Find Amplitude, Period, and Phase Shift y=-5cos(pi/8x)+3

$y = - 5 \cos (\frac{π}{8} x) + 3$

Step 1

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

$a = - 5$

$b = \frac{π}{8}$

$c = 0$

$d = 3$

Step 2

Find the amplitude $| a |$ .

Amplitude: $5$

Step 3

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

Find the period of $- 5 \cos (\frac{π x}{8})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{8}$ in the formula for period.

$\frac{2 π}{| \frac{π}{8} |}$

$\frac{π}{8}$ is approximately $0.39269908$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{8}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{8}{π}$

Cancel the common factor.

$π \cdot 2 \frac{8}{π}$

Rewrite the expression.

$2 \cdot 8$

Multiply $2$ by $8$ .

$16$

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 $\frac{π}{8}$ in the formula for period.

$\frac{2 π}{| \frac{π}{8} |}$

$\frac{π}{8}$ is approximately $0.39269908$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{8}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{8}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{8}{π}$

Cancel the common factor.

$π \cdot 2 \frac{8}{π}$

Rewrite the expression.

$2 \cdot 8$

Multiply $2$ by $8$ .

$16$

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

$16$

Step 4

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: $\frac{0}{\frac{π}{8}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{8}{π})$

Multiply $0$ by $\frac{8}{π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $5$

Period: $16$

Phase Shift: None

Vertical Shift: $3$

Step 6

Find Amplitude, Period, and Phase Shift y=-5cos(pi/8x)+3

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