Re: Algebraic (transformation) and trigonometric model
reg said:
... i just don't know what they mean by a trig model ...
Hi reg:
A model that uses one of the trigonometry functions (eg: sin, cos, tan, sec, csc, cot) is a "trig model".
For example, if the electrical voltage of some circuit at time T is given by the expression
16000 * sin(T+0.5)
then we could say that this expression models the voltage at time T. Because this expression contains a trigonometry function, we could further say that it's a trig model.
Since they already gave you the form h = a * cos(b * t + c) + d
it's kinda redundant for them to instruct you to come up with a trig model. They've already done that for you by providing an expression for h that includes a trigonometry function (cosine).
reg said:
I need to use the algebraic (transformation) method ... i just don't know what they mean by ... the algebraic method.
Transformations have to do with the relationship between changes made to a function's DEFINITION and the resulting effects that those changes make in the function's GRAPH.
For example, you know what the graph of sin(x) looks like. If we change the expression to 4 * sin(x), then the graph becomes "transformed". Instead of a wave between y = -1 and y = 1, the transformed graph has a sine wave between y = -4 and y = 4.
By altering the number in front of sin(x), we can transform the graph's wave to be as short or as tall as we like.
(1/4) * sin(x) will make the amplitude of the graph smaller.
100 * sin(x) will make the amplitude of the graph much larger.
There are many different types of transformations.
-sin(x) transforms the graph of sin(x) by reflecting it across the x-axis.
sin(x + Pi/2) transforms the graph of sin(x) by shifting it Pi/2 units to the LEFT.
sin(x - 10) transforms the graph of sin(x) by shifting it 10 units to the RIGHT
sin(x) + 5 transforms the graph of sin(x) by shifting the entire graph UP by 5 units.
sin(x) - 500 transforms the graph of sin(x) by shifting the entire graph DOWN by 500 units.
sin(3x) transforms the graph of sin(x) by compressing it by a factor of 3; in other words, you can fit three periods of the graph of sin(3x) into the same space as one period of the graph of sin(x).
reg said:
... where i should start?
If you understand the various transformations (i.e., how changing the values of a, b, c, and d affect the amplitude, horizontal shifting, vertical shifting, reflection [if any], and compression or stretching of the graph of h), then I would start by graphing your data. Then determine the values of a, b, c, and d needed to get the graph of h to roughly match the graph of your data.
If you do not understand what happens to the graph of h when you change the value of a, b, c, or d, then I would start by reviewing that material. If you have a graphing calculator, then use it to plot families of h as you review.
For example, if you're reviewing how the value of c shifts the graph of h horizontally, then plot the following expressions for h:
cos(t + 0)
cos(t + 1)
cos(t + 2)
cos(t + 3)
(in this example, a=1, b=1, and d=0).
Try changing the value of a, and see what happens to the graph of h.
-2 * cos(t)
-1 * cos(t)
cos(t)
2 * cos(t)
(here, the values are b=1, c=0, and d=0).
If you still do not have any idea on how to proceed to make the graph of h match the graph of your data, then please let us know. It's very helpful to us if you provide us with your thoughts on why you're stuck.
Cheers,
~ Mark
