Find Amplitude, Period, and Phase Shift f(t)=3sec(t-pi/4)-2

$f (t) = 3 \sec (t - \frac{π}{4}) - 2$

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

$a = 3$

$b = 1$

$c = \frac{π}{4}$

$d = - 2$

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

Amplitude: None

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

Find the period of $3 \sec (t - \frac{π}{4})$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Find the period of $- 2$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$2 π$

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: $\frac{\frac{π}{4}}{1}$

Divide $\frac{π}{4}$ by $1$ .

Phase Shift: $\frac{π}{4}$

List the properties of the trigonometric function.

Amplitude: None

Period: $2 π$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $- 2$