Find Amplitude, Period, and Phase Shift f(x)=tan((pix)/2)-3

$f (x) = \tan (\frac{π x}{2}) - 3$

Step 1

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

$a = 1$

$b = \frac{π}{2}$

$c = 0$

$d = - 3$

Step 2

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

Amplitude: None

Step 3

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

Find the period of $\tan (\frac{π x}{2})$ .

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

$\frac{π}{| b |}$

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{π}{| \frac{π}{2} |}$

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$π \frac{2}{π}$

Rewrite the expression.

$2$

Find the period of $- 3$ .

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

$\frac{π}{| b |}$

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{π}{| \frac{π}{2} |}$

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$π \frac{2}{π}$

Rewrite the expression.

$2$

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

$2$

Step 4

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

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

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

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

Phase Shift: $\frac{0}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{π})$

Multiply $0$ by $\frac{2}{π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $2$

Phase Shift: None

Vertical Shift: $- 3$

Step 6

Find Amplitude, Period, and Phase Shift f(x)=tan((pix)/2)-3

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