The only part of graphing that we haven't yet drawn is phase shifts. The next example includes this aspect.
The amplitide of this graph is going to be the same as for regular sine waves, because there's an "understood" 1 multiplied on the sine.
But the midline of the graph is going to be at y = 3 instead of y = 0 (that is, the x-axis), because of the "+3" at the end of the function.
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The regular period for sine waves is 2π, but the variable in this function is multiplied by π; doing the division, the period of this particular function is going to be . Since I have to graph "at least two periods" of this function, I'll need my x-axis to be at least four units wide.
Now, the new part of graphing: the phase shift. Looking inside the argument, I see that there's something multiplied on the variable, and also that something is added onto it. To figure out the actual phase shift, I'll have to factor out the multiplier, π, on the variable. The argument factors as . Now I can see that there's a added to the variable, so the graph will be shifted units to the left.
I'll start with the usual graph:
I know that this graph has a vertical shift upwards of three units. But, instead of shifting the graphed sine wave three units up, I'll add room underneath my current graph, shift the horizontal axis three units down, and then re-number the y-axis:
The above is the same graph; all I've really done is moved the x-axis down three units, redrawn and relabelled it, and then renumbered the y-axis. I haven't touched the graphed sine wave, drawn in blue.
The regular period for sine waves goes from 0 to 2π; this one goes from 0 to 2, so I'll re-number the x-axis:
All I've done in the above graph is relabel the x-axis from π, 2π, 3π, and 4π to 1, 2, 3, and 4, respectively. I haven't touched my blue graphed line.
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From the phase-shift computations, I know that the graph is shifted to the left by , so I'll shift the y-axis to the right by and re-number the x-axis again. This is the last bit of computation, so this is my final graph:
All I did to get the last graph above was erase the existing y-axis, redraw it half a unit to the right, and then erase and rewrite the labels on the x-axis, again half a unit to the right. I never touched my blue-line graph.
Can you see why I used pencil and did a lot of erasing when I was doing graphing?
My best advice regarding doing these graphs is to practice, practice, practice. You don't want to freeze or have a "brain-fart" on the test, and this is pretty straightforward once you get the feel of it. So keep doing extra graphing, until you are feeling comfortable and confident in your skills.
You can use the Mathway widget below to practice finding the amplitude, period, and phase shift. Try the entered exercise, or type in your own exercise. Then click the button, selecting "Find Amplitude, Period, and Phase Shift" from the list of options, to compare your answer to Mathway's. (You can also select "Graph".)
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URL: https://www.purplemath.com/modules/grphtrig3.htm
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