Find the Maximum/Minimum Value s(t)=-16t^2+28t+8

$s (t) = - 16 t^{2} + 28 t + 8$

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 $t = - \frac{b}{2 a}$ .

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

$t = - \frac{28}{2 (- 16)}$

Remove parentheses.

$t = - \frac{28}{2 (- 16)}$

Simplify $- \frac{28}{2 (- 16)}$ .

Cancel the common factor of $28$ and $2$ .

Factor $2$ out of $28$ .

$t = - \frac{2 \cdot 14}{2 \cdot - 16}$

Cancel the common factors.

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

$t = - \frac{2 \cdot 14}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 14}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{14}{- 16}$

Cancel the common factor of $14$ and $- 16$ .

Factor $2$ out of $14$ .

$t = - \frac{2 (7)}{- 16}$

Cancel the common factors.

Factor $2$ out of $- 16$ .

$t = - \frac{2 \cdot 7}{2 \cdot - 8}$

Cancel the common factor.

$t = - \frac{2 \cdot 7}{2 \cdot - 8}$

Rewrite the expression.

$t = - \frac{7}{- 8}$

Move the negative in front of the fraction.

$t = - - \frac{7}{8}$

Multiply $- - \frac{7}{8}$ .

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{7}{8})$

Multiply $\frac{7}{8}$ by $1$ .

$t = \frac{7}{8}$

Step 3

Evaluate $f (\frac{7}{8})$ .

Replace the variable $t$ with $\frac{7}{8}$ in the expression.

$s (\frac{7}{8}) = - 16 {(\frac{7}{8})}^{2} + 28 (\frac{7}{8}) + 8$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{7}{8}$ .

$s (\frac{7}{8}) = - 16 \frac{7^{2}}{8^{2}} + 28 (\frac{7}{8}) + 8$

Raise $7$ to the power of $2$ .

$s (\frac{7}{8}) = - 16 \frac{49}{8^{2}} + 28 (\frac{7}{8}) + 8$

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

$s (\frac{7}{8}) = - 16 (\frac{49}{64}) + 28 (\frac{7}{8}) + 8$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$s (\frac{7}{8}) = 16 (- 1) (\frac{49}{64}) + 28 (\frac{7}{8}) + 8$

Factor $16$ out of $64$ .

$s (\frac{7}{8}) = 16 \cdot (- 1 \frac{49}{16 \cdot 4}) + 28 (\frac{7}{8}) + 8$

Cancel the common factor.

$s (\frac{7}{8}) = 16 \cdot (- 1 \frac{49}{16 \cdot 4}) + 28 (\frac{7}{8}) + 8$

Rewrite the expression.

$s (\frac{7}{8}) = - 1 (\frac{49}{4}) + 28 (\frac{7}{8}) + 8$

Rewrite $- 1 (\frac{49}{4})$ as $- (\frac{49}{4})$ .

$s (\frac{7}{8}) = - (\frac{49}{4}) + 28 (\frac{7}{8}) + 8$

Cancel the common factor of $4$ .

Factor $4$ out of $28$ .

$s (\frac{7}{8}) = - \frac{49}{4} + 4 (7) (\frac{7}{8}) + 8$

Factor $4$ out of $8$ .

$s (\frac{7}{8}) = - \frac{49}{4} + 4 \cdot (7 (\frac{7}{4 \cdot 2})) + 8$

Cancel the common factor.

$s (\frac{7}{8}) = - \frac{49}{4} + 4 \cdot (7 (\frac{7}{4 \cdot 2})) + 8$

Rewrite the expression.

$s (\frac{7}{8}) = - \frac{49}{4} + 7 (\frac{7}{2}) + 8$

Combine $7$ and $\frac{7}{2}$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{7 \cdot 7}{2} + 8$

Multiply $7$ by $7$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49}{2} + 8$

Find the common denominator.

Multiply $\frac{49}{2}$ by $\frac{2}{2}$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49}{2} \cdot \frac{2}{2} + 8$

Multiply $\frac{49}{2}$ by $\frac{2}{2}$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + 8$

Write $8$ as a fraction with denominator $1$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{8}{1}$

Multiply $\frac{8}{1}$ by $\frac{4}{4}$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{8}{1} \cdot \frac{4}{4}$

Multiply $\frac{8}{1}$ by $\frac{4}{4}$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49 \cdot 2}{2 \cdot 2} + \frac{8 \cdot 4}{4}$

Multiply $2$ by $2$ .

$s (\frac{7}{8}) = - \frac{49}{4} + \frac{49 \cdot 2}{4} + \frac{8 \cdot 4}{4}$

Combine the numerators over the common denominator.

$s (\frac{7}{8}) = \frac{- 49 + 49 \cdot 2 + 8 \cdot 4}{4}$

Simplify each term.

Multiply $49$ by $2$ .

$s (\frac{7}{8}) = \frac{- 49 + 98 + 8 \cdot 4}{4}$

Multiply $8$ by $4$ .

$s (\frac{7}{8}) = \frac{- 49 + 98 + 32}{4}$

Simplify by adding numbers.

Add $- 49$ and $98$ .

$s (\frac{7}{8}) = \frac{49 + 32}{4}$

Add $49$ and $32$ .

$s (\frac{7}{8}) = \frac{81}{4}$

The final answer is $\frac{81}{4}$ .

$\frac{81}{4}$

Step 4

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

$(\frac{7}{8}, \frac{81}{4})$

Step 5

Find the Maximum/Minimum Value s(t)=-16t^2+28t+8

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