Precalculus Examples

$f (x) = \frac{1}{2} \sin (\frac{1}{3} x + \frac{π}{2})$

Step 1

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

$a = \frac{1}{2}$

$b = \frac{1}{3}$

$c = - \frac{π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

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

Step 3

Find the period of $\frac{\sin (\frac{x}{3} + \frac{π}{2})}{2}$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

Step 3.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Step 3.5

Multiply $3$ by $2$ .

$6 π$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

Phase Shift: $\frac{- \frac{π}{2}}{\frac{1}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot 3$

Step 4.4

Multiply $- \frac{π}{2} \cdot 3$ .

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Step 4.4.1

Multiply $3$ by $- 1$ .

Phase Shift: $- 3 \frac{π}{2}$

Step 4.4.2

Combine $- 3$ and $\frac{π}{2}$ .

Phase Shift: $\frac{- 3 π}{2}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{3 π}{2}$

Step 5

List the properties of the trigonometric function.

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

Period: $6 π$

Phase Shift: $- \frac{3 π}{2}$ ( $\frac{3 π}{2}$ to the left)

Vertical Shift: None

Step 6