Precalculus Examples

$f (x) = x^{2} + 18 x$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $x^{2} + 18 x$ .

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Step 1.1.1.1

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

$a = 1$

$b = 18$

$c = 0$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{18}{2 \cdot 1}$

Step 1.1.1.3.2

Cancel the common factor of $18$ and $2$ .

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Step 1.1.1.3.2.1

Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Step 1.1.1.3.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 9}{2 (1)}$

Step 1.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Step 1.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{9}{1}$

Step 1.1.1.3.2.2.4

Divide $9$ by $1$ .

$d = 9$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = 0 - \frac{18^{2}}{4 \cdot 1}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raise $18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{324}{4}$

Step 1.1.1.4.2.1.3

Divide $324$ by $4$ .

$e = 0 - 1 \cdot 81$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 1.1.1.4.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 9)}^{2} - 81$ .

${(x + 9)}^{2} - 81$

Step 1.1.2

Set $y$ equal to the new right side.

$y = {(x + 9)}^{2} - 81$

Step 1.2

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

$a = 1$

$h = - 9$

$k = - 81$

Step 1.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 9, - 81)$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Cancel the common factor of $1$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(- 9, - \frac{323}{4})$

Step 1.7

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

$x = - 9$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$y = - \frac{325}{4}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 9, - 81)$

Focus: $(- 9, - \frac{323}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{325}{4}$

Direction: Opens Up

Vertex: $(- 9, - 81)$

Focus: $(- 9, - \frac{323}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{325}{4}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 2.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = {(- 10)}^{2} + 18 (- 10)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raise $- 10$ to the power of $2$ .

$f (- 10) = 100 + 18 (- 10)$

Step 2.2.1.2

Multiply $18$ by $- 10$ .

$f (- 10) = 100 - 180$

Step 2.2.2

Subtract $180$ from $100$ .

$f (- 10) = - 80$

Step 2.2.3

The final answer is $- 80$ .

$- 80$

Step 2.3

The $y$ value at $x = - 10$ is $- 80$ .

$y = - 80$

Step 2.4

Replace the variable $x$ with $- 11$ in the expression.

$f (- 11) = {(- 11)}^{2} + 18 (- 11)$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raise $- 11$ to the power of $2$ .

$f (- 11) = 121 + 18 (- 11)$

Step 2.5.1.2

Multiply $18$ by $- 11$ .

$f (- 11) = 121 - 198$

Step 2.5.2

Subtract $198$ from $121$ .

$f (- 11) = - 77$

Step 2.5.3

The final answer is $- 77$ .

$- 77$

Step 2.6

The $y$ value at $x = - 11$ is $- 77$ .

$y = - 77$

Step 2.7

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = {(- 8)}^{2} + 18 (- 8)$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Raise $- 8$ to the power of $2$ .

$f (- 8) = 64 + 18 (- 8)$

Step 2.8.1.2

Multiply $18$ by $- 8$ .

$f (- 8) = 64 - 144$

Step 2.8.2

Subtract $144$ from $64$ .

$f (- 8) = - 80$

Step 2.8.3

The final answer is $- 80$ .

$- 80$

Step 2.9

The $y$ value at $x = - 8$ is $- 80$ .

$y = - 80$

Step 2.10

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = {(- 7)}^{2} + 18 (- 7)$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

Raise $- 7$ to the power of $2$ .

$f (- 7) = 49 + 18 (- 7)$

Step 2.11.1.2

Multiply $18$ by $- 7$ .

$f (- 7) = 49 - 126$

Step 2.11.2

Subtract $126$ from $49$ .

$f (- 7) = - 77$

Step 2.11.3

The final answer is $- 77$ .

$- 77$

Step 2.12

The $y$ value at $x = - 7$ is $- 77$ .

$y = - 77$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 11 & - 77 \\ - 10 & - 80 \\ - 9 & - 81 \\ - 8 & - 80 \\ - 7 & - 77 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 9, - 81)$

Focus: $(- 9, - \frac{323}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{325}{4}$

$\begin{array}{c} x & y \\ - 11 & - 77 \\ - 10 & - 80 \\ - 9 & - 81 \\ - 8 & - 80 \\ - 7 & - 77 \end{array}$

Step 4