Question

How do you determine the amplitude, period, and shifts to graph `f(x)= 10 cos 3x`?

Answer 1

Amplitude `= 10`
Period `= 120^@ ((2pi)/3)`
Phase Shift `= 0`
Vertical Shift `= 0`

Explanation:

The general equation for a cosine function is:

`f(x)=acos(k(x-d))+c`

The amplitude is the peak height subtract the trough height divided by `2`. It can also be described as the height from the centre line (of the graph) to the peak (or trough).
Additionally, the amplitude is also the absolute value found before `cos` in the equation. In this case, the amplitude is `10`. A general formula to find the amplitude is:

`Amplitude=|a|`

The period is the length from one point to the next matching point. It can also be described as the change in the independent variable (`x`) in one cycle.
Additionally, the period is also `360^@` (`2pi`) divided by `|k|`. In this case, the period is `120^@` (`(2pi)/3`). A general formula to find the amplitude is:

`Period=360^@/|k|` or `Period=(2pi)/|k|`

The phase shift is the length that the transformed graph has shifted horizontally to the left or right compared to its parent function. In this case, `d` is `0` in the equation, so there is no phase shift.

The vertical shift is the length that the transformed graph has shifted vertically up or down compared to its parent function.
Additionally, the vertical shift is also the maximum height plus the minimum height divided by `2`. In this case, `c` is `0` in the equation, so there is no vertical shift. A general formula to find the vertical shift is:

`"Vertical shift"=("maximum y"+"minimum y")/2`