Trigonometry Examples

$y = 4 (x - 13) (x + 5)$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $4 (x - 13) (x + 5)$ .

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Step 1.1.1.1

Simplify the expression.

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Step 1.1.1.1.1

Apply the distributive property.

$(4 x + 4 \cdot - 13) (x + 5)$

Step 1.1.1.1.2

Multiply $4$ by $- 13$ .

$(4 x - 52) (x + 5)$

Step 1.1.1.1.3

Expand $(4 x - 52) (x + 5)$ using the FOIL Method.

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Step 1.1.1.1.3.1

Apply the distributive property.

$4 x (x + 5) - 52 (x + 5)$

Step 1.1.1.1.3.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 5 - 52 (x + 5)$

Step 1.1.1.1.3.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 5 - 52 x - 52 \cdot 5$

Step 1.1.1.1.4

Simplify and combine like terms.

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Step 1.1.1.1.4.1

Simplify each term.

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Step 1.1.1.1.4.1.1

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

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Step 1.1.1.1.4.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot 5 - 52 x - 52 \cdot 5$

Step 1.1.1.1.4.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot 5 - 52 x - 52 \cdot 5$

Step 1.1.1.1.4.1.2

Multiply $5$ by $4$ .

$4 x^{2} + 20 x - 52 x - 52 \cdot 5$

Step 1.1.1.1.4.1.3

Multiply $- 52$ by $5$ .

$4 x^{2} + 20 x - 52 x - 260$

Step 1.1.1.1.4.2

Subtract $52 x$ from $20 x$ .

$4 x^{2} - 32 x - 260$

Step 1.1.1.2

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

$a = 4$

$b = - 32$

$c = - 260$

Step 1.1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$d = \frac{- 32}{2 \cdot 4}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Cancel the common factor of $- 32$ and $2$ .

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Step 1.1.1.4.2.1.1

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

Step 1.1.1.4.2.1.2

Cancel the common factors.

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Step 1.1.1.4.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 16}{2 (4)}$

Step 1.1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot 4}$

Step 1.1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 16}{4}$

Step 1.1.1.4.2.2

Cancel the common factor of $- 16$ and $4$ .

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Step 1.1.1.4.2.2.1

Factor $4$ out of $- 16$ .

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

Step 1.1.1.4.2.2.2

Cancel the common factors.

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Step 1.1.1.4.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.1.1.4.2.2.2.2

Cancel the common factor.

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

Step 1.1.1.4.2.2.2.3

Rewrite the expression.

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

Step 1.1.1.4.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 1.1.1.5

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

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Step 1.1.1.5.1

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

$e = - 260 - \frac{{(- 32)}^{2}}{4 \cdot 4}$

Step 1.1.1.5.2

Simplify the right side.

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Step 1.1.1.5.2.1

Simplify each term.

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Step 1.1.1.5.2.1.1

Raise $- 32$ to the power of $2$ .

$e = - 260 - \frac{1024}{4 \cdot 4}$

Step 1.1.1.5.2.1.2

Multiply $4$ by $4$ .

$e = - 260 - \frac{1024}{16}$

Step 1.1.1.5.2.1.3

Divide $1024$ by $16$ .

$e = - 260 - 1 \cdot 64$

Step 1.1.1.5.2.1.4

Multiply $- 1$ by $64$ .

$e = - 260 - 64$

Step 1.1.1.5.2.2

Subtract $64$ from $- 260$ .

$e = - 324$

Step 1.1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 4)}^{2} - 324$ .

$4 {(x - 4)}^{2} - 324$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 4 {(x - 4)}^{2} - 324$

Step 1.2

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

$a = 4$

$h = 4$

$k = - 324$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(4, - 324)$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Multiply $4$ by $4$ .

$\frac{1}{16}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(4, - \frac{5183}{16})$

Step 1.7

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

$x = 4$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$y = - \frac{5185}{16}$

Step 1.9

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

Direction: Opens Up

Vertex: $(4, - 324)$

Focus: $(4, - \frac{5183}{16})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5185}{16}$

Direction: Opens Up

Vertex: $(4, - 324)$

Focus: $(4, - \frac{5183}{16})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5185}{16}$

Step 2

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 2.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 4 ((3) - 13) ((3) + 5)$

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $13$ from $3$ .

$f (3) = 4 \cdot (- 10 ((3) + 5))$

Step 2.2.2

Multiply $4$ by $- 10$ .

$f (3) = - 40 ((3) + 5)$

Step 2.2.3

Add $3$ and $5$ .

$f (3) = - 40 \cdot 8$

Step 2.2.4

Multiply $- 40$ by $8$ .

$f (3) = - 320$

Step 2.2.5

The final answer is $- 320$ .

$- 320$

Step 2.3

The $y$ value at $x = 3$ is $- 320$ .

$y = - 320$

Step 2.4

Replace the variable $x$ with $2$ in the expression.

$f (2) = 4 ((2) - 13) ((2) + 5)$

Step 2.5

Simplify the result.

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Step 2.5.1

Subtract $13$ from $2$ .

$f (2) = 4 \cdot (- 11 ((2) + 5))$

Step 2.5.2

Multiply $4$ by $- 11$ .

$f (2) = - 44 ((2) + 5)$

Step 2.5.3

Add $2$ and $5$ .

$f (2) = - 44 \cdot 7$

Step 2.5.4

Multiply $- 44$ by $7$ .

$f (2) = - 308$

Step 2.5.5

The final answer is $- 308$ .

$- 308$

Step 2.6

The $y$ value at $x = 2$ is $- 308$ .

$y = - 308$

Step 2.7

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

$f (5) = 4 ((5) - 13) ((5) + 5)$

Step 2.8

Simplify the result.

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Step 2.8.1

Subtract $13$ from $5$ .

$f (5) = 4 \cdot (- 8 ((5) + 5))$

Step 2.8.2

Multiply $4$ by $- 8$ .

$f (5) = - 32 ((5) + 5)$

Step 2.8.3

Add $5$ and $5$ .

$f (5) = - 32 \cdot 10$

Step 2.8.4

Multiply $- 32$ by $10$ .

$f (5) = - 320$

Step 2.8.5

The final answer is $- 320$ .

$- 320$

Step 2.9

The $y$ value at $x = 5$ is $- 320$ .

$y = - 320$

Step 2.10

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

$f (6) = 4 ((6) - 13) ((6) + 5)$

Step 2.11

Simplify the result.

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Step 2.11.1

Subtract $13$ from $6$ .

$f (6) = 4 \cdot (- 7 ((6) + 5))$

Step 2.11.2

Multiply $4$ by $- 7$ .

$f (6) = - 28 ((6) + 5)$

Step 2.11.3

Add $6$ and $5$ .

$f (6) = - 28 \cdot 11$

Step 2.11.4

Multiply $- 28$ by $11$ .

$f (6) = - 308$

Step 2.11.5

The final answer is $- 308$ .

$- 308$

Step 2.12

The $y$ value at $x = 6$ is $- 308$ .

$y = - 308$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & - 308 \\ 3 & - 320 \\ 4 & - 324 \\ 5 & - 320 \\ 6 & - 308 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 324)$

Focus: $(4, - \frac{5183}{16})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{5185}{16}$

$\begin{array}{c} x & y \\ 2 & - 308 \\ 3 & - 320 \\ 4 & - 324 \\ 5 & - 320 \\ 6 & - 308 \end{array}$

Step 4