Precalculus Examples

$f (t) = 100 - 49 t - 7 t^{2}$

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 $b$ .

$t = - \frac{- 49}{2 (- 7)}$

Step 2.2

Remove parentheses.

$t = - \frac{- 49}{2 (- 7)}$

Step 2.3

Cancel the common factor of $- 49$ and $- 7$ .

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Step 2.3.1

Factor $- 7$ out of $- 49$ .

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $- 7$ out of $2 (- 7)$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 3

Evaluate $f (- \frac{7}{2})$ .

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Step 3.1

Replace the variable $t$ with $- \frac{7}{2}$ in the expression.

$f (- \frac{7}{2}) = 100 - 49 (- \frac{7}{2}) - 7 {(- \frac{7}{2})}^{2}$

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

Multiply $- 49 (- \frac{7}{2})$ .

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Step 3.2.1.1.1

Multiply $- 1$ by $- 49$ .

$f (- \frac{7}{2}) = 100 + 49 (\frac{7}{2}) - 7 {(- \frac{7}{2})}^{2}$

Step 3.2.1.1.2

Combine $49$ and $\frac{7}{2}$ .

$f (- \frac{7}{2}) = 100 + \frac{49 \cdot 7}{2} - 7 {(- \frac{7}{2})}^{2}$

Step 3.2.1.1.3

Multiply $49$ by $7$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 {(- \frac{7}{2})}^{2}$

Step 3.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.2.1

Apply the product rule to $- \frac{7}{2}$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 ({(- 1)}^{2} {(\frac{7}{2})}^{2})$

Step 3.2.1.2.2

Apply the product rule to $\frac{7}{2}$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 ({(- 1)}^{2} (\frac{7^{2}}{2^{2}}))$

Step 3.2.1.3

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

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 (1 (\frac{7^{2}}{2^{2}}))$

Step 3.2.1.4

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 \frac{7^{2}}{2^{2}}$

Step 3.2.1.5

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

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 \frac{49}{2^{2}}$

Step 3.2.1.6

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

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - 7 (\frac{49}{4})$

Step 3.2.1.7

Multiply $- 7 (\frac{49}{4})$ .

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Step 3.2.1.7.1

Combine $- 7$ and $\frac{49}{4}$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} + \frac{- 7 \cdot 49}{4}$

Step 3.2.1.7.2

Multiply $- 7$ by $49$ .

$f (- \frac{7}{2}) = 100 + \frac{343}{2} + \frac{- 343}{4}$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{7}{2}) = 100 + \frac{343}{2} - \frac{343}{4}$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $100$ as a fraction with denominator $1$ .

$f (- \frac{7}{2}) = \frac{100}{1} + \frac{343}{2} - \frac{343}{4}$

Step 3.2.2.2

Multiply $\frac{100}{1}$ by $\frac{4}{4}$ .

$f (- \frac{7}{2}) = \frac{100}{1} \cdot \frac{4}{4} + \frac{343}{2} - \frac{343}{4}$

Step 3.2.2.3

Multiply $\frac{100}{1}$ by $\frac{4}{4}$ .

$f (- \frac{7}{2}) = \frac{100 \cdot 4}{4} + \frac{343}{2} - \frac{343}{4}$

Step 3.2.2.4

Multiply $\frac{343}{2}$ by $\frac{2}{2}$ .

$f (- \frac{7}{2}) = \frac{100 \cdot 4}{4} + \frac{343}{2} \cdot \frac{2}{2} - \frac{343}{4}$

Step 3.2.2.5

Multiply $\frac{343}{2}$ by $\frac{2}{2}$ .

$f (- \frac{7}{2}) = \frac{100 \cdot 4}{4} + \frac{343 \cdot 2}{2 \cdot 2} - \frac{343}{4}$

Step 3.2.2.6

Multiply $2$ by $2$ .

$f (- \frac{7}{2}) = \frac{100 \cdot 4}{4} + \frac{343 \cdot 2}{4} - \frac{343}{4}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{7}{2}) = \frac{100 \cdot 4 + 343 \cdot 2 - 343}{4}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $100$ by $4$ .

$f (- \frac{7}{2}) = \frac{400 + 343 \cdot 2 - 343}{4}$

Step 3.2.4.2

Multiply $343$ by $2$ .

$f (- \frac{7}{2}) = \frac{400 + 686 - 343}{4}$

Step 3.2.5

Simplify by adding and subtracting.

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Step 3.2.5.1

Add $400$ and $686$ .

$f (- \frac{7}{2}) = \frac{1086 - 343}{4}$

Step 3.2.5.2

Subtract $343$ from $1086$ .

$f (- \frac{7}{2}) = \frac{743}{4}$

Step 3.2.6

The final answer is $\frac{743}{4}$ .

$\frac{743}{4}$

Step 4

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

$(- \frac{7}{2}, \frac{743}{4})$

Step 5