Trigonometry Examples

f (x) = 5 \sec (x - 2) + 1

$f (x) = 5 \sec (x - 2) + 1$

Step 1

The parent function is the simplest form of the type of function given.

g (x) = \sec (x)

$g (x) = \sec (x)$

Step 2

Use the form

a \sec (b x - c) + d

$a \sec (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = 5

$a = 5$

b = 1

$b = 1$

c = 2

$c = 2$

d = 1

$d = 1$

Step 3

Since the graph of the function

s e c

$s e c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period using the formula

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

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Step 4.1

Find the period of

5 \sec (x - 2)

$5 \sec (x - 2)$ .

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Step 4.1.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 4.1.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 4.1.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 4.2

Find the period of

1

$1$ .

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Step 4.2.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 4.2.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 4.2.3

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 4.2.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 4.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

2 π

$2 π$

2 π

$2 π$

Step 5

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 5.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{2}{1}

$\frac{2}{1}$

Step 5.3

Divide

2

$2$ by

1

$1$ .

Phase Shift:

2

$2$

Phase Shift:

2

$2$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period:

2 π

$2 π$

Phase Shift:

2

$2$ (

2

$2$ to the right)

Vertical Shift:

1

$1$

Step 7