

$\frac{529}{4}$

Step 4

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

$(\frac{23}{8}, \frac{529}{4})$

Step 5