Trigonometry Examples

${(y - 3)}^{2} = 8 (x + 2)$

Step 1

Rewrite the equation as $8 (x + 2) = {(y - 3)}^{2}$ .

$8 (x + 2) = {(y - 3)}^{2}$

Step 2

Divide each term in $8 (x + 2) = {(y - 3)}^{2}$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 (x + 2) = {(y - 3)}^{2}$ by $8$ .

$\frac{8 (x + 2)}{8} = \frac{{(y - 3)}^{2}}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 (x + 2)}{8} = \frac{{(y - 3)}^{2}}{8}$

Step 2.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{{(y - 3)}^{2}}{8}$

Step 3

Subtract $2$ from both sides of the equation.

$x = \frac{{(y - 3)}^{2}}{8} - 2$

Step 4

Find the properties of the given parabola.

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Step 4.1

Reorder terms.

$x = \frac{1}{8} \cdot {(y - 3)}^{2} - 2$

Step 4.2

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

$a = \frac{1}{8}$

$h = - 2$

$k = 3$

Step 4.3

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

Opens Right

Step 4.4

Find the vertex $(h, k)$ .

$(- 2, 3)$

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.

$\frac{1}{4 \cdot \frac{1}{8}}$

Step 4.5.3

Simplify.

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Step 4.5.3.1

Combine $4$ and $\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Step 4.5.3.2

Cancel the common factor of $4$ and $8$ .

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Step 4.5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Step 4.5.3.2.2

Cancel the common factors.

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Step 4.5.3.2.2.1

Factor $4$ out of $8$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 4.5.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 4.5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{2}}$

Step 4.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 4.5.3.4

Multiply $2$ by $1$ .

$2$

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.

$(0, 3)$

Step 4.7

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

$y = 3$

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.

$x = - 4$

Step 4.9

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

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(0, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - 4$

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(0, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - 4$

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 $- 1$ into $f (x) = 2 \sqrt{2 (x + 2)} + 3$ . In this case, the point is $(- 1, 5.82842712)$ .

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Step 5.1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 2 \sqrt{2 ((- 1) + 2)} + 3$

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

Add $- 1$ and $2$ .

$f (- 1) = 2 \sqrt{2 \cdot 1} + 3$

Step 5.1.2.1.2

Multiply $2$ by $1$ .

$f (- 1) = 2 \sqrt{2} + 3$

Step 5.1.2.2

The final answer is $2 \sqrt{2} + 3$ .

$y = 2 \sqrt{2} + 3$

Step 5.1.3

Convert $2 \sqrt{2} + 3$ to decimal.

$= 5.82842712$

Step 5.2

Substitute the $x$ value $- 1$ into $f (x) = - 2 \sqrt{2 (x + 2)} + 3$ . In this case, the point is $(- 1, 0.17157287)$ .

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Step 5.2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - 2 \sqrt{2 ((- 1) + 2)} + 3$

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

Add $- 1$ and $2$ .

$f (- 1) = - 2 \sqrt{2 \cdot 1} + 3$

Step 5.2.2.1.2

Multiply $2$ by $1$ .

$f (- 1) = - 2 \sqrt{2} + 3$

Step 5.2.2.2

The final answer is $- 2 \sqrt{2} + 3$ .

$y = - 2 \sqrt{2} + 3$

Step 5.2.3

Convert $- 2 \sqrt{2} + 3$ to decimal.

$= 0.17157287$

Step 5.3

Substitute the $x$ value $0$ into $f (x) = 2 \sqrt{2 (x + 2)} + 3$ . In this case, the point is $(0, 7)$ .

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Step 5.3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 \sqrt{2 ((0) + 2)} + 3$

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Add $0$ and $2$ .

$f (0) = 2 \sqrt{2 \cdot 2} + 3$

Step 5.3.2.1.2

Multiply $2$ by $2$ .

$f (0) = 2 \sqrt{4} + 3$

Step 5.3.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (0) = 2 \sqrt{2^{2}} + 3$

Step 5.3.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 2 \cdot 2 + 3$

Step 5.3.2.1.5

Multiply $2$ by $2$ .

$f (0) = 4 + 3$

Step 5.3.2.2

Add $4$ and $3$ .

$f (0) = 7$

Step 5.3.2.3

The final answer is $7$ .

$y = 7$

Step 5.3.3

Convert $7$ to decimal.

$= 7$

Step 5.4

Substitute the $x$ value $0$ into $f (x) = - 2 \sqrt{2 (x + 2)} + 3$ . In this case, the point is $(0, - 1)$ .

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Step 5.4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 2 \sqrt{2 ((0) + 2)} + 3$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Add $0$ and $2$ .

$f (0) = - 2 \sqrt{2 \cdot 2} + 3$

Step 5.4.2.1.2

Multiply $2$ by $2$ .

$f (0) = - 2 \sqrt{4} + 3$

Step 5.4.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (0) = - 2 \sqrt{2^{2}} + 3$

Step 5.4.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = - 2 \cdot 2 + 3$

Step 5.4.2.1.5

Multiply $- 2$ by $2$ .

$f (0) = - 4 + 3$

Step 5.4.2.2

Add $- 4$ and $3$ .

$f (0) = - 1$

Step 5.4.2.3

The final answer is $- 1$ .

$y = - 1$

Step 5.4.3

Convert $- 1$ to decimal.

$= - 1$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 5.83 \\ - 1 & 0.17 \\ 0 & 7 \\ 0 & - 1 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(0, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - 4$

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 5.83 \\ - 1 & 0.17 \\ 0 & 7 \\ 0 & - 1 \end{array}$

Step 7