Trigonometry Examples

$y = \frac{2}{5} \cdot {(x - 4)}^{2} - 3$

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 $k$ .

$a = \frac{2}{5}$

$h = 4$

$k = - 3$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(4, - 3)$

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.

$\frac{1}{4 \cdot \frac{2}{5}}$

Step 1.4.3

Simplify.

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Step 1.4.3.1

Combine $4$ and $\frac{2}{5}$ .

$\frac{1}{\frac{4 \cdot 2}{5}}$

Step 1.4.3.2

Multiply $4$ by $2$ .

$\frac{1}{\frac{8}{5}}$

Step 1.4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{5}{8})$

Step 1.4.3.4

Multiply $\frac{5}{8}$ by $1$ .

$\frac{5}{8}$

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.

$(4, - \frac{19}{8})$

Step 1.6

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

$x = 4$

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.

$y = - \frac{29}{8}$

Step 1.8

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

Direction: Opens Up

Vertex: $(4, - 3)$

Focus: $(4, - \frac{19}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{29}{8}$

Direction: Opens Up

Vertex: $(4, - 3)$

Focus: $(4, - \frac{19}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{29}{8}$

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 $3$ in the expression.

$f (3) = \frac{2 {(3)}^{2}}{5} - \frac{16 (3)}{5} + \frac{17}{5}$

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

$f (3) = \frac{2 {(3)}^{2} - 16 \cdot 3 + 17}{5}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

$f (3) = \frac{2 \cdot 9 - 16 \cdot 3 + 17}{5}$

Step 2.2.2.2

Multiply $2$ by $9$ .

$f (3) = \frac{18 - 16 \cdot 3 + 17}{5}$

Step 2.2.2.3

Multiply $- 16$ by $3$ .

$f (3) = \frac{18 - 48 + 17}{5}$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Subtract $48$ from $18$ .

$f (3) = \frac{- 30 + 17}{5}$

Step 2.2.3.2

Add $- 30$ and $17$ .

$f (3) = \frac{- 13}{5}$

Step 2.2.3.3

Move the negative in front of the fraction.

$f (3) = - \frac{13}{5}$

Step 2.2.4

The final answer is $- \frac{13}{5}$ .

$- \frac{13}{5}$

Step 2.3

The $y$ value at $x = 3$ is $- \frac{13}{5}$ .

$y = - \frac{13}{5}$

Step 2.4

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

$f (2) = \frac{2 {(2)}^{2}}{5} - \frac{16 (2)}{5} + \frac{17}{5}$

Step 2.5

Simplify the result.

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Step 2.5.1

Combine the numerators over the common denominator.

$f (2) = \frac{2 {(2)}^{2} - 16 \cdot 2 + 17}{5}$

Step 2.5.2

Simplify each term.

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Step 2.5.2.1

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

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Step 2.5.2.1.1

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

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Step 2.5.2.1.1.1

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

$f (2) = \frac{2 {(2)}^{2} - 16 \cdot 2 + 17}{5}$

Step 2.5.2.1.1.2

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

$f (2) = \frac{2^{1 + 2} - 16 \cdot 2 + 17}{5}$

Step 2.5.2.1.2

Add $1$ and $2$ .

$f (2) = \frac{2^{3} - 16 \cdot 2 + 17}{5}$

Step 2.5.2.2

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

$f (2) = \frac{8 - 16 \cdot 2 + 17}{5}$

Step 2.5.2.3

Multiply $- 16$ by $2$ .

$f (2) = \frac{8 - 32 + 17}{5}$

Step 2.5.3

Simplify the expression.

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Step 2.5.3.1

Subtract $32$ from $8$ .

$f (2) = \frac{- 24 + 17}{5}$

Step 2.5.3.2

Add $- 24$ and $17$ .

$f (2) = \frac{- 7}{5}$

Step 2.5.3.3

Move the negative in front of the fraction.

$f (2) = - \frac{7}{5}$

Step 2.5.4

The final answer is $- \frac{7}{5}$ .

$- \frac{7}{5}$

Step 2.6

The $y$ value at $x = 2$ is $- \frac{7}{5}$ .

$y = - \frac{7}{5}$

Step 2.7

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

$f (5) = \frac{2 {(5)}^{2}}{5} - \frac{16 (5)}{5} + \frac{17}{5}$

Step 2.8

Simplify the result.

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Step 2.8.1

Combine the numerators over the common denominator.

$f (5) = \frac{2 {(5)}^{2} - 16 \cdot 5 + 17}{5}$

Step 2.8.2

Simplify each term.

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Step 2.8.2.1

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

$f (5) = \frac{2 \cdot 25 - 16 \cdot 5 + 17}{5}$

Step 2.8.2.2

Multiply $2$ by $25$ .

$f (5) = \frac{50 - 16 \cdot 5 + 17}{5}$

Step 2.8.2.3

Multiply $- 16$ by $5$ .

$f (5) = \frac{50 - 80 + 17}{5}$

Step 2.8.3

Simplify the expression.

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Step 2.8.3.1

Subtract $80$ from $50$ .

$f (5) = \frac{- 30 + 17}{5}$

Step 2.8.3.2

Add $- 30$ and $17$ .

$f (5) = \frac{- 13}{5}$

Step 2.8.3.3

Move the negative in front of the fraction.

$f (5) = - \frac{13}{5}$

Step 2.8.4

The final answer is $- \frac{13}{5}$ .

$- \frac{13}{5}$

Step 2.9

The $y$ value at $x = 5$ is $- \frac{13}{5}$ .

$y = - \frac{13}{5}$

Step 2.10

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

$f (6) = \frac{2 {(6)}^{2}}{5} - \frac{16 (6)}{5} + \frac{17}{5}$

Step 2.11

Simplify the result.

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Step 2.11.1

Combine the numerators over the common denominator.

$f (6) = \frac{2 {(6)}^{2} - 16 \cdot 6 + 17}{5}$

Step 2.11.2

Simplify each term.

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Step 2.11.2.1

Raise $6$ to the power of $2$ .

$f (6) = \frac{2 \cdot 36 - 16 \cdot 6 + 17}{5}$

Step 2.11.2.2

Multiply $2$ by $36$ .

$f (6) = \frac{72 - 16 \cdot 6 + 17}{5}$

Step 2.11.2.3

Multiply $- 16$ by $6$ .

$f (6) = \frac{72 - 96 + 17}{5}$

Step 2.11.3

Simplify the expression.

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Step 2.11.3.1

Subtract $96$ from $72$ .

$f (6) = \frac{- 24 + 17}{5}$

Step 2.11.3.2

Add $- 24$ and $17$ .

$f (6) = \frac{- 7}{5}$

Step 2.11.3.3

Move the negative in front of the fraction.

$f (6) = - \frac{7}{5}$

Step 2.11.4

The final answer is $- \frac{7}{5}$ .

$- \frac{7}{5}$

Step 2.12

The $y$ value at $x = 6$ is $- \frac{7}{5}$ .

$y = - \frac{7}{5}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & - \frac{7}{5} \\ 3 & - \frac{13}{5} \\ 4 & - 3 \\ 5 & - \frac{13}{5} \\ 6 & - \frac{7}{5} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 3)$

Focus: $(4, - \frac{19}{8})$

Axis of Symmetry: $x = 4$

Directrix: $y = - \frac{29}{8}$

$\begin{array}{c} x & y \\ 2 & - \frac{7}{5} \\ 3 & - \frac{13}{5} \\ 4 & - 3 \\ 5 & - \frac{13}{5} \\ 6 & - \frac{7}{5} \end{array}$

Step 4