Precalculus Examples

$f (x) = \frac{- 1}{3} \sin (π - \frac{2 x}{5})$

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.

$a = - \frac{1}{3}$

$b = - \frac{2}{5}$

$c = - π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{3}$

Step 3

Find the period of $- \frac{\sin (π - \frac{2 x}{5})}{3}$ .

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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 $- \frac{2}{5}$ in the formula for period.

$\frac{2 π}{| - \frac{2}{5} |}$

Step 3.3

$- \frac{2}{5}$ is approximately $- 0.4$ which is negative so negate $- \frac{2}{5}$ and remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Step 3.5.2

Cancel the common factor.

$2 π \frac{5}{2}$

Step 3.5.3

Rewrite the expression.

$π \cdot 5$

Step 3.6

Move $5$ to the left of $π$ .

$5 π$

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: $\frac{- π}{- \frac{2}{5}}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift: $\frac{π}{\frac{2}{5}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{5}{2})$

Step 4.5

Combine $π$ and $\frac{5}{2}$ .

Phase Shift: $\frac{π \cdot 5}{2}$

Step 4.6

Move $5$ to the left of $π$ .

Phase Shift: $\frac{5 π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{3}$

Period: $5 π$

Phase Shift: $\frac{5 π}{2}$ ( $\frac{5 π}{2}$ to the right)

Vertical Shift: None

Step 6