Find Amplitude, Period, and Phase Shift f(t)=120cos(pi/4(t-3))+30

$f (t) = 120 \cos (\frac{π}{4} (t - 3)) + 30$

Step 1

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

$a = 120$

$b = \frac{π}{4}$

$c = \frac{3 π}{4}$

$d = 30$

Step 2

Find the amplitude $| a |$ .

Amplitude: $120$

Step 3

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

Find the period of $120 \cos (\frac{π t}{4} - \frac{3 π}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

Find the period of $30$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

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

$8$

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: $\frac{\frac{3 π}{4}}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{4} \cdot \frac{4}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $3 π$ .

Phase Shift: $\frac{π \cdot 3}{4} \cdot \frac{4}{π}$

Cancel the common factor.

Phase Shift: $\frac{π \cdot 3}{4} \cdot \frac{4}{π}$

Rewrite the expression.

Phase Shift: $\frac{3}{4} \cdot 4$

Cancel the common factor of $4$ .

Cancel the common factor.

Phase Shift: $\frac{3}{4} \cdot 4$

Rewrite the expression.

Phase Shift: $3$

Step 5

List the properties of the trigonometric function.

Amplitude: $120$

Period: $8$

Phase Shift: $3$ ( $3$ to the right)

Vertical Shift: $30$

Step 6

Find Amplitude, Period, and Phase Shift f(t)=120cos(pi/4(t-3))+30

Related Questions

Support LCMGCF.com by disabling your adblocker.