In this discussion, we are going to graph the cosecant function with multiple transformations.
Key things to consider:
- Observe the Parent Graph of Cosecant using the Sine Graph
- Replace the cosecant function with the Sine Function
- Put the Sine Function in Standard Form
- Compute the Period of the Sine Function
- Compute the x-scale of the Sine Function
- Draw the Transformation of Sine Graph
- Draw the Transformation of Cosecant Graph
First, we begin with the parent graph of y=csc(x)
To begin with the parent graph of y= csc(x),
we observe the parent graph of y=sin(x), because sin(x) = 1/csc(x)
The parent graph of csc(x)
Beginning at zero and traveling to the right sin(x) has:
a value of 0 at x=0
a value of 1 at x=pi/2
a value of 0 at x=pi
a value of -1 at x= 3pi/2
a value of 0 at x=2pi
Beginning at zero and traveling to the left sin(x) has:
a value of 0 at x=0
a value of -1 at x= -pi/2
a value of 0 at x= -pi
a value of 1 at x= -3pi/2
a value of 0 at x= -2pi
The horizontal midline for sine is at y=0.
This pattern continues to the right and to the left.
Now, sin(x) = 1/csc(x)
Thus, for csc(x) we simply take the reciprocal of all the values of sin(x).
All the values of sin(x) that are equal to 1 and -1
will remain the same for csc(x) because
1/1=1
and
1/-1 =-1.
However, all the values of sin(x) that are equal to 0
will become undefined for csc(x)
because
1/0 is undefined.
Thus, we draw vertical asymptotes everywhere sine is equal to zero, and points everywhere
sine values are equal to 1 and -1.
Draw the graph of cosecant, starting from the max and min values of sine, away from the graph of sine and towards the asymptotes.
Now, our given trig function is of the form:
y=acsc[b(x-c)]+d, where
a is the vertical stretch/compress
b is the horizontal stretch/compress
c tells you how many units to shift right/left
d tells you how many units to shift up/down
Note that if we have “-a” we will simply reflect the x-axis.
To find the exact placements on the graph, we need to compute the period.
period = 2pi/b
Next, we compute the x-scale.
The x-scale is the increment value that we will add along x-axis.
x-scale = period/4
Now we begin at c and then add increments of our x-scale.
Draw your new graph, modeling the parent graph, but with the new transformations.
Now let us look at an example.
Example
We have
y=csc(3x) + 1.
Observe the Original Graph of Cosecant using the Sine Graph
Replace the Cosecant Function with the Sine Function
Since it is easier to begin with the graph of sine, change the function from cosecant to sine.
Thus, we have,
y=sin(3x) + 1.
Put the Sine Function in Standard Form
Next, we want the function in standard form:
y=asin[b(x-c)] + d
So, we factor out a 3 inside of the function,
put a “1” in place of “a”,
a “0” in place of c,
and a “1” in place of “d.”
Thus, we have
y=1sin[3(x-0)] +1
We see that
|a|=1 b=3 c=0 d=1
In our example, we will mirror the parent graph of sine, but we will:
compress horizontally by a factor of b (our b is 3)
Shift up be a factor of d (our d is 1)
Since our c is 0, we do not need to shift right or left
Since our a is 1, we do not have to stretch/compress vertically
To find the exact placements on the graph, we need to compute the period.
Compute the Period of the Sine Function
period = 2pi/b = 2pi/3 =2pi/3
Compute the x-scale of the Sine Function
Next, we compute the x-scale
x-scale = period/4 = (2pi/3)/4 = 2pi/12 = pi/6
Now we begin at c (0) and then add increments of pi/6.
Here are the values:
0
0 + pi/6 = pi/6
pi/6 + pi/6 = pi/3
pi/3 + pi/6 = pi/2
pi/2 + pi/6 = 2pi/3
Going the other way, we have:
04 – pi/6 = -pi/6
-pi/6 – pi/6 = -pi/3
-2pi/6 – pi/6 = – pi/2
-pi/2 – pi/6 = -2pi/3
Since |a|=1, our max and min values (before we shift up) for sine will be 1 and -1 respectively.
Since d=1, we will shift the graph up 1 unit.
Thus, the max and min value for sine becomes 2 and 0 respectively and the horizontal midline will becomes y=1.
Now there are points at:
(0, 1)
(pi/6,2)
(pi/3, 1)
(pi/2, 0)
(2pi/3,1)
(-pi/6, 0)
(-pi/3,1)
(-pi/2, 2)
(-2pi/3, 1)
and horizontal midline at y=1.
Graph the Transformations of the Sine Function
Draw the graph of sine, connecting the point and modeling the parent graph of sine, but with the new transformations.
Graph the Transformations of the Cosecant
Draw vertical asymptotes where sine is equal to one and points where sine has
minimum and maximum values.
Draw the cosecant graph, starting from the max and min points of sine, away from the sine graph, and towards the asymptotes.
Now, we have the graph of y= csc3x.
We are done!
If you have any questions, or if this reading helps let me know in the comments!
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