Find Amplitude, Period, and Phase Shift 3+0.5sec(2x+1)

$3 + 0.5 \sec (2 x + 1)$

Step 1

Rewrite the expression as $0.5 \sec (2 x + 1) + 3$ .

$0.5 \sec (2 x + 1) + 3$

Step 2

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

$a = 0.5$

$b = 2$

$c = - 1$

$d = 3$

Step 3

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

Step 4

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

Find the period of $0.5 \sec (2 x + 1)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Find the period of $3$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

$π$

Step 5

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: $\frac{- 1}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{2}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $π$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: $3$

Step 7

Find Amplitude, Period, and Phase Shift 3+0.5sec(2x+1)

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