Question

I don't understand how to do this question. The equation is
`g(x) = 4 \sin( (1)/(2)(x - 5\pi) )`

Answer 1

We are given the following sinusoidal function g(x)

`g(x)=4\sin ((1)/(2)(x-5\pi))`

Recall that the standard form of a sinusoidal function is given by

`f(x)=A\sin (B(x-C))+D`

Where A is the amplitude, B is the frequency, C is the phase shift, D is the vertical translation.

Let us compare the given function with the standard form.

Amplitude:

As you can see, the amplitude is 4

A = 4

The amplitude is the vertical distance measured from the midline to the highest or lowest peak.

Frequency:

As you can see, the frequency is 1/2

B = 1/2 Hz

The frequency is the number of cycles in a given interval. It is measured in cycles per second or Hertz.

Phase shift:

The phase shift is also known as the horizontal shift.

C = 5π

This means that the function is shifted to the left by 5π

Vertical translation:

For the given case, there is no vertical translation.

D = 0

This means that how much function is shifted upward or downward.

Equation of midline:

This is the horizontal center line about which the function is symmetrical.

For the given case, the midline is y = 0 since the function is not shifted vertically

Let us graph the equation function and the function f(x) = sin(x)

The graph in red is the function g(x) and the graph in blue is f(x)

As you can see, g(x) has a greater amplitude than f(x)

f(x) has a greater frequency than g(x)

g(x) starts earlier than f(x) due to phase shift