Algebra Examples

$y = 5 \sin (6 x - π)$

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

$a = 5$

$b = 6$

$c = π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $5$

Find the period of $5 \sin (6 x - π)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $6$ in the formula for period.

$\frac{2 π}{| 6 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{2 π}{6}$

Cancel the common factor of $2$ and $6$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6}$

Cancel the common factors.

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Factor $2$ out of $6$ .

$\frac{2 π}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Rewrite the expression.

$\frac{π}{3}$

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

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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: $\frac{π}{6}$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $\frac{π}{3}$

Phase Shift: $\frac{π}{6}$ ( $\frac{π}{6}$ to the right)

Vertical Shift: None