Trigonometry Examples

$(2 x - 32 x - 32 x - 3) (x)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $(2 x - 32 x - 32 x - 3) (x)$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Subtract $32 x$ from $2 x$ .

$(- 30 x - 32 x - 3) x$

Step 1.1.1.2

Subtract $32 x$ from $- 30 x$ .

$(- 62 x - 3) x$

Step 1.1.1.3

Apply the distributive property.

$- 62 x \cdot x - 3 x$

Step 1.1.1.4

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

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Step 1.1.1.4.1

Move $x$ .

$- 62 (x \cdot x) - 3 x$

Step 1.1.1.4.2

Multiply $x$ by $x$ .

$- 62 x^{2} - 3 x$

Step 1.1.2

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

$a = - 62$

$b = - 3$

$c = 0$

Step 1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.4

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

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Step 1.1.4.1

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

$d = \frac{- 3}{2 \cdot - 62}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Multiply $2$ by $- 62$ .

$d = \frac{- 3}{- 124}$

Step 1.1.4.2.2

Dividing two negative values results in a positive value.

$d = \frac{3}{124}$

Step 1.1.5

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

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Step 1.1.5.1

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

$e = 0 - \frac{{(- 3)}^{2}}{4 \cdot - 62}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

Raise $- 3$ to the power of $2$ .

$e = 0 - \frac{9}{4 \cdot - 62}$

Step 1.1.5.2.1.2

Multiply $4$ by $- 62$ .

$e = 0 - \frac{9}{- 248}$

Step 1.1.5.2.1.3

Move the negative in front of the fraction.

$e = 0 - - \frac{9}{248}$

Step 1.1.5.2.1.4

Multiply $- - \frac{9}{248}$ .

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Step 1.1.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{9}{248})$

Step 1.1.5.2.1.4.2

Multiply $\frac{9}{248}$ by $1$ .

$e = 0 + \frac{9}{248}$

Step 1.1.5.2.2

Add $0$ and $\frac{9}{248}$ .

$e = \frac{9}{248}$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 62 {(x + \frac{3}{124})}^{2} + \frac{9}{248}$ .

$- 62 {(x + \frac{3}{124})}^{2} + \frac{9}{248}$

Step 1.2

Set $y$ equal to the new right side.

$y = - 62 {(x + \frac{3}{124})}^{2} + \frac{9}{248}$

Step 2

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

$a = - 62$

$h = - \frac{3}{124}$

$k = \frac{9}{248}$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- \frac{3}{124}, \frac{9}{248})$

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.

$\frac{1}{4 \cdot - 62}$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 62$ .

$\frac{1}{- 248}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{248}$

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.

$(- \frac{3}{124}, \frac{1}{31})$

Step 7

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

$x = - \frac{3}{124}$

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.

$y = \frac{5}{124}$

Step 9

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

Direction: Opens Down

Vertex: $(- \frac{3}{124}, \frac{9}{248})$

Focus: $(- \frac{3}{124}, \frac{1}{31})$

Axis of Symmetry: $x = - \frac{3}{124}$

Directrix: $y = \frac{5}{124}$

Step 10