Precalculus Examples

$f (x) = - \frac{4}{5} x^{2} + \frac{48}{5} x - \frac{114}{5}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x^{2}$ and $\frac{4}{5}$ .

$y = - \frac{x^{2} \cdot 4}{5} + \frac{48}{5} x - \frac{114}{5}$

Step 1.1.2

Move $4$ to the left of $x^{2}$ .

$y = - \frac{4 \cdot x^{2}}{5} + \frac{48}{5} x - \frac{114}{5}$

Step 1.1.3

Combine $\frac{48}{5}$ and $x$ .

$y = - \frac{4 x^{2}}{5} + \frac{48 x}{5} - \frac{114}{5}$

Step 1.2

Complete the square for $- \frac{4 x^{2}}{5} + \frac{48 x}{5} - \frac{114}{5}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{4}{5}$

$b = \frac{48}{5}$

$c = - \frac{114}{5}$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{48}{5}}{2 (- \frac{4}{5})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{48}{5} \cdot \frac{1}{2 (- \frac{4}{5})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $48$ .

$d = \frac{2 (24)}{5} \cdot \frac{1}{2 (- \frac{4}{5})}$

Step 1.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 24}{5} \cdot \frac{1}{2 (- \frac{4}{5})}$

Step 1.2.3.2.2.3

Rewrite the expression.

$d = \frac{24}{5} \cdot \frac{1}{- \frac{4}{5}}$

Step 1.2.3.2.3

Multiply $\frac{24}{5}$ by $\frac{1}{- \frac{4}{5}}$ .

$d = \frac{24}{5 (- \frac{4}{5})}$

Step 1.2.3.2.4

Multiply $- 1$ by $5$ .

$d = \frac{24}{- 5 (\frac{4}{5})}$

Step 1.2.3.2.5

Combine $- 5$ and $\frac{4}{5}$ .

$d = \frac{24}{\frac{- 5 \cdot 4}{5}}$

Step 1.2.3.2.6

Multiply $- 5$ by $4$ .

$d = \frac{24}{\frac{- 20}{5}}$

Step 1.2.3.2.7

Cancel the common factor of $- 20$ and $5$ .

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Step 1.2.3.2.7.1

Factor $5$ out of $- 20$ .

$d = \frac{24}{\frac{5 \cdot - 4}{5}}$

Step 1.2.3.2.7.2

Cancel the common factors.

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Step 1.2.3.2.7.2.1

Factor $5$ out of $5$ .

$d = \frac{24}{\frac{5 \cdot - 4}{5 (1)}}$

Step 1.2.3.2.7.2.2

Cancel the common factor.

$d = \frac{24}{\frac{5 \cdot - 4}{5 \cdot 1}}$

Step 1.2.3.2.7.2.3

Rewrite the expression.

$d = \frac{24}{\frac{- 4}{1}}$

Step 1.2.3.2.7.2.4

Divide $- 4$ by $1$ .

$d = \frac{24}{- 4}$

Step 1.2.3.2.8

Cancel the common factor of $24$ and $- 4$ .

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Step 1.2.3.2.8.1

Factor $4$ out of $24$ .

$d = \frac{4 (6)}{- 4}$

Step 1.2.3.2.8.2

Move the negative one from the denominator of $\frac{6}{- 1}$ .

$d = - 1 \cdot 6$

Step 1.2.3.2.9

Multiply $- 1$ by $6$ .

$d = - 6$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - \frac{114}{5} - \frac{{(\frac{48}{5})}^{2}}{4 (- \frac{4}{5})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{48}{5}$ .

$e = - \frac{114}{5} - \frac{\frac{48^{2}}{5^{2}}}{4 (- \frac{4}{5})}$

Step 1.2.4.2.1.1.2

Raise $48$ to the power of $2$ .

$e = - \frac{114}{5} - \frac{\frac{2304}{5^{2}}}{4 (- \frac{4}{5})}$

Step 1.2.4.2.1.1.3

Raise $5$ to the power of $2$ .

$e = - \frac{114}{5} - \frac{\frac{2304}{25}}{4 (- \frac{4}{5})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{114}{5} - \frac{\frac{2304}{25}}{- 4 (\frac{4}{5})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{4}{5}$ .

$e = - \frac{114}{5} - \frac{\frac{2304}{25}}{\frac{- 4 \cdot 4}{5}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $4$ .

$e = - \frac{114}{5} - \frac{\frac{2304}{25}}{\frac{- 16}{5}}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = - \frac{114}{5} - \frac{\frac{2304}{25}}{- \frac{16}{5}}$

Step 1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{114}{5} - (\frac{2304}{25} (- \frac{5}{16}))$

Step 1.2.4.2.1.6

Cancel the common factor of $16$ .

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Step 1.2.4.2.1.6.1

Move the leading negative in $- \frac{5}{16}$ into the numerator.

$e = - \frac{114}{5} - (\frac{2304}{25} \cdot \frac{- 5}{16})$

Step 1.2.4.2.1.6.2

Factor $16$ out of $2304$ .

$e = - \frac{114}{5} - (\frac{16 (144)}{25} \cdot \frac{- 5}{16})$

Step 1.2.4.2.1.6.3

Cancel the common factor.

$e = - \frac{114}{5} - (\frac{16 \cdot 144}{25} \cdot \frac{- 5}{16})$

Step 1.2.4.2.1.6.4

Rewrite the expression.

$e = - \frac{114}{5} - (\frac{144}{25} \cdot - 5)$

Step 1.2.4.2.1.7

Cancel the common factor of $5$ .

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Step 1.2.4.2.1.7.1

Factor $5$ out of $25$ .

$e = - \frac{114}{5} - (\frac{144}{5 (5)} \cdot - 5)$

Step 1.2.4.2.1.7.2

Factor $5$ out of $- 5$ .

$e = - \frac{114}{5} - (\frac{144}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.4.2.1.7.3

Cancel the common factor.

$e = - \frac{114}{5} - (\frac{144}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.4.2.1.7.4

Rewrite the expression.

$e = - \frac{114}{5} - (\frac{144}{5} \cdot - 1)$

Step 1.2.4.2.1.8

Combine $\frac{144}{5}$ and $- 1$ .

$e = - \frac{114}{5} - \frac{144 \cdot - 1}{5}$

Step 1.2.4.2.1.9

Multiply $144$ by $- 1$ .

$e = - \frac{114}{5} - \frac{- 144}{5}$

Step 1.2.4.2.1.10

Move the negative in front of the fraction.

$e = - \frac{114}{5} - - \frac{144}{5}$

Step 1.2.4.2.1.11

Multiply $- - \frac{144}{5}$ .

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Step 1.2.4.2.1.11.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{114}{5} + 1 (\frac{144}{5})$

Step 1.2.4.2.1.11.2

Multiply $\frac{144}{5}$ by $1$ .

$e = - \frac{114}{5} + \frac{144}{5}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 114 + 144}{5}$

Step 1.2.4.2.3

Add $- 114$ and $144$ .

$e = \frac{30}{5}$

Step 1.2.4.2.4

Divide $30$ by $5$ .

$e = 6$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{4}{5} {(x - 6)}^{2} + 6$ .

$- \frac{4}{5} {(x - 6)}^{2} + 6$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{4}{5} \cdot {(x - 6)}^{2} + 6$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{4}{5}$

$h = 6$

$k = 6$

Step 3

Find the vertex $(h, k)$ .

$(6, 6)$

Step 4