Examples

$f (x) = - x^{2} - 5 x - 5$

Step 1

Write $f (x) = - x^{2} - 5 x - 5$ as an equation.

$y = - x^{2} - 5 x - 5$

Step 2

Rewrite the equation in vertex form.

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Step 2.1

Complete the square for $- x^{2} - 5 x - 5$ .

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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 $c$ .

$a = - 1$

$b = - 5$

$c = - 5$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 5}{2 \cdot - 1}$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Multiply $2$ by $- 1$ .

$d = \frac{- 5}{- 2}$

Step 2.1.3.2.2

Dividing two negative values results in a positive value.

$d = \frac{5}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 5 - \frac{{(- 5)}^{2}}{4 \cdot - 1}$

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

$e = - 5 - \frac{25}{4 \cdot - 1}$

Step 2.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 5 - \frac{25}{- 4}$

Step 2.1.4.2.1.3

Move the negative in front of the fraction.

$e = - 5 - - \frac{25}{4}$

Step 2.1.4.2.1.4

Multiply $- - \frac{25}{4}$ .

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Step 2.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = - 5 + 1 (\frac{25}{4})$

Step 2.1.4.2.1.4.2

Multiply $\frac{25}{4}$ by $1$ .

$e = - 5 + \frac{25}{4}$

Step 2.1.4.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.1.4.2.3

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

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

Step 2.1.4.2.4

Combine the numerators over the common denominator.

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

Step 2.1.4.2.5

Simplify the numerator.

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Step 2.1.4.2.5.1

Multiply $- 5$ by $4$ .

$e = \frac{- 20 + 25}{4}$

Step 2.1.4.2.5.2

Add $- 20$ and $25$ .

$e = \frac{5}{4}$

Step 2.1.5

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

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

Step 2.2

Set $y$ equal to the new right side.

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

Step 3

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

$a = - 1$

$h = - \frac{5}{2}$

$k = \frac{5}{4}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- \frac{5}{2}, \frac{5}{4})$

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.

$(- \frac{5}{2}, 1)$

Step 8

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

$x = - \frac{5}{2}$

Step 9

Find the directrix.

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Step 9.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 9.2

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

$y = \frac{3}{2}$

Step 10

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(- \frac{5}{2}, \frac{5}{4})$

Focus: $(- \frac{5}{2}, 1)$

Axis of Symmetry: $x = - \frac{5}{2}$

Directrix: $y = \frac{3}{2}$

Step 11

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