| Graphing Techniques |
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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.
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cos(x-/2) = sin(x)
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| Transforming Sine and Cosine |
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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 |
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We now turn our attention to functions of the form
| 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.
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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|
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| 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. |
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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.
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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:
| Amplitude, Period and Phase Shift |
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| Sine and cosine graphs |
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