Trigonometry Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

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

$a = 2$

$h = 3$

$k = 4$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(3, 4)$

Step 1.4

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

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Step 1.4.1

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

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of $a$ into the formula.

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

Step 1.4.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

Step 1.5

Find the focus.

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Step 1.5.1

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 1.5.2

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

$(3, \frac{33}{8})$

Step 1.6

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

$x = 3$

Step 1.7

Find the directrix.

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Step 1.7.1

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 1.7.2

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

$y = \frac{31}{8}$

Step 1.8

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

Direction: Opens Up

Vertex: $(3, 4)$

Focus: $(3, \frac{33}{8})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{31}{8}$

Direction: Opens Up

Vertex: $(3, 4)$

Focus: $(3, \frac{33}{8})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{31}{8}$

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.

$f (2) = 2 {(2)}^{2} - 12 \cdot 2 + 22$

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

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 2.2.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 2.2.1.1.1.1

Raise $2$ to the power of $1$ .

$f (2) = 2 {(2)}^{2} - 12 \cdot 2 + 22$

Step 2.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = 2^{1 + 2} - 12 \cdot 2 + 22$

Step 2.2.1.1.2

Add $1$ and $2$ .

$f (2) = 2^{3} - 12 \cdot 2 + 22$

Step 2.2.1.2

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

$f (2) = 8 - 12 \cdot 2 + 22$

Step 2.2.1.3

Multiply $- 12$ by $2$ .

$f (2) = 8 - 24 + 22$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $24$ from $8$ .

$f (2) = - 16 + 22$

Step 2.2.2.2

Add $- 16$ and $22$ .

$f (2) = 6$

Step 2.2.3

The final answer is $6$ .

$6$

Step 2.3

The $y$ value at $x = 2$ is $6$ .

$y = 6$

Step 2.4

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

$f (1) = 2 {(1)}^{2} - 12 \cdot 1 + 22$

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

One to any power is one.

$f (1) = 2 \cdot 1 - 12 \cdot 1 + 22$

Step 2.5.1.2

Multiply $2$ by $1$ .

$f (1) = 2 - 12 \cdot 1 + 22$

Step 2.5.1.3

Multiply $- 12$ by $1$ .

$f (1) = 2 - 12 + 22$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $12$ from $2$ .

$f (1) = - 10 + 22$

Step 2.5.2.2

Add $- 10$ and $22$ .

$f (1) = 12$

Step 2.5.3

The final answer is $12$ .

$12$

Step 2.6

The $y$ value at $x = 1$ is $12$ .

$y = 12$

Step 2.7

Replace the variable $x$ with $4$ in the expression.

$f (4) = 2 {(4)}^{2} - 12 \cdot 4 + 22$

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

Raise $4$ to the power of $2$ .

$f (4) = 2 \cdot 16 - 12 \cdot 4 + 22$

Step 2.8.1.2

Multiply $2$ by $16$ .

$f (4) = 32 - 12 \cdot 4 + 22$

Step 2.8.1.3

Multiply $- 12$ by $4$ .

$f (4) = 32 - 48 + 22$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $48$ from $32$ .

$f (4) = - 16 + 22$

Step 2.8.2.2

Add $- 16$ and $22$ .

$f (4) = 6$

Step 2.8.3

The final answer is $6$ .

$6$

Step 2.9

The $y$ value at $x = 4$ is $6$ .

$y = 6$

Step 2.10

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

$f (5) = 2 {(5)}^{2} - 12 \cdot 5 + 22$

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

Raise $5$ to the power of $2$ .

$f (5) = 2 \cdot 25 - 12 \cdot 5 + 22$

Step 2.11.1.2

Multiply $2$ by $25$ .

$f (5) = 50 - 12 \cdot 5 + 22$

Step 2.11.1.3

Multiply $- 12$ by $5$ .

$f (5) = 50 - 60 + 22$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $60$ from $50$ .

$f (5) = - 10 + 22$

Step 2.11.2.2

Add $- 10$ and $22$ .

$f (5) = 12$

Step 2.11.3

The final answer is $12$ .

$12$

Step 2.12

The $y$ value at $x = 5$ is $12$ .

$y = 12$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 12 \\ 2 & 6 \\ 3 & 4 \\ 4 & 6 \\ 5 & 12 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(3, 4)$

Focus: $(3, \frac{33}{8})$

Axis of Symmetry: $x = 3$

Directrix: $y = \frac{31}{8}$

$\begin{array}{c} x & y \\ 1 & 12 \\ 2 & 6 \\ 3 & 4 \\ 4 & 6 \\ 5 & 12 \end{array}$

Step 4