Trigonometry Examples

$y = - x^{2} - 2 x + 24$

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 $- x^{2} - 2 x + 24$ .

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Step 1.1.1.1

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

$a = - 1$

$b = - 2$

$c = 24$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

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

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Step 1.1.1.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 1.1.1.3.2.1.2

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

$d = - 1 \cdot - 1$

Step 1.1.1.3.2.2

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$d = - - 1$

Step 1.1.1.3.2.3

Multiply $- 1$ by $- 1$ .

$d = 1$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = 24 - \frac{{(- 2)}^{2}}{4 \cdot - 1}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

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

$e = 24 - \frac{4}{4 \cdot - 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = 24 - \frac{4}{- 4}$

Step 1.1.1.4.2.1.3

Divide $4$ by $- 4$ .

$e = 24 - - 1$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 24 + 1$

Step 1.1.1.4.2.2

Add $24$ and $1$ .

$e = 25$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 1)}^{2} + 25$ .

$- {(x + 1)}^{2} + 25$

Step 1.1.2

Set $y$ equal to the new right side.

$y = - {(x + 1)}^{2} + 25$

Step 1.2

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

$a = - 1$

$h = - 1$

$k = 25$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(- 1, 25)$

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.

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

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.

$(- 1, \frac{99}{4})$

Step 1.7

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

$x = - 1$

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.

$y = \frac{101}{4}$

Step 1.9

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

Direction: Opens Down

Vertex: $(- 1, 25)$

Focus: $(- 1, \frac{99}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{101}{4}$

Direction: Opens Down

Vertex: $(- 1, 25)$

Focus: $(- 1, \frac{99}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{101}{4}$

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 $- 2$ in the expression.

$f (- 2) = - {(- 2)}^{2} - 2 \cdot - 2 + 24$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 2) = - 1 \cdot 4 - 2 \cdot - 2 + 24$

Step 2.2.1.2

Multiply $- 1$ by $4$ .

$f (- 2) = - 4 - 2 \cdot - 2 + 24$

Step 2.2.1.3

Multiply $- 2$ by $- 2$ .

$f (- 2) = - 4 + 4 + 24$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $- 4$ and $4$ .

$f (- 2) = 0 + 24$

Step 2.2.2.2

Add $0$ and $24$ .

$f (- 2) = 24$

Step 2.2.3

The final answer is $24$ .

$24$

Step 2.3

The $y$ value at $x = - 2$ is $24$ .

$y = 24$

Step 2.4

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

$f (- 3) = - {(- 3)}^{2} - 2 \cdot - 3 + 24$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$f (- 3) = - 1 \cdot 9 - 2 \cdot - 3 + 24$

Step 2.5.1.2

Multiply $- 1$ by $9$ .

$f (- 3) = - 9 - 2 \cdot - 3 + 24$

Step 2.5.1.3

Multiply $- 2$ by $- 3$ .

$f (- 3) = - 9 + 6 + 24$

Step 2.5.2

Simplify by adding numbers.

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Step 2.5.2.1

Add $- 9$ and $6$ .

$f (- 3) = - 3 + 24$

Step 2.5.2.2

Add $- 3$ and $24$ .

$f (- 3) = 21$

Step 2.5.3

The final answer is $21$ .

$21$

Step 2.6

The $y$ value at $x = - 3$ is $21$ .

$y = 21$

Step 2.7

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

$f (0) = - {(0)}^{2} - 2 \cdot 0 + 24$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = - 0 - 2 \cdot 0 + 24$

Step 2.8.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 - 2 \cdot 0 + 24$

Step 2.8.1.3

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 + 24$

Step 2.8.2

Simplify by adding numbers.

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Step 2.8.2.1

Add $0$ and $0$ .

$f (0) = 0 + 24$

Step 2.8.2.2

Add $0$ and $24$ .

$f (0) = 24$

Step 2.8.3

The final answer is $24$ .

$24$

Step 2.9

The $y$ value at $x = 0$ is $24$ .

$y = 24$

Step 2.10

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

$f (1) = - {(1)}^{2} - 2 \cdot 1 + 24$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

One to any power is one.

$f (1) = - 1 \cdot 1 - 2 \cdot 1 + 24$

Step 2.11.1.2

Multiply $- 1$ by $1$ .

$f (1) = - 1 - 2 \cdot 1 + 24$

Step 2.11.1.3

Multiply $- 2$ by $1$ .

$f (1) = - 1 - 2 + 24$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $2$ from $- 1$ .

$f (1) = - 3 + 24$

Step 2.11.2.2

Add $- 3$ and $24$ .

$f (1) = 21$

Step 2.11.3

The final answer is $21$ .

$21$

Step 2.12

The $y$ value at $x = 1$ is $21$ .

$y = 21$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 21 \\ - 2 & 24 \\ - 1 & 25 \\ 0 & 24 \\ 1 & 21 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, 25)$

Focus: $(- 1, \frac{99}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{101}{4}$

$\begin{array}{c} x & y \\ - 3 & 21 \\ - 2 & 24 \\ - 1 & 25 \\ 0 & 24 \\ 1 & 21 \end{array}$

Step 4