Precalculus Examples

$f (x) = 9 x^{2} - 9 x$

Step 1

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{- 9}{2 (9)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 9}{2 (9)}$

Step 2.3

Simplify $- \frac{- 9}{2 (9)}$ .

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Step 2.3.1

Cancel the common factor of $- 9$ and $9$ .

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Step 2.3.1.1

Factor $9$ out of $- 9$ .

$x = - \frac{9 \cdot - 1}{2 \cdot 9}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $9$ out of $2 \cdot 9$ .

$x = - \frac{9 \cdot - 1}{9 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{9 \cdot - 1}{9 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Move the negative in front of the fraction.

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

Step 2.3.3

Multiply $- - \frac{1}{2}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.2

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

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

Step 3

Evaluate $f (\frac{1}{2})$ .

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Step 3.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 9 {(\frac{1}{2})}^{2} - 9 (\frac{1}{2})$

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

Apply the product rule to $\frac{1}{2}$ .

$f (\frac{1}{2}) = 9 (\frac{1^{2}}{2^{2}}) - 9 (\frac{1}{2})$

Step 3.2.1.2

One to any power is one.

$f (\frac{1}{2}) = 9 (\frac{1}{2^{2}}) - 9 (\frac{1}{2})$

Step 3.2.1.3

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

$f (\frac{1}{2}) = 9 (\frac{1}{4}) - 9 (\frac{1}{2})$

Step 3.2.1.4

Combine $9$ and $\frac{1}{4}$ .

$f (\frac{1}{2}) = \frac{9}{4} - 9 (\frac{1}{2})$

Step 3.2.1.5

Combine $- 9$ and $\frac{1}{2}$ .

$f (\frac{1}{2}) = \frac{9}{4} + \frac{- 9}{2}$

Step 3.2.1.6

Move the negative in front of the fraction.

$f (\frac{1}{2}) = \frac{9}{4} - \frac{9}{2}$

Step 3.2.2

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{1}{2}) = \frac{9}{4} - \frac{9}{2} \cdot \frac{2}{2}$

Step 3.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.3.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$f (\frac{1}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{2 \cdot 2}$

Step 3.2.3.2

Multiply $2$ by $2$ .

$f (\frac{1}{2}) = \frac{9}{4} - \frac{9 \cdot 2}{4}$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{9 - 9 \cdot 2}{4}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 9$ by $2$ .

$f (\frac{1}{2}) = \frac{9 - 18}{4}$

Step 3.2.5.2

Subtract $18$ from $9$ .

$f (\frac{1}{2}) = \frac{- 9}{4}$

Step 3.2.6

Move the negative in front of the fraction.

$f (\frac{1}{2}) = - \frac{9}{4}$

Step 3.2.7

The final answer is $- \frac{9}{4}$ .

$- \frac{9}{4}$

Step 4

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

$(\frac{1}{2}, - \frac{9}{4})$

Step 5