Precalculus Examples

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

Step 1

Write $f (x) = - x^{2} - 4 x - 1$ as an equation.

$y = - x^{2} - 4 x - 1$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- x^{2} - 4 x - 1 = y$

Step 3

Add $1$ to both sides of the equation.

$- x^{2} - 4 x = y + 1$

Step 4

Complete the square.

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Step 4.1

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

$a = - 1$

$b = - 4$

$c = 0 = y + 1$

Step 4.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 1$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Factor $2$ out of $- 4$ .

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

Step 4.3.2.1.2

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

$d = - 1 \cdot - 2$

Step 4.3.2.2

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

$d = - - 2$

Step 4.3.2.3

Multiply $- 1$ by $- 2$ .

$d = 2$

Step 4.4

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

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Step 4.4.1

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

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot - 1}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

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

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Step 4.4.2.1.1.1

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

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 1}$

Step 4.4.2.1.1.2

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

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 1}$

Step 4.4.2.1.1.3

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot - 1}$

Step 4.4.2.1.1.4

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

$e = 0 - \frac{4^{2}}{4 \cdot - 1}$

Step 4.4.2.1.1.5

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

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 4.4.2.1.1.6

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

$e = 0 - (- 1 \cdot 4)$

Step 4.4.2.1.2

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

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Step 4.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - - 4$

Step 4.4.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 4.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 4.5

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

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

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Subtract $4$ from both sides of the equation.

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

Step 5.2

Subtract $4$ from $1$ .

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

Step 6

Divide each term in $- {(x + 2)}^{2} = y - 3$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- {(x + 2)}^{2} = y - 3$ by $- 1$ .

$\frac{- {(x + 2)}^{2}}{- 1} = \frac{y}{- 1} + \frac{- 3}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{{(x + 2)}^{2}}{1} = \frac{y}{- 1} + \frac{- 3}{- 1}$

Step 6.2.2

Divide ${(x + 2)}^{2}$ by $1$ .

${(x + 2)}^{2} = \frac{y}{- 1} + \frac{- 3}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

${(x + 2)}^{2} = - 1 \cdot y + \frac{- 3}{- 1}$

Step 6.3.1.2

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

${(x + 2)}^{2} = - y + \frac{- 3}{- 1}$

Step 6.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 7

Factor.

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Step 7.1

Factor $- 1$ out of $- y$ .

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

Step 7.2

Rewrite $3$ as $- 1 (- 3)$ .

${(x + 2)}^{2} = - (y) - 1 (- 3)$

Step 7.3

Factor $- 1$ out of $- (y) - 1 (- 3)$ .

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