Trigonometry Examples

${(y + 6)}^{2} = 4 (x - 4)$

Step 1

Rewrite the equation as $4 (x - 4) = {(y + 6)}^{2}$ .

$4 (x - 4) = {(y + 6)}^{2}$

Step 2

Divide each term in $4 (x - 4) = {(y + 6)}^{2}$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 (x - 4) = {(y + 6)}^{2}$ by $4$ .

$\frac{4 (x - 4)}{4} = \frac{{(y + 6)}^{2}}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 (x - 4)}{4} = \frac{{(y + 6)}^{2}}{4}$

Step 2.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{{(y + 6)}^{2}}{4}$

Step 3

Add $4$ to both sides of the equation.

$x = \frac{{(y + 6)}^{2}}{4} + 4$

Step 4

Find the properties of the given parabola.

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Step 4.1

Reorder terms.

$x = \frac{1}{4} \cdot {(y + 6)}^{2} + 4$

Step 4.2

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

$a = \frac{1}{4}$

$h = 4$

$k = - 6$

Step 4.3

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

Opens Right

Step 4.4

Find the vertex $(h, k)$ .

$(4, - 6)$

Step 4.5

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

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Step 4.5.1

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

$\frac{1}{4 a}$

Step 4.5.2

Substitute the value of $a$ into the formula.

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

Step 4.5.3

Simplify.

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Step 4.5.3.1

Combine $4$ and $\frac{1}{4}$ .

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

Step 4.5.3.2

Simplify by dividing numbers.

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Step 4.5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 4.5.3.2.2

Divide $1$ by $1$ .

$1$

Step 4.6

Find the focus.

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Step 4.6.1

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 4.6.2

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

$(5, - 6)$

Step 4.7

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

$y = - 6$

Step 4.8

Find the directrix.

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Step 4.8.1

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 4.8.2

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

$x = 3$

Step 4.9

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

Direction: Opens Right

Vertex: $(4, - 6)$

Focus: $(5, - 6)$

Axis of Symmetry: $y = - 6$

Directrix: $x = 3$

Direction: Opens Right

Vertex: $(4, - 6)$

Focus: $(5, - 6)$

Axis of Symmetry: $y = - 6$

Directrix: $x = 3$

Step 5

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 5.1

Substitute the $x$ value $5$ into $f (x) = 2 \sqrt{x - 4} - 6$ . In this case, the point is $(5, - 4)$ .

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Step 5.1.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = 2 \sqrt{(5) - 4} - 6$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Subtract $4$ from $5$ .

$f (5) = 2 \sqrt{1} - 6$

Step 5.1.2.1.2

Any root of $1$ is $1$ .

$f (5) = 2 \cdot 1 - 6$

Step 5.1.2.1.3

Multiply $2$ by $1$ .

$f (5) = 2 - 6$

Step 5.1.2.2

Subtract $6$ from $2$ .

$f (5) = - 4$

Step 5.1.2.3

The final answer is $- 4$ .

$y = - 4$

Step 5.1.3

Convert $- 4$ to decimal.

$= - 4$

Step 5.2

Substitute the $x$ value $5$ into $f (x) = - 2 \sqrt{x - 4} - 6$ . In this case, the point is $(5, - 8)$ .

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Step 5.2.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - 2 \sqrt{(5) - 4} - 6$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Subtract $4$ from $5$ .

$f (5) = - 2 \sqrt{1} - 6$

Step 5.2.2.1.2

Any root of $1$ is $1$ .

$f (5) = - 2 \cdot 1 - 6$

Step 5.2.2.1.3

Multiply $- 2$ by $1$ .

$f (5) = - 2 - 6$

Step 5.2.2.2

Subtract $6$ from $- 2$ .

$f (5) = - 8$

Step 5.2.2.3

The final answer is $- 8$ .

$y = - 8$

Step 5.2.3

Convert $- 8$ to decimal.

$= - 8$

Step 5.3

Substitute the $x$ value $6$ into $f (x) = 2 \sqrt{x - 4} - 6$ . In this case, the point is $(6, - 3.17157287)$ .

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Step 5.3.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 2 \sqrt{(6) - 4} - 6$

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Subtract $4$ from $6$ .

$f (6) = 2 \sqrt{2} - 6$

Step 5.3.2.2

The final answer is $2 \sqrt{2} - 6$ .

$y = 2 \sqrt{2} - 6$

Step 5.3.3

Convert $2 \sqrt{2} - 6$ to decimal.

$= - 3.17157287$

Step 5.4

Substitute the $x$ value $6$ into $f (x) = - 2 \sqrt{x - 4} - 6$ . In this case, the point is $(6, - 8.82842712)$ .

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Step 5.4.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = - 2 \sqrt{(6) - 4} - 6$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Subtract $4$ from $6$ .

$f (6) = - 2 \sqrt{2} - 6$

Step 5.4.2.2

The final answer is $- 2 \sqrt{2} - 6$ .

$y = - 2 \sqrt{2} - 6$

Step 5.4.3

Convert $- 2 \sqrt{2} - 6$ to decimal.

$= - 8.82842712$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & - 6 \\ 5 & - 4 \\ 5 & - 8 \\ 6 & - 3.17 \\ 6 & - 8.83 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(4, - 6)$

Focus: $(5, - 6)$

Axis of Symmetry: $y = - 6$

Directrix: $x = 3$

$\begin{array}{c} x & y \\ 4 & - 6 \\ 5 & - 4 \\ 5 & - 8 \\ 6 & - 3.17 \\ 6 & - 8.83 \end{array}$

Step 7