LauraDickerson

To examine the graph of y = cos x, I will examine y = A cos (Bx + C) for different values of A, B, & C. This will allow me to make generalizations for the effects of changes in parameters A, B, and C & thus I will know how lớn graph a function y = Acos(Bx + C) quickly.

Let us first look at the graph y = cos x. This is where A = 1, B = 1, and C = 0. This is the "parent" cosine graph that we will compare to lớn other graphs. The parametersA, B, & C transformthe graph to related graphs.

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Now, let us change the values of A. Multiple graphs are produced on the same axes, with A = 1, 2, 3, -1,-2, and -32. In all of these cases B = 1 & C = 0.

The differences shown in the graphs above show the graphs change as the value of "A" in y = A cos x changes. The "A" is a scalar multiple of our function. For any f(x) the value of Af(x) changes the size & is thus a similarity transformation. We define |A| in this circular function to the Amplitude of the curve. It is one half of the peak khổng lồ peak variation of the curve and is given by the equation

The sign of A is of special importance. Notice that the last three graphs show negative values for A.The -1 times the amplitude acts as a separate scalar multiple & is a reflection in the x-axis.

Now let us change the values of B.

y = cos x y = cos ( -x) y = cos 2x y = cos ( -2x) y = cos 3x y = cos ( -3x)

It appears that B affects the period of the curve. To lớn see if this is true, lets graph some curves where the value of B is less than zero.

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y = cos x y = cos ( 50% x) y = cos ( 1/3 x)

We can see that in fact, B does affect the period of the curve. It takes 1/B times lớn complete a period of a curve. If B is equal to lớn 1, then it takes 2pi lớn complete a period. If B is equal to lớn 2, then it takes only pi to complete a period. If B is equal khổng lồ 1/2, then it takes only 4pi to complete a period, twice as long as a normal period. Once again we also see that a negative only reflects the curve about the x-axis.

Now let us change the values of C. It is easy to characterize the effect of changes in C as a mapping (x, y) to (x + C, y) và this is a translation either moving the graph to lớn the left if C is positive, or to the right if C is negative. The translation is a mapping that preserves distance & so the image curve is congruent -- there is no change in the shape or the kích cỡ of the curve.

y = cos x y = cos (x + 1) y = cos (x + 2) y = cos (x - 1) y = cos (x - 2)

Automatically we can see that the actual picture of the graph does not change, it only shifts. If C is positive it shifts lớn the right, if C is negative it shifts lớn the left. Thus C affects the horizontal displacement (or shift) of the graph.