Trigonometry Examples

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

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.

$a = - 1$

$b = 3$

$c = - π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $- \sin (3 x + π)$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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: $\frac{- π}{3}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2 π}{3}$

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

Vertical Shift: None

Step 6