Trigonometry Examples

${(x + 1)}^{2} + {(v - 4)}^{2} = {(x + 3)}^{2}$

Step 1

Solve for $y$ .

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Step 1.1

Subtract ${(v - 4)}^{2}$ from both sides of the equation.

${(x + 1)}^{2} = {(x + 3)}^{2} - {(v - 4)}^{2}$

Step 1.2

Subtract ${(x + 3)}^{2}$ from both sides of the equation.

${(x + 1)}^{2} - {(x + 3)}^{2} = - {(v - 4)}^{2}$

Step 1.3

Simplify $- {(v - 4)}^{2}$ .

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Step 1.3.1

Rewrite ${(v - 4)}^{2}$ as $(v - 4) (v - 4)$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - ((v - 4) (v - 4))$

Step 1.3.2

Expand $(v - 4) (v - 4)$ using the FOIL Method.

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Step 1.3.2.1

Apply the distributive property.

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v (v - 4) - 4 (v - 4))$

Step 1.3.2.2

Apply the distributive property.

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v \cdot v + v \cdot - 4 - 4 (v - 4))$

Step 1.3.2.3

Apply the distributive property.

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v \cdot v + v \cdot - 4 - 4 v - 4 \cdot - 4)$

Step 1.3.3

Simplify and combine like terms.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $v$ by $v$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v^{2} + v \cdot - 4 - 4 v - 4 \cdot - 4)$

Step 1.3.3.1.2

Move $- 4$ to the left of $v$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v^{2} - 4 \cdot v - 4 v - 4 \cdot - 4)$

Step 1.3.3.1.3

Multiply $- 4$ by $- 4$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v^{2} - 4 v - 4 v + 16)$

Step 1.3.3.2

Subtract $4 v$ from $- 4 v$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - (v^{2} - 8 v + 16)$

Step 1.3.4

Apply the distributive property.

${(x + 1)}^{2} - {(x + 3)}^{2} = - v^{2} - (- 8 v) - 1 \cdot 16$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Multiply $- 8$ by $- 1$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - v^{2} + 8 v - 1 \cdot 16$

Step 1.3.5.2

Multiply $- 1$ by $16$ .

${(x + 1)}^{2} - {(x + 3)}^{2} = - v^{2} + 8 v - 16$

Step 1.4

Move all terms to the left side of the equation and simplify.

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Step 1.4.1

Move all the expressions to the left side of the equation.

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Step 1.4.1.1

Add $v^{2}$ to both sides of the equation.

${(x + 1)}^{2} - {(x + 3)}^{2} + v^{2} = 8 v - 16$

Step 1.4.1.2

Subtract $8 v$ from both sides of the equation.

${(x + 1)}^{2} - {(x + 3)}^{2} + v^{2} - 8 v = - 16$

Step 1.4.1.3

Add $16$ to both sides of the equation.

${(x + 1)}^{2} - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2

Simplify ${(x + 1)}^{2} - {(x + 3)}^{2} + v^{2} - 8 v + 16$ .

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

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

$(x + 1) (x + 1) - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.2

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

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Step 1.4.2.1.2.1

Apply the distributive property.

$x (x + 1) + 1 (x + 1) - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1) - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.3

Simplify and combine like terms.

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Step 1.4.2.1.3.1

Simplify each term.

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Step 1.4.2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.3.1.2

Multiply $x$ by $1$ .

$x^{2} + x + 1 x + 1 \cdot 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.3.1.3

Multiply $x$ by $1$ .

$x^{2} + x + x + 1 \cdot 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.3.1.4

Multiply $1$ by $1$ .

$x^{2} + x + x + 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.3.2

Add $x$ and $x$ .

$x^{2} + 2 x + 1 - {(x + 3)}^{2} + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.4

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$x^{2} + 2 x + 1 - ((x + 3) (x + 3)) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.5

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.4.2.1.5.1

Apply the distributive property.

$x^{2} + 2 x + 1 - (x (x + 3) + 3 (x + 3)) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.5.2

Apply the distributive property.

$x^{2} + 2 x + 1 - (x \cdot x + x \cdot 3 + 3 (x + 3)) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.5.3

Apply the distributive property.

$x^{2} + 2 x + 1 - (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.6

Simplify and combine like terms.

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Step 1.4.2.1.6.1

Simplify each term.

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Step 1.4.2.1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + 2 x + 1 - (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.6.1.2

Move $3$ to the left of $x$ .

$x^{2} + 2 x + 1 - (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.6.1.3

Multiply $3$ by $3$ .

$x^{2} + 2 x + 1 - (x^{2} + 3 x + 3 x + 9) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.6.2

Add $3 x$ and $3 x$ .

$x^{2} + 2 x + 1 - (x^{2} + 6 x + 9) + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.7

Apply the distributive property.

$x^{2} + 2 x + 1 - x^{2} - (6 x) - 1 \cdot 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.8

Simplify.

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Step 1.4.2.1.8.1

Multiply $6$ by $- 1$ .

