Pre-Algebra Examples

$h (x) = {(1 - 9 x)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Reorder terms.

$y = {(- 9 x + 1)}^{2}$

Step 1.2

Complete the square for ${(- 9 x + 1)}^{2}$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Rewrite ${(- 9 x + 1)}^{2}$ as $(- 9 x + 1) (- 9 x + 1)$ .

$(- 9 x + 1) (- 9 x + 1)$

Step 1.2.1.2

Expand $(- 9 x + 1) (- 9 x + 1)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$- 9 x (- 9 x + 1) + 1 (- 9 x + 1)$

Step 1.2.1.2.2

Apply the distributive property.

$- 9 x (- 9 x) - 9 x \cdot 1 + 1 (- 9 x + 1)$

Step 1.2.1.2.3

Apply the distributive property.

$- 9 x (- 9 x) - 9 x \cdot 1 + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$- 9 \cdot - 9 x \cdot x - 9 x \cdot 1 + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.3.1.2.1

Move $x$ .

$- 9 \cdot - 9 (x \cdot x) - 9 x \cdot 1 + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3.1.2.2

Multiply $x$ by $x$ .

$- 9 \cdot - 9 x^{2} - 9 x \cdot 1 + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3.1.3

Multiply $- 9$ by $- 9$ .

$81 x^{2} - 9 x \cdot 1 + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3.1.4

Multiply $- 9$ by $1$ .

$81 x^{2} - 9 x + 1 (- 9 x) + 1 \cdot 1$

Step 1.2.1.3.1.5

Multiply $- 9 x$ by $1$ .

$81 x^{2} - 9 x - 9 x + 1 \cdot 1$

Step 1.2.1.3.1.6

Multiply $1$ by $1$ .

$81 x^{2} - 9 x - 9 x + 1$

Step 1.2.1.3.2

Subtract $9 x$ from $- 9 x$ .

$81 x^{2} - 18 x + 1$

Step 1.2.2

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

$a = 81$

$b = - 18$

$c = 1$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{- 18}{2 \cdot 81}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 81}$

Step 1.2.4.2.1.2

Cancel the common factors.

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Step 1.2.4.2.1.2.1

Factor $2$ out of $2 \cdot 81$ .

$d = \frac{2 \cdot - 9}{2 (81)}$

Step 1.2.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 81}$

Step 1.2.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 9}{81}$

Step 1.2.4.2.2

Cancel the common factor of $- 9$ and $81$ .

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Step 1.2.4.2.2.1

Factor $9$ out of $- 9$ .

$d = \frac{9 (- 1)}{81}$

Step 1.2.4.2.2.2

Cancel the common factors.

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Step 1.2.4.2.2.2.1

Factor $9$ out of $81$ .

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

Step 1.2.4.2.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.2.2.3

Rewrite the expression.

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

Step 1.2.4.2.3

Move the negative in front of the fraction.

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

Step 1.2.5

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

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Step 1.2.5.1

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

$e = 1 - \frac{{(- 18)}^{2}}{4 \cdot 81}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

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

$e = 1 - \frac{324}{4 \cdot 81}$

Step 1.2.5.2.1.2

Multiply $4$ by $81$ .

$e = 1 - \frac{324}{324}$

Step 1.2.5.2.1.3

Cancel the common factor of $324$ .

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Step 1.2.5.2.1.3.1

Cancel the common factor.

$e = 1 - \frac{324}{324}$

Step 1.2.5.2.1.3.2

Rewrite the expression.

$e = 1 - 1 \cdot 1$

Step 1.2.5.2.1.4

Multiply $- 1$ by $1$ .

$e = 1 - 1$

Step 1.2.5.2.2

Subtract $1$ from $1$ .

$e = 0$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $81 {(x - \frac{1}{9})}^{2} + 0$ .

$81 {(x - \frac{1}{9})}^{2} + 0$

Step 1.3

Set $y$ equal to the new right side.

$y = 81 {(x - \frac{1}{9})}^{2} + 0$

Step 2

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

$a = 81$

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

$k = 0$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(\frac{1}{9}, 0)$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 81}$

Step 5.3

Multiply $4$ by $81$ .

$\frac{1}{324}$

Step 6

Find the focus.

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Step 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 6.2

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

$(\frac{1}{9}, \frac{1}{324})$

Step 7

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

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

Step 8

Find the directrix.

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Step 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 8.2

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

$y = - \frac{1}{324}$

Step 9

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

Direction: Opens Up

Vertex: $(\frac{1}{9}, 0)$

Focus: $(\frac{1}{9}, \frac{1}{324})$

Axis of Symmetry: $x = \frac{1}{9}$

Directrix: $y = - \frac{1}{324}$

Step 10