Graphing Techniques

We have seen that once a graph of a function is determined we can obtain the graphs of related functions through scaling and shifting. We restate these rules assuming we know the graph of the function f.

 Modified function How the graph is obtained Af(x) The graph of f with y values scaled by a factor of A f(x-c) The graph of f shifted |c| units to the right if c > 0 and to the left if c < 0 f(kx) The graph of f scaled horizontally be a factor of k f(x)+b The graph of f moved vertically b units For example, the graph of y = 3sin(x)+1 is the graph of sin(x) scaled vertically by a factor of 3 and then moved vertically 1 unit. Its graph (in blue) and that of sin(x) (in red) are shown.

Does it make a difference in the order in which we perform the operations? That is, could we have shifted vertically one unit and then vertically scaled by 3? The answer is no but you should sketch the corresponding graph to see the difference. What about the graph of f(x) = cos(x- /2)? This then is the graph of cos(x) shifted to the right /2 units and its graph is shown. Note that its graph is identical to the graph of y = sin(x). In other words, the two functions have identical values at all values of x and so the two functions are equal. This gives us the following identity.

 cos(x- /2) = sin(x)

Transforming Sine and Cosine
In this demonstration, you are given a base graph of sine or cosine. Move the scrollbars on the right to scale and shift the curve vertically or horizontally. The formula of the corresponding function will be displayed.

You should check the above identity
cos(x- /2) = sin(x)
using this applet. Can you come up with some other identities?

Similar behavior is exhibited with the other four trigonometric functions. The graph of 3tan(x)+1, for example, is that of tan(x) scaled vertically by a factor of 3 and shifted vertically 1 unit.

Amplitude, Phase Shift and Period

We now turn our attention to functions of the form

y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d
where A and k are non-zero real numbers.

 Period In the demonstration above, you will probably have noticed that as you alter the horizontal scaling factor, the period of the curve changed. That is, the least positive real number for the function to repeat changes. For example, the graph of sin(2x) repeats every units whereas the graph of sin(x/2) repeats every 4 units. Also, the period is unchanged by vertical scaling or shifting or by horizontal shifting.

 The period of y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d where k is a non-zero real number, is 2 /|k|

 Amplitude The sine and cosine functions take on values between -1 and 1. By scaling vertically either function by a factor of A, the values of the function lie between -A and A. We define the amplitude to be one-half of the difference of the greatest value the function and the least value of the function. For example, consider 3sin(x). The greatest value this expression attains is 3 and its least value is -3. Its amplitude is (3 - (-3))/2 or 3. Likewise, the function y = -2sin(x) + 5 has amplitude 2 since its least value is 3 and its greatest value is 7. Note that vertical shifts do alter the greatest and least values that the function attains but do not alter the amplitude. Also note that horizontal scaling and shifting do not affect the amplitude.

 The amplitude of y = Asin(k(x - c)) + d, y = Acos(k(x - c)) + d where k is a non-zero real number, is |A|

 Phase shift The real number c is called the phase shift. It is a measure of the horizontal shift. If the phase shift is positive, there has been a horizontal shift to the right and if it is negative, there has been a horizontal shift to the left. In reading off the phase shift, make sure you have the function in the form above. For example, the phase shift of y = sin(2x - ) is NOT . Rewrite the expression for the function in the form required to get y = sin(2(x - /2)). Now we see the correct phase shift, namely /2.

Although we have considered only transformations of sine and cosine, the same rules apply to all the trigonometric functions. However recall that the period of tangent and cotangent is . So for example the period of y = tan(3x) is /3 and not 2 /3. In order to read off the phase shift of transformations of the other trigonometric functions remember to write the function in the appropriate form, namely:

y = Atan(k(x - c)) + d, y = Acot(k(x - c)) + d
y = Acot(k(x - c)) + d, y = Acsc(k(x - c)) + d
where A and k are non-zero real numbers.

Amplitude, Period and Phase Shift
In this exercise you are asked to find the amplitude, period or phase shift of the given trigonometric function. For the phase shift, remember to write the function in the form discussed in this section.

Sine and cosine graphs
In the applet you are given the expression of a sine or cosine function. Use the scrollbars to manipulate the base graph until you get the graph of the given function. When you think you have the correct graph, press the "Check" button. The "Reset" button will put the curve back to the original position. You should try several of these problems.

Difficulty level 0 involves graphs with only one transformation, difficulty level 1 involves graphs with at most two transformations, and difficulty level 2 gives graphs with possibly all transformations.