Precalculus Examples

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

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

Step 1

The minimum of a quadratic function occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ . If

a

$a$ is positive, the minimum value of the function is

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

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

f_{min}

$f_{min}$

x = a x^{2} + b x + c

$x = a x^{2} + b x + c$ occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$

Step 2

Find the value of

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ .

Tap for more steps...

Step 2.1

Substitute in the values of

a

$a$ and

b

$b$ .

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

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

Step 2.2

Remove parentheses.

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

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

Step 2.3

Multiply

2

$2$ by

2

$2$ .

x = - \frac{3}{4}

$x = - \frac{3}{4}$

x = - \frac{3}{4}

$x = - \frac{3}{4}$

Step 3

Evaluate

f (- \frac{3}{4})

$f (- \frac{3}{4})$ .

Tap for more steps...

Step 3.1

Replace the variable

x

$x$ with

- \frac{3}{4}

$- \frac{3}{4}$ in the expression.

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 3.2.1.1.1

Apply the product rule to

- \frac{3}{4}

$- \frac{3}{4}$ .

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

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

Step 3.2.1.1.2

Apply the product rule to

\frac{3}{4}

$\frac{3}{4}$ .

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

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

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

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

Step 3.2.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 3.2.1.3

Multiply

\frac{3^{2}}{4^{2}}

$\frac{3^{2}}{4^{2}}$ by

1

$1$ .

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

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

Step 3.2.1.4

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 3.2.1.5

Raise

4

$4$ to the power of

2

$2$ .

f (- \frac{3}{4}) = 2 (\frac{9}{16}) + 3 (- \frac{3}{4}) - 4

$f (- \frac{3}{4}) = 2 (\frac{9}{16}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.6

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.2.1.6.1

Factor

2

$2$ out of

16

$16$ .

f (- \frac{3}{4}) = 2 (\frac{9}{2 (8)}) + 3 (- \frac{3}{4}) - 4

$f (- \frac{3}{4}) = 2 (\frac{9}{2 (8)}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{3}{4}) = 2 (\frac{9}{2 \cdot 8}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{3}{4}) = \frac{9}{8} + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.7

Multiply

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

Tap for more steps...

Step 3.2.1.7.1

Multiply

$- 1$ by

$3$ .

$f (- \frac{3}{4}) = \frac{9}{8} - 3 (\frac{3}{4}) - 4$

Step 3.2.1.7.2

Combine

$- 3$ and

$\frac{3}{4}$ .

$f (- \frac{3}{4}) = \frac{9}{8} + \frac{- 3 \cdot 3}{4} - 4$

Step 3.2.1.7.3

Multiply

$- 3$ by

$3$ .

$f (- \frac{3}{4}) = \frac{9}{8} + \frac{- 9}{4} - 4$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9}{4} - 4$

Step 3.2.2

Find the common denominator.

Tap for more steps...

Step 3.2.2.1

Multiply

$\frac{9}{4}$ by

$\frac{2}{2}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - (\frac{9}{4} \cdot \frac{2}{2}) - 4$

Step 3.2.2.2

Multiply

$\frac{9}{4}$ by

$\frac{2}{2}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} - 4$

Step 3.2.2.3

Write

$- 4$ as a fraction with denominator

$1$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4}{1}$

Step 3.2.2.4

Multiply

$\frac{- 4}{1}$ by

$\frac{8}{8}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4}{1} \cdot \frac{8}{8}$

Step 3.2.2.5

Multiply

$\frac{- 4}{1}$ by

$\frac{8}{8}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4 \cdot 8}{8}$

Step 3.2.2.6

Reorder the factors of

$4 \cdot 2$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{2 \cdot 4} + \frac{- 4 \cdot 8}{8}$

Step 3.2.2.7

Multiply

$2$ by

$4$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{8} + \frac{- 4 \cdot 8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{3}{4}) = \frac{9 - 9 \cdot 2 - 4 \cdot 8}{8}$

Step 3.2.4

Simplify each term.

Tap for more steps...

Step 3.2.4.1

Multiply

$- 9$ by

$2$ .

$f (- \frac{3}{4}) = \frac{9 - 18 - 4 \cdot 8}{8}$

Step 3.2.4.2

Multiply

$- 4$ by

$8$ .

$f (- \frac{3}{4}) = \frac{9 - 18 - 32}{8}$

Step 3.2.5

Simplify the expression.

Tap for more steps...

Step 3.2.5.1

Subtract

$18$ from

$9$ .

$f (- \frac{3}{4}) = \frac{- 9 - 32}{8}$

Step 3.2.5.2

Subtract

$32$ from

$- 9$ .

$f (- \frac{3}{4}) = \frac{- 41}{8}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{3}{4}) = - \frac{41}{8}$

Step 3.2.6

The final answer is

$- \frac{41}{8}$ .

$- \frac{41}{8}$

Step 4

Use the

$x$ and

$y$ values to find where the minimum occurs.

$(- \frac{3}{4}, - \frac{41}{8})$

Step 5

Enter YOUR Problem