Precalculus Examples

$f (x) = 2 x - x^{2}$

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Remove parentheses.

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

Step 2.3

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

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Step 2.3.1

Cancel the common factor of $2$ .

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Step 2.3.1.1

Cancel the common factor.

$x = - \frac{2}{2 \cdot - 1}$

Step 2.3.1.2

Rewrite the expression.

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

Step 2.3.1.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$x = - (- 1 \cdot 1)$

Step 2.3.2

Multiply $- (- 1 \cdot 1)$ .

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Step 2.3.2.1

Multiply $- 1$ by $1$ .

$x = - - 1$

Step 2.3.2.2

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 3

Evaluate $f (1)$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 (1) - {(1)}^{2}$

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 $2$ by $1$ .

$f (1) = 2 - {(1)}^{2}$

Step 3.2.1.2

One to any power is one.

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

Step 3.2.1.3

Multiply $- 1$ by $1$ .

$f (1) = 2 - 1$

Step 3.2.2

Subtract $1$ from $2$ .

$f (1) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 4

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

$(1, 1)$

Step 5