Precalculus Examples

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

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 $b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Simplify $- \frac{- 4}{2 (3)}$ .

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Step 2.3.1

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 4$ .

$x = - \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 3$ .

$x = - \frac{2 \cdot - 2}{2 (3)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 2}{2 \cdot 3}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 2}{3}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{2}{3}$

Step 2.3.3

Multiply $- - \frac{2}{3}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{2}{3})$

Step 2.3.3.2

Multiply $\frac{2}{3}$ by $1$ .

$x = \frac{2}{3}$

Step 3

Evaluate $f (\frac{2}{3})$ .

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Step 3.1

Replace the variable $x$ with $\frac{2}{3}$ in the expression.

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

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 $\frac{2}{3}$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Raise $3$ to the power of $2$ .

$f (\frac{2}{3}) = 3 (\frac{4}{9}) - 4 (\frac{2}{3}) - 2$

Step 3.2.1.4

Cancel the common factor of $3$ .

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Step 3.2.1.4.1

Factor $3$ out of $9$ .

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

Step 3.2.1.4.2

Cancel the common factor.

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

Step 3.2.1.4.3

Rewrite the expression.

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

Step 3.2.1.5

Multiply $- 4 (\frac{2}{3})$ .

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Step 3.2.1.5.1

Combine $- 4$ and $\frac{2}{3}$ .

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

Step 3.2.1.5.2

Multiply $- 4$ by $2$ .

$f (\frac{2}{3}) = \frac{4}{3} + \frac{- 8}{3} - 2$

Step 3.2.1.6

Move the negative in front of the fraction.

$f (\frac{2}{3}) = \frac{4}{3} - \frac{8}{3} - 2$

Step 3.2.2

Combine fractions.

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Step 3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{2}{3}) = - 2 + \frac{4 - 8}{3}$

Step 3.2.2.2

Simplify the expression.

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Step 3.2.2.2.1

Subtract $8$ from $4$ .

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

Step 3.2.2.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.2.4

Combine $- 2$ and $\frac{3}{3}$ .

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 2$ by $3$ .

$f (\frac{2}{3}) = \frac{- 6 - 4}{3}$

Step 3.2.6.2

Subtract $4$ from $- 6$ .

$f (\frac{2}{3}) = \frac{- 10}{3}$

Step 3.2.7

Move the negative in front of the fraction.

$f (\frac{2}{3}) = - \frac{10}{3}$

Step 3.2.8

The final answer is $- \frac{10}{3}$ .

$- \frac{10}{3}$

Step 4

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

$(\frac{2}{3}, - \frac{10}{3})$

Step 5