Precalculus Examples

$f (x) = - x^{2} - 4 x + 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- x^{2} - 4 x + 1$ .

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Step 1.1.1

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

$a = - 1$

$b = - 4$

$c = 1$

Step 1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.3

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

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Step 1.1.3.1

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

$d = \frac{- 4}{2 \cdot - 1}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

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

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Step 1.1.3.2.1.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot - 1}$

Step 1.1.3.2.1.2

Move the negative one from the denominator of $\frac{- 2}{- 1}$ .

$d = - 1 \cdot - 2$

Step 1.1.3.2.2

Rewrite $- 1 \cdot - 2$ as $- - 2$ .

$d = - - 2$

Step 1.1.3.2.3

Multiply $- 1$ by $- 2$ .

$d = 2$

Step 1.1.4

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

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Step 1.1.4.1

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

$e = 1 - \frac{{(- 4)}^{2}}{4 \cdot - 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.1.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 1 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 1}$

Step 1.1.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 1 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 1}$

Step 1.1.4.2.1.1.3

Raise $- 1$ to the power of $2$ .

$e = 1 - \frac{1 \cdot 4^{2}}{4 \cdot - 1}$

Step 1.1.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 1 - \frac{4^{2}}{4 \cdot - 1}$

Step 1.1.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 1 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 1.1.4.2.1.1.6

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 1 - (- 1 \cdot 4)$

Step 1.1.4.2.1.2

Multiply $- (- 1 \cdot 4)$ .

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Step 1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 1 - - 4$

Step 1.1.4.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 1 + 4$

Step 1.1.4.2.2

Add $1$ and $4$ .

$e = 5$

Step 1.1.5

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

$- {(x + 2)}^{2} + 5$

Step 1.2

Set $y$ equal to the new right side.

$y = - {(x + 2)}^{2} + 5$

Step 2

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

$a = - 1$

$h = - 2$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 5)$

Step 4