Algebra and Trig. I
4.6 – Graphs of Other Trigonometric Functions
The graph of
The graph of
. The trigonometric functions can be
graphed in a rectangular coordinate system by plotting points
whose coordinates satisfy the function. Thus, we graph
by listing some points on the graph because the period of the
tangent function is π, we will graph the function on the interval
[0, ]. The rest of the graph is made up of repetitions of this period
x
0
y=tanx
0
undefined
The graph of the tangent function allows us to visualize some of
the properties of the tangent function:
1. The tangent function is an odd function
The graph is symmetric with respect to the origin
2. The period is π. The graph’s pattern repeats in every interval
of length π.
3. The function is undefined at so that there is a vertical
asymptote at
(The tangent function will be undefined at
all odd multiples of )
Below we plot the points from above and we know that the graph
is symmetric to the origin so we can fill in the rest of the
interval
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Properties of
1. Period π
2. Domain: all real numbers except odd multiples of
3. Range: all real numbers
4. Vertical Asymptote at odd multiples of
5. An x-intercept occurs midway between consecutive vertical
asymptotes
6. Odd function with origin symmetry
7. Points on the graph a (1/4)th and (3/4)th of the way between
consecutive asymptotes have a y value of 1 and -1.
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The graph of
Bx-C=
y-cord. = A
Bx-C=
x-int
y-cord. = -A
Graphing
1. Find two consecutive asymptotes by finding an interval
containing one period
a pair of consecutive asymptotes occur at
2. Identify an x-intercept, midway between the consecutive
asymptotes
3. Find the points on the graph (1/4)th and (3/4)th of the way
between the consecutive asymptotes. These points have ycoordinates –A and A.
4. Use steps 1-3 to graph one full period of the function.
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Example – Graph
1. Find two consecutive asymptotes
2. Find the midway point between your asymptotes
3. Find points
4. Graph the function
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Example – Graph two full periods
1. Find two consecutive asymptotes
2. Find the midway point between your asymptotes
3. Find points
4. Graph the function
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The graph of
Bx-C=0
y-cord. = A
Bx-C=
x-int
y-cord. = -A
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Hannah Province – Mathematics Department – Southwest Tennessee Community College
Properties of
1. Period π
2. Domain: all real numbers except integer multiples of
3. Range: all real numbers
4. Vertical Asymptote at integer multiples of
5. An x-intercept occurs midway between consecutive vertical
asymptotes
6. Odd function with origin symmetry
7. Points on the graph a (1/4)th and (3/4)th of the way between
consecutive asymptotes have a y value of 1 and -1.
Graphing
1. Find two consecutive asymptotes by finding an interval
containing one period
a pair of consecutive asymptotes occur at
2. Identify an x-intercept, midway between the consecutive
asymptotes
3. Find the points on the graph (1/4)th and (3/4)th of the way
between the consecutive asymptotes. These points have ycoordinates –A and A.
4. Use steps 1-3 to graph one full period of the function.
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Example – Graph
1. Find two consecutive asymptotes
2. Find the midway point between your asymptotes
3. Find points
4. Graph the function
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The graph of y = csc(x) and y = sec(x)
We obtain the graph of the graph of y = csc(x) and y = sec(x)
by noting that
and sec(x) = 1/cos(x)
We draw the graph of y = cos x first, and consider the reciprocals of
the y-values:
The graph of y = sec x: So we will have asymptotes where cos x has
value zero, that is:
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We recall that
So we will have asymptotes where sin x has value zero, that is:
We draw the graph of y = sin x first, and consider the reciprocals
of the y-values:
The graph of y = csc x:
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Properties of
1. Period 2π
2. Domain: all real numbers except integer multiples of
3. Range: all real numbers y such that
4. Vertical Asymptote at integer multiples of
5. Odd function with origin symmetry
Properties of
1. Period 2π
2. Domain: all real numbers except odd multiples of
3. Range: all real numbers y such that
4. Vertical Asymptote at odd multiples of
5. Even function with y-axis symmetry
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Example – Use the graph of y=2sin(2x) to obtain the graph of
y=2csc(2x)
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Example – Graph
for –π<x<5π
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Hannah Province – Mathematics Department – Southwest Tennessee Community College