Precalculus Examples

$y = \csc (x) - 9$

Step 1

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

$y = \csc (x)$

Step 2

Assume that $y = \csc (x)$ is $f (x) = \csc (x)$ and $y = \csc (x) - 9$ is $g (x) = \csc (x) - 9$ .

$f (x) = \csc (x)$

$g (x) = \csc (x) - 9$

Step 3

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

$a = 1$

$b = 1$

$c = 0$

$d = - 9$

Step 4

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

Amplitude: None

Step 5

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 5.1

Find the period of $\csc (x)$ .

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Step 5.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.1.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2

Find the period of $- 9$ .

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Step 5.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.3

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

$2 π$

Step 6

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 6.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 6.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{1}$

Step 6.3

Divide $0$ by $1$ .

Phase Shift: $0$

Step 7

List the properties of the trigonometric function.

Amplitude: None

Period: $2 π$

Phase Shift: None

Vertical Shift: $- 9$

Step 8