$x^{2} + 2 x + 1 - x^{2} - 6 x - 1 \cdot 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.1.8.2

Multiply $- 1$ by $9$ .

$x^{2} + 2 x + 1 - x^{2} - 6 x - 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.2

Combine the opposite terms in $x^{2} + 2 x + 1 - x^{2} - 6 x - 9 + v^{2} - 8 v + 16$ .

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Step 1.4.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$2 x + 1 + 0 - 6 x - 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.2.2

Add $2 x + 1$ and $0$ .

$2 x + 1 - 6 x - 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.3

Subtract $6 x$ from $2 x$ .

$- 4 x + 1 - 9 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.4

Subtract $9$ from $1$ .

$- 4 x - 8 + v^{2} - 8 v + 16 = 0$

Step 1.4.2.5

Add $- 8$ and $16$ .

$- 4 x + v^{2} - 8 v + 8 = 0$

Step 1.5

Move all terms not containing $x$ to the right side of the equation.

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Step 1.5.1

Subtract $v^{2}$ from both sides of the equation.

$- 4 x - 8 v + 8 = - v^{2}$

Step 1.5.2

Add $8 v$ to both sides of the equation.

$- 4 x + 8 = - v^{2} + 8 v$

Step 1.5.3

Subtract $8$ from both sides of the equation.

$- 4 x = - v^{2} + 8 v - 8$

Step 1.6

Divide each term in $- 4 x = - v^{2} + 8 v - 8$ by $- 4$ and simplify.

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Step 1.6.1

Divide each term in $- 4 x = - v^{2} + 8 v - 8$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- v^{2}}{- 4} + \frac{8 v}{- 4} + \frac{- 8}{- 4}$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $- 4$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- v^{2}}{- 4} + \frac{8 v}{- 4} + \frac{- 8}{- 4}$

Step 1.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- v^{2}}{- 4} + \frac{8 v}{- 4} + \frac{- 8}{- 4}$

Step 1.6.3

Simplify the right side.

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Step 1.6.3.1

Simplify each term.

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Step 1.6.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{v^{2}}{4} + \frac{8 v}{- 4} + \frac{- 8}{- 4}$

Step 1.6.3.1.2

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

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Step 1.6.3.1.2.1

Factor $4$ out of $8 v$ .

$x = \frac{v^{2}}{4} + \frac{4 (2 v)}{- 4} + \frac{- 8}{- 4}$

Step 1.6.3.1.2.2

Move the negative one from the denominator of $\frac{2 v}{- 1}$ .

$x = \frac{v^{2}}{4} - 1 \cdot (2 v) + \frac{- 8}{- 4}$

Step 1.6.3.1.3

Rewrite $- 1 \cdot (2 v)$ as $- (2 v)$ .

$x = \frac{v^{2}}{4} - (2 v) + \frac{- 8}{- 4}$

Step 1.6.3.1.4

Multiply $2$ by $- 1$ .

$x = \frac{v^{2}}{4} - 2 v + \frac{- 8}{- 4}$

Step 1.6.3.1.5

Divide $- 8$ by $- 4$ .

$x = \frac{v^{2}}{4} - 2 v + 2$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $- v^{2} + 8 v - 16$ .

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Step 2.1.1.1

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

$a = - 1$

$b = 8$

$c = - 16$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$d = \frac{8}{2 \cdot - 1}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $8$ and $2$ .

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Step 2.1.1.3.2.1.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot - 1}$

Step 2.1.1.3.2.1.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$d = - 1 \cdot 4$

Step 2.1.1.3.2.2

Multiply $- 1$ by $4$ .

$d = - 4$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = - 16 - \frac{8^{2}}{4 \cdot - 1}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Raise $8$ to the power of $2$ .

$e = - 16 - \frac{64}{4 \cdot - 1}$

Step 2.1.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = - 16 - \frac{64}{- 4}$

Step 2.1.1.4.2.1.3

Divide $64$ by $- 4$ .

$e = - 16 - - 16$

Step 2.1.1.4.2.1.4

Multiply $- 1$ by $- 16$ .

$e = - 16 + 16$

Step 2.1.1.4.2.2

Add $- 16$ and $16$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(v - 4)}^{2} + 0$ .

$- {(v - 4)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - {(v - 4)}^{2} + 0$

Step 2.2

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

$a = - 1$

$h = 4$

$k = 0$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(4, 0)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.5.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot - 1}$

Step 2.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(4, - \frac{1}{4})$

Step 2.7

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

$v = 4$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

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

Step 2.9

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

Direction: Opens Down

Vertex: $(4, 0)$

Focus: $(4, - \frac{1}{4})$

Axis of Symmetry: $v = 4$

Directrix: $y = \frac{1}{4}$

Direction: Opens Down

Vertex: $(4, 0)$

Focus: $(4, - \frac{1}{4})$

Axis of Symmetry: $v = 4$

Directrix: $y = \frac{1}{4}$

Step 3