4.6 Graphs of Other Trigonometric FUNctions
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Transcript of 4.6 Graphs of Other Trigonometric FUNctions
4.6 Graphs of Other Trigonometric FUNctions
How can I sketch the graphs of all of the cool quadratic FUNctions?
Graph of the tangent FUNction
• The tangent FUNction is odd and periodic with period π.
• As we saw in Section 2.6, FUNctions that are fractions can have vertical asymptotes where the denominator is zero and the numerator is not.
• Therefore, since , the graph of will have vertical asymptotes at , where n is an integer.
xxx
cossintan xy tan
n22
Let’s graph y = tan x.
• The tangent graph is so much easier to work with then the sine graph or the cosine graph.– We know the asymptotes.– We know the x-intercepts.
x
y
y = 2 tan (2x)
• Now, our period will be
• Additionally, the graph will get larger twice as quickly.
• The asymptotes will be at • The x-intercept will be (0,0)
2
b
4
x
y
• The period is 2π.• The asymptotes are at ±π.• The x-intercept is (0,0).
2
tan xy
x
y
Graph of a Cotangent FUNction
• Like the tangent FUNction, the cotangent FUNction is – odd.– periodic.– has a period of π.
• Unlike the tangent FUNction, the cotangent FUNction has– asymptotes at period πn.
y = cot x
• The asymptotes are at ±πn.• There is an x-intercept at
0,2
x
y
y = -2 cot (2x)
• The period is
• There is an x-intercept at
• There is an asymptote at
2
0,4
2
x
y
Graphs of the Reciprocal FUNctions
• Just a reminder – the sine and cosecant FUNctions are reciprocal
FUNctions– the cosine and secant FUNctions are reciprocal
FUNctions• So….– where the sine FUNction is zero, the cosecant
FUNction has a vertical asymptote– where the cosine FUNction is zero, the secant
FUNction has a vertical asymptote
• And…– where the sine FUNction has a relative minimum,
the cosecant FUNction has a relative maximum– where the sine FUNction has a relative maximum,
the cosecant FUNction has a relative minimum– the same is true for the cosine and secant
FUNctions
• Let’s graph y = csc x
x
y
x
y
x
y
Now, let’s graph y = sec x
x
y
x
y
x
y
Now, you try your own….
• Just graph the FUNction as if it were a sine or cosine FUNction, then make the changes we have already made.
xy
xy
sec
2csc2
x
y
x
y
Damped Trigonometric Graphs (Just for Fun!)
• Some FUNctions, when multiplied by a sine or cosine FUNction, become damping factors.
• We use the properties of both FUNctions to graph the new FUNction.
• For more fun on damping FUNctions, please read p 339 in your textbook.
• For a nifty summary of the trigonometric FUNctions, please check out page 340.
• As a matter of fact, I would make sure I memorized all of the information on page 340.
Writing About Math
• Please turn to page 340 and complete the Writing About Math – Combining Trigonometric Functions.
• You may work with your group.• This activity is due at the end of the class.