Finite Math Examples

$y = - \frac{1}{8} x^{2} + x$

Step 1

Combine $x^{2}$ and $\frac{1}{8}$ .

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

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{1}{2 (- 0.125)}$

Step 3.2

Remove parentheses.

$x = - \frac{1}{2 (- 0.125)}$

Step 3.3

Simplify $- \frac{1}{2 (- 0.125)}$ .

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Step 3.3.1

Multiply $2$ by $- 0.125$ .

$x = - \frac{1}{- 0.25}$

Step 3.3.2

Divide $1$ by $- 0.25$ .

$x = - - 4$

Step 3.3.3

Multiply $- 1$ by $- 4$ .

$x = 4$

Step 4

Evaluate $f (4)$ .

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Step 4.1

Replace the variable $x$ with $4$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Raise $4$ to the power of $2$ .

$f (4) = - \frac{16}{8} + 4$

Step 4.2.2.2

Divide $16$ by $8$ .

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

Step 4.2.2.3

Multiply $- 1$ by $2$ .

$f (4) = - 2 + 4$

Step 4.2.3

Add $- 2$ and $4$ .

$f (4) = 2$

Step 4.2.4

The final answer is $2$ .

$2$

Step 5

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

$(4, 2)$

Step 6