Question

How do you determine the amplitude and period for `y=-5 cos 6x`?

Answer 1

Look at the coefficients of the function and the function variable respectively.

Explanation:

The "period" is the distance or time it takes for the function to complete one full cycle - `360^o`. The "amplitude" is the height of the curve, or the distance from the baseline (midpoint) to the highest or lowest point.

The amplitude is determined by the coefficient on the function - `5` in this case. The sign (direction) does not affect the amplitude. In this case, the curve will cycle between `-5` and `5` instead of between `-1` and `1`.

The period is determined by the coefficient on the `'x'` term of the function - `6` in this case. The higher the number, the shorter the frequency, or number of times the function completes a full cycle for a given range. In this case, the period will be `1/6` of the normal cosine curve. Six "cycles" of cosine from `0 - 360^o` will occur in the same space that one cycle would for a standard cosine curve.

Further explanation, examples and graphics here:
http://www.purplemath.com/modules/grphtrig.htm

Additional examples and practice problems here:
http://www.mathwarehouse.com/trigonometry/period-sine-cosine/how-equation-effects-graph.php