Algebra Examples

$f (x) = - \frac{1}{4} x^{2} + 2 x - 3$

Step 1

Combine $x^{2}$ and $\frac{1}{4}$ .

$f (x) = - \frac{x^{2}}{4} + 2 x - 3$

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 $b$ .

$x = - \frac{2}{2 (- 0.25)}$

Step 3.2

Remove parentheses.

$x = - \frac{2}{2 (- 0.25)}$

Step 3.3

Simplify $- \frac{2}{2 (- 0.25)}$ .

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

$x = - \frac{2}{2 \cdot - 0.25}$

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Divide $1$ by $- 0.25$ .

$x = - - 4$

Step 3.3.2.2

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.

$f (4) = - \frac{{(4)}^{2}}{4} + 2 (4) - 3$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of ${(4)}^{2}$ and $4$ .

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Step 4.2.1.1.1

Factor $4$ out of ${(4)}^{2}$ .

$f (4) = - \frac{4 \cdot 4}{4} + 2 (4) - 3$

Step 4.2.1.1.2

Cancel the common factors.

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Step 4.2.1.1.2.1

Factor $4$ out of $4$ .

$f (4) = - \frac{4 \cdot 4}{4 (1)} + 2 (4) - 3$

Step 4.2.1.1.2.2

Cancel the common factor.

$f (4) = - \frac{4 \cdot 4}{4 \cdot 1} + 2 (4) - 3$

Step 4.2.1.1.2.3

Rewrite the expression.

$f (4) = - \frac{4}{1} + 2 (4) - 3$

Step 4.2.1.1.2.4

Divide $4$ by $1$ .

$f (4) = - 1 \cdot 4 + 2 (4) - 3$

Step 4.2.1.2

Multiply $- 1$ by $4$ .

$f (4) = - 4 + 2 (4) - 3$

Step 4.2.1.3

Multiply $2$ by $4$ .

$f (4) = - 4 + 8 - 3$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $- 4$ and $8$ .

$f (4) = 4 - 3$

Step 4.2.2.2

Subtract $3$ from $4$ .

$f (4) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 5

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

$(4, 1)$

Step 6