Question

Determine the direction and magnitude of the vertical shift Determine the midline, amplitude period, and phase shift of the function

Answer 1

GIVEN:

We are given the cosine function below;

`y=(1)/(2)cos((x)/(3)-(\pi)/(3))`

Required;

To determine the midline, amplitude period and phase shift of the function given.

Step-by-step solution;

The midline of a sinusodial graph is the horizontal center line between the maximum and minimum of the graph. It is affected by any vertical translations to the graph. For the graph given;

`y=(1)/(2)cos((x)/(3)-(\pi)/(3))+0`

The midline is at;

`midline:y=0`

The amplitude is determined as one-half of the positive difference between the maximum and minimum values of the function. Simply put, the amplitude is the distance from the midline to the maximum (or the midline to the minimum).

For the function given as,

`y=Acos(Bx-C)+D`

The amplitude is determined as;

`Amplitude=|A|`

Therefore, the amplitude of this function is

`\begin{gathered} Amplitude=(1)/(2) \\ \\ OR \\ \\ Amplitude=0.5 \end{gathered}`

To determine the phase shift for the function;

`y=Acos(Bx-C)+D`

The phase shift is determined as

`Phase\text{ }shift=(C)/(B)`

Therefore we have;

`\begin{gathered} Phase\text{ }shift=(\pi)/(3)/(1)/(3) \\ \\ Phase\text{ }shift=(\pi)/(3)*(3)/(1) \\ \\ Phase\text{ }shift=\pi \end{gathered}`

Therefore,

ANSWER;

`\begin{gathered} midline:y=0 \\ \\ Amplitude=0.5 \\ \\ Phase\text{ }shift=\pi \end{gathered}`