Precalculus Examples

${(y - 2)}^{2} = 4 (1.5) (x - 4.5)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as $4 (1.5) (x - 4.5) = {(y - 2)}^{2}$ .

$4 (1.5) (x - 4.5) = {(y - 2)}^{2}$

Step 1.1.2

Multiply $4$ by $1.5$ .

$6 (x - 4.5) = {(y - 2)}^{2}$

Step 1.1.3

Divide each term in $6 (x - 4.5) = {(y - 2)}^{2}$ by $6$ and simplify.

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Step 1.1.3.1

Divide each term in $6 (x - 4.5) = {(y - 2)}^{2}$ by $6$ .

$\frac{6 (x - 4.5)}{6} = \frac{{(y - 2)}^{2}}{6}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $6$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{6 (x - 4.5)}{6} = \frac{{(y - 2)}^{2}}{6}$

Step 1.1.3.2.1.2

Divide $x - 4.5$ by $1$ .

$x - 4.5 = \frac{{(y - 2)}^{2}}{6}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Multiply by $1$ .

$x - 4.5 = \frac{1 {(y - 2)}^{2}}{6}$

Step 1.1.3.3.2

Factor $6$ out of $6$ .

$x - 4.5 = \frac{1 {(y - 2)}^{2}}{6 (1)}$

Step 1.1.3.3.3

Separate fractions.

$x - 4.5 = \frac{1}{6} \cdot \frac{{(y - 2)}^{2}}{1}$

Step 1.1.3.3.4

Divide $1$ by $6$ .

$x - 4.5 = 0.1\overline{6} \frac{{(y - 2)}^{2}}{1}$

Step 1.1.3.3.5

Divide ${(y - 2)}^{2}$ by $1$ .

$x - 4.5 = 0.1\overline{6} {(y - 2)}^{2}$

Step 1.1.4

Add $4.5$ to both sides of the equation.

$x = 0.1\overline{6} {(y - 2)}^{2} + 4.5$

Step 1.2

Complete the square for $0.1\overline{6} {(y - 2)}^{2} + 4.5$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(y - 2)}^{2}$ as $(y - 2) (y - 2)$ .

$0.1\overline{6} ((y - 2) (y - 2)) + 4.5$

Step 1.2.1.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$0.1\overline{6} (y (y - 2) - 2 (y - 2)) + 4.5$

Step 1.2.1.1.2.2

Apply the distributive property.

$0.1\overline{6} (y \cdot y + y \cdot - 2 - 2 (y - 2)) + 4.5$

Step 1.2.1.1.2.3

Apply the distributive property.

$0.1\overline{6} (y \cdot y + y \cdot - 2 - 2 y - 2 \cdot - 2) + 4.5$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$0.1\overline{6} (y^{2} + y \cdot - 2 - 2 y - 2 \cdot - 2) + 4.5$

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $y$ .

$0.1\overline{6} (y^{2} - 2 \cdot y - 2 y - 2 \cdot - 2) + 4.5$

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$0.1\overline{6} (y^{2} - 2 y - 2 y + 4) + 4.5$

Step 1.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

$0.1\overline{6} (y^{2} - 4 y + 4) + 4.5$

Step 1.2.1.1.4

Apply the distributive property.

$0.1\overline{6} y^{2} + 0.1\overline{6} (- 4 y) + 0.1\overline{6} \cdot 4 + 4.5$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 4$ by $0.1\overline{6}$ .

$0.1\overline{6} y^{2} - 0.\overline{6} y + 0.1\overline{6} \cdot 4 + 4.5$

Step 1.2.1.1.5.2

Multiply $0.1\overline{6}$ by $4$ .

$0.1\overline{6} y^{2} - 0.\overline{6} y + 0.\overline{6} + 4.5$

Step 1.2.1.2

Add $0.\overline{6}$ and $4.5$ .

$0.1\overline{6} y^{2} - 0.\overline{6} y + 5.1\overline{6}$

Step 1.2.2

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

$a = 0.1\overline{6}$

$b = - 0.\overline{6}$

$c = 5.1\overline{6}$

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

$d = \frac{- 0.\overline{6}}{2 \cdot 0.1\overline{6}}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $2$ by $0.1\overline{6}$ .

$d = \frac{- 0.\overline{6}}{0.\overline{3}}$

Step 1.2.4.2.2

Divide $- 0.\overline{6}$ by $0.\overline{3}$ .

$d = - 2$

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

$e = 5.1\overline{6} - \frac{{(- 0.\overline{6})}^{2}}{4 \cdot 0.1\overline{6}}$

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 $- 0.\overline{6}$ to the power of $2$ .

$e = 5.1\overline{6} - \frac{0.\overline{4}}{4 \cdot 0.1\overline{6}}$

Step 1.2.5.2.1.2

Multiply $4$ by $0.1\overline{6}$ .

$e = 5.1\overline{6} - \frac{0.\overline{4}}{0.\overline{6}}$

Step 1.2.5.2.1.3

Divide $0.\overline{4}$ by $0.\overline{6}$ .

$e = 5.1\overline{6} - 1 \cdot 0.66666666$

Step 1.2.5.2.1.4

Multiply $- 1$ by $0.66666666$ .

$e = 5.1\overline{6} - 0.66666666$

Step 1.2.5.2.2

Subtract $0.66666666$ from $5.1\overline{6}$ .

$e = 4.5$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $0.1\overline{6} {(y - 2)}^{2} + 4.5$ .

$0.1\overline{6} {(y - 2)}^{2} + 4.5$

Step 1.3

Set $x$ equal to the new right side.

$x = 0.1\overline{6} {(y - 2)}^{2} + 4.5$

Step 2

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

$a = 0.1\overline{6}$

$h = 4.5$

$k = 2$

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(4.5, 2)$

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.

$\frac{1}{4 \cdot 0.1\overline{6}}$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $0.1\overline{6}$ .

$\frac{1}{0.\overline{6}}$

Step 5.3.2

Divide $1$ by $0.\overline{6}$ .

$1.4\overline{9}$

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 x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

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

$(6, 2)$

Step 7

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

$y = 2$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 8.2

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

$x = 3$

Step 9

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

Direction: Opens Right

Vertex: $(4.5, 2)$

Focus: $(6, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = 3$

Step 10