If you know the basic graphs, then the more-complicated graphs can be fairly easy to draw. Here's how it works:
The "minus" sign tells me that the graph is upside down. Since the multiplier out front is an "understood" –1, the amplitude is unchanged. The argument (the 3x inside the cosine) is growing three times as fast as usual, because of the 3 multiplied on the variable, so the period is one-third as long. The period for this graph will be (2/3)π.
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Here is the regular graph of cosine:
I need to flip this upside down, so I'll swap the +1 and –1 points on the graph:
...and then I'll fill in the rest of the graph. (The original, "regular", graph is shown in gray below; my new, flipped, graph is shown in blue.)
Okay, that takes care of the amplitude. Now I need to change the period.
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Rather than trying to figure out the points for the graph on the regular axis, I'll instead re-number the axis, which is a lot easier. The regular period is from 0 to 2π, but this graph's period goes from 0 to (2π)/3. Then the midpoint of the period is going to be (1/2)(2π)/3 = π/3, and the zeroes will be midway between the peaks (the high points) and the troughs (the low points). So I'll erase the x-axis values from the regular graph, and re-number the axis.
The following is my final (hand-in) graph:
Notice how I changed the axis instead of the graph. You'll quickly get pretty good at drawing a regular sine or cosine, but the shifted and transformed graphs can prove difficult. Instead of trying to figure out all of the changes to the graph, just tweak the axis system.
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The previous example showed how to change things around for the amplitude and the period. The next example shows how to move things around for a vertical shift.
The regular tangent looks like this:
The graph for tan(θ) – 1 is the same shape as the regular tangent graph, because nothing is multiplied onto the tangent.
But this graph is shifted down by one unit. In other words, instead of the graph's midline being the x-axis, it's going to be the line y = -1.
Rather than trying to figure out the points for moving the tangent curve one unit lower, I'll just erase the original horizontal axis and re-draw the axis one unit higher. Then my final (hand-in) graph looks like this:
You may, at first, want to use scratch-paper for the various changes (flipping graphs upside-down, moving axes up and down, changing the measurements on the x-axis, etc), so your hand-in homework isn't full of erasures. But get used to working neatly, from start to finish, on the hand-in sheet, so your work on the next test will be acceptable.
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Hint: Start by drawing lightly in pencil, and have a good eraser (like from an office supply or craft store). Be sure you're using a ruler for your final drawing. Also, it can be helpful to use a regular pencil for the temporary "regular" graph, but then use colored pencils for your final version.
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