Question Find the frequency of the function $f(x) = -3 \sin(4x) + 9$ .

Studdy Solution

STEP 1

Assumptions
1. The function given is $f(x) = -3 \sin(4x) + 9$ .
2. The frequency of a function is the number of cycles it completes in a unit interval.
3. The frequency is related to the coefficient of $x$ in the sine function.

STEP 2

Identify the standard form of a sine function which is $f(x) = A \sin(Bx + C) + D$ where:
1. $A$ is the amplitude.
2. $B$ is the coefficient that affects the period of the function.
3. $C$ is the phase shift.
4. $D$ is the vertical shift.

STEP 3

Compare the given function $f(x) = -3 \sin(4x) + 9$ with the standard form to find the value of $B$ .
$B = 4$

STEP 4

The period of the sine function is given by $T = \frac{2\pi}{|B|}$ .

STEP 5

Calculate the period using the value of $B$ from STEP_3.
$T = \frac{2\pi}{|4|}$

STEP 6

Simplify the expression for the period.
$T = \frac{2\pi}{4}$

STEP 7

Further simplify the period.
$T = \frac{\pi}{2}$

STEP 8

The frequency $f$ is the reciprocal of the period $T$ .
$f = \frac{1}{T}$

STEP 9

Substitute the value of $T$ from STEP_7 to find the frequency.
$f = \frac{1}{\frac{\pi}{2}}$

STEP 10

Simplify the expression for frequency.
$f = \frac{2}{\pi}$
The frequency of the function $f(x) = -3 \sin(4x) + 9$ is $\frac{2}{\pi}$ .

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