Precalculus Examples

$f (x) = 2 x^{2} - 6 x + k$

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 $b$ .

$x = - \frac{- 6}{2 (2)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 6}{2 (2)}$

Step 2.3

Simplify $- \frac{- 6}{2 (2)}$ .

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Step 2.3.1

Cancel the common factor of $- 6$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 6$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Move the negative in front of the fraction.

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

Step 2.3.3

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

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.3.2

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

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

Step 3

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

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Step 3.1

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

$f (\frac{3}{2}) = 2 {(\frac{3}{2})}^{2} - 6 (\frac{3}{2}) + k$

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 $\frac{3}{2}$ .

$f (\frac{3}{2}) = 2 (\frac{3^{2}}{2^{2}}) - 6 (\frac{3}{2}) + k$

Step 3.2.1.2

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

$f (\frac{3}{2}) = 2 (\frac{9}{2^{2}}) - 6 (\frac{3}{2}) + k$

Step 3.2.1.3

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

$f (\frac{3}{2}) = 2 (\frac{9}{4}) - 6 (\frac{3}{2}) + k$

Step 3.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.1.4.1

Factor $2$ out of $4$ .

$f (\frac{3}{2}) = 2 (\frac{9}{2 (2)}) - 6 (\frac{3}{2}) + k$

Step 3.2.1.4.2

Cancel the common factor.

$f (\frac{3}{2}) = 2 (\frac{9}{2 \cdot 2}) - 6 (\frac{3}{2}) + k$

Step 3.2.1.4.3

Rewrite the expression.

$f (\frac{3}{2}) = \frac{9}{2} - 6 (\frac{3}{2}) + k$

Step 3.2.1.5

Cancel the common factor of $2$ .

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Step 3.2.1.5.1

Factor $2$ out of $- 6$ .

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

Step 3.2.1.5.2

Cancel the common factor.

$f (\frac{3}{2}) = \frac{9}{2} + 2 \cdot (- 3 (\frac{3}{2})) + k$

Step 3.2.1.5.3

Rewrite the expression.

$f (\frac{3}{2}) = \frac{9}{2} - 3 \cdot 3 + k$

Step 3.2.1.6

Multiply $- 3$ by $3$ .

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

Step 3.2.2

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

$f (\frac{3}{2}) = \frac{9}{2} - 9 \cdot \frac{2}{2} + k$

Step 3.2.3

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

$f (\frac{3}{2}) = \frac{9}{2} + \frac{- 9 \cdot 2}{2} + k$

Step 3.2.4

Combine the numerators over the common denominator.

$f (\frac{3}{2}) = \frac{9 - 9 \cdot 2}{2} + k$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 9$ by $2$ .

$f (\frac{3}{2}) = \frac{9 - 18}{2} + k$

Step 3.2.5.2

Subtract $18$ from $9$ .

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

Step 3.2.6

Move the negative in front of the fraction.

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

Step 3.2.7

The final answer is $- \frac{9}{2} + k$ .

$- \frac{9}{2} + k$

Step 4

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

$(\frac{3}{2}, - \frac{9}{2} + k)$

Step 5