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For trig functions there are some important quantities that give us information about the shape of the graphs, and we should figure out how the various transformations effect these quatities. The period of a trig function is the minimum a such that for all we have. If you have seen waves in other contexts, like a Physics class, you are also probably familiar with the notion of "frequency" of a wave.
Period and frequency contain the same information about a wave. They are inverses of each other. Frequency tells you number of cycles per second while period tells you number of seconds per cycle. The red curve is obtained by shifting the black curve to the right by.
Transformations of Trigonometric Functions
It makes sense to say that the graph of the red function has a phase shift of. The red curve is the graph of , it does not have an obvious in it, we ge the by taking. The green curve is obtained by shifting the black curve to the right by. The green curve is the graph of and we get the by taking the in the formula and dividing by 2. Amplitude is affected by the multiplication, but not by translation.
Transforming Trigonometric Functions
Now the crests go up to 6, but the troughs only go down to 2 so the difference is 4, and half of that difference is the amplitude. Important: You can not determine amplitude just by looking at the maximum height in a graph, you have to consider maximum change.
The amplitude of is 2, even though the maximum height of the graph is 6! The input is now. So completes one cycle as goes from 0 to. Thus the period of is. We can achieve a horizontal shift by replacing with.
Trigonometry - Graphing transformations of sin and cos
This requires some care if there is also a multiplication. Consider again.
Transformation of Trigonometric Graphs
To shift right by replace by and get. It is important to understand that is NOT shifted by. As we just said, to shift by you would need. See the discussion of Phase above for more examples of phase shift. So given a formula we can write down the amplitude, vertical shift, phase shift, and period of the function, and from this information we can draw a graph, which we will do below.
Conversely, given a graph we can figure out the amplitude, period, phase shift, and vertical shift, and from this information we can figure out a formula for the graph.
Transforming the Graphs of Trigonometry Functions
This is the equilibrium line. Here it is at height 1, which tells us that the vertical shift is 1. Step 3: Draw vertical lines marking one complete cycle of the curve, beginning where the curve crosses the equilibrium line going up.
In this figure the starting line is at and the finish line is at. It would be hard to tell from the picture exactly where these lines should go, but suppose we know that our curve crosses the equilibrium line at exactly and.
Trigonometry Notes: Transformations of Trig Functions
The distance between these two lines is the period, so in this case the period is. Also, the position of the starting line tells us the phase shift. Here the phase shift is. All that remains is to find and. From the period we have:. From the phase shift we know and since that means. Step 1: Draw a horizontal line at the height of the vertical shift, in this case at height This is the equilibrium line:. Step 2: Draw horizontal lines above and below the equilibrium line at distance equal to the amplitude.
In this case the amplitude is , so we draw horizontal lines at heights and at. These are the max and min lines. Step 3: Draw a vertical starting line at coordinate equal to the phase shift, in this case at. Step 4: Draw a vertical finish line at -coordinate equal to the phase shift plus the period, so at. Step 5: So far we've drawn a box which should contain one complete cycle of the sine function. Divide the box into four boxes of equal width. The contents of the site have not been reviewed or approved by the University of Utah.
We can do all the usual tranformations to the graphs of the trig functions: add or multiply by a constant either before or after taking the function. Wave Quantities Amplitude Sine and cosine look like waves.
T-Charts for the Six Trigonometric Functions
Amplitude tells you how large those waves are. Like waves, the graphs have high points, the crests or peaks, and low points, the troughs. Amplitude is half total height difference between a crest and a trough. Alternatively, amplitude is the height of a crest compared to the average height of the wave, which is the same as the depth of a trough compared to the average height of a wave.
For sine and cosine the amplitude is 1. Using the first definition, the crests have height 1 and the troughs have height -1, so the difference is 2 and half of 2 is the amplitude, 1. Alternatively, the average height is zero, and the crests are height 1 abouve zero while the troughs are height 1 below zero.
Period The trig functions are periodic.
Sine and cosine are -periodic: The same is true for secant and cosecant. Tangent and cotangent are -periodic: The period of a trig function is the minimum a such that for all we have. Phase Phase is a concept of horizontal shift for a wave.
There is some disagreement about how to define phase shift. We will follow the definition in the text, which says that has a phase shift of. The red curve is the graph of. The green curve is the graph of. Standard Transformations Vertical Transformations Applying operations outside of the trig function produces a vertical change.
Adding a constant gives a vertical shift, multiplying by a constant gives a vertical stretch. For example the graph of would look like the graph of , but would be stretched vertically by a factor of 2 and then the whole graph would be shifted up 4.
These transformations only change the graph vertically, so phase and period are unaffected.
Horizontal Transformations Multiplication inside the function produces horizontal stretching. Multiplication by causes the curve to oscillate faster for. This changes the period. Figure it out. There are two things we would like to be able to do. One is to look at a graph of a trig function and be able to right down a formula for it.
Writing Equations from Transformed Graphs for Sin and Cos
The other is to look at a formula and be able to draw its graph. We can do this for anything of the form: by using what we learned about transformations of trig functions. We know that: So given a formula we can write down the amplitude, vertical shift, phase shift, and period of the function, and from this information we can draw a graph, which we will do below. Going from graph to fomula Here's the graph of some sine curve: Step 1: Draw horizontal lines at the maximum and minimum heights.
The distance between these two lines is 4, so the amplitude is 2. Step 2: Draw a horizontal line midway between the max line and the min line. The distance between these two lines is the period, so in this case the period is Also, the position of the starting line tells us the phase shift. Now we know: All that remains is to find and. From the period we have: From the phase shift we know and since that means.
Now we plug these values in to the formula So a formula matching this graph is:. Suppose we hav a formula like: From the formula we can write down: Now we want to draw a graph with these properties.
Transformation of trigonometric functions pdf
This is the equilibrium line: Step 2: Draw horizontal lines above and below the equilibrium line at distance equal to the amplitude. Step 6: Draw in the pieces of the sine curve. Step 7: We've drawn one cycle of the curve. Now use periodicity to make copies of this cycle.