Precalculus Examples

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

Step 1

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

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

Step 2

Rewrite the equation in vertex form.

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Step 2.1

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

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Step 2.1.1

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

$a = - 1$

$b = - 2$

$c = - 1$

Step 2.1.2

Consider the vertex form of a parabola.

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

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

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Step 2.1.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 2.1.3.2.1.2

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

$d = - 1 \cdot - 1$

Step 2.1.3.2.2

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

$d = - - 1$

Step 2.1.3.2.3

Multiply $- 1$ by $- 1$ .

$d = 1$

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify each term.

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Step 2.1.4.2.1.1

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

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

Step 2.1.4.2.1.2

Multiply $4$ by $- 1$ .

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

Step 2.1.4.2.1.3

Divide $4$ by $- 4$ .

$e = - 1 - - 1$

Step 2.1.4.2.1.4

Multiply $- 1$ by $- 1$ .

$e = - 1 + 1$

Step 2.1.4.2.2

Add $- 1$ and $1$ .

$e = 0$

Step 2.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 1)}^{2} + 0$ .

$- {(x + 1)}^{2} + 0$

Step 2.2

Set $y$ equal to the new right side.

$y = - {(x + 1)}^{2} + 0$

Step 3

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

$a = - 1$

$h = - 1$

$k = 0$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 1, 0)$

Step 6

Find $p$ , the distance from the vertex to the focus.

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Step 6.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 6.2

Substitute the value of $a$ into the formula.

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

Step 6.3

Cancel the common factor of $1$ and $- 1$ .

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Step 6.3.1

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

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 6.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7

Find the focus.

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Step 7.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 7.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 1, - \frac{1}{4})$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = - 1$

Step 9