In this discussion, we graph the secant function with multiple transformations.
Key things to consider:
- Observe the Parent Graph of Secant using the Cosine Graph
- Replace the Secant function with the Cosine Function
- Put the Cosine Function in Standard Form
- Compute the Period of the Cosine Function
- Compute the x-scale of the Cosine Function
- Draw the Transformation of Cosine Graph
- Draw the Transformation of Secant Graph
Let us discuss concepts, then work an example.
First, begin with the parent graph of y=sec(x) using the parent graph of y=cos(x), because cos(x) = 1/sec(x)
The parent graph of cos(x)
Beginning at zero and traveling to the right cos(x) has:
a value of 1 at x=0
a value of 0 at x=pi/2
a value of -1 at x=pi
a value of 0 at x= 3pi/2
a value of 1 at x=2pi
Beginning at zero and traveling to the left cos(x) has:
a value of 1 at x=0
a value of 0 at x= -pi/2
a value of -1 at x= -pi
a value of 0 at x= -3pi/2
a value of 1 at x= -2pi
This pattern continues to the right and to the left.
Now, cos(x) = 1/sec(x)
Thus, for sec(x) we simply take the reciprocal of all the values of cos(x).
All values of cos(x) that are equal to 1 and -1
will remain the same for sec(x) because 1/1=1 and -1/1 =-1.
However, all values of cos(x) that are equal to 0 will become undefined for sec(x) because1/0 is undefined.
Thus, draw vertical asymptotes everywhere cosine is equal to zero, and points everywhere cosine values are equal to 1 and -1.
Draw the graph of secant, starting from the max and min values, away from the graph of cosine and towards the asymptotes.
Now, the function is of the form:
y=asec[b(x-c)]+d, where
a is the vertical stretch/compress
b is the horizontal stretch/compress
c indicates many units to shift right/left
d indicates how many units to shift up/down
Note that if there is negative “a”, simply reflect the x-axis.
To find the exact placements on the graph, compute the period.
period = 2pi/b
Next, we compute the x-scale.
The x-scale is the increment value that is added along x-axis.
x-scale = period/4
Now begin at c and then add increments of the x-scale.
Draw your new graph, modeling the parent graph, but with the new transformations.
Now let us look at an example.
Example
Let’s say we have
y=sec[4x-pi].
Replace the Secant Function with the Cosine Function
Since it is easier to begin with the graph of cosine, change the function from secant to cosine.
Thus, we have,
y=cos[4x-pi].
Put the Cosine Function in Standard Form
Put the function in standard form to get:
y=acos[b(x-c)] + d
Factor out a 4 inside the function,
put a “1” in place of “a”,
and a “0” in place of “d.”
y=1cos[4(x-pi/4)] +0
We see that
|a|=1 b=4 c=pi/4 d=0
We will mirror the parent graph of cosine, but:
shift all x-values c units to the right (our c is pi/4)
compress horizontally by a factor of b (our b is 4)
Since a is 1, we do not have to stretch/compress vertically.
Compute the Period of the Cosine Function
To find the exact placements on the graph, compute the period.
period = 2pi/b = 2pi/4 =pi/2
Compute the x-scale of the Cosine Function
Next, compute the x-scale
x-scale = period/4 = (pi/2)/4 = pi/8
Now begin at c (pi/4) and then add increments pi/8.
Here are the values:
pi/4
pi/4 + pi/8 = 3pi/8
3pi/8 + pi/8 = pi/2
pi/2 + pi/8 = 5pi/8
5pi/8 + pi/8 = 3pi/4
Going the other way, we have:
pi/4 – pi/8 = pi/8
pi/8 – pi/8 = 0
0 – pi/8 = – pi/8
-pi/8 – pi/8 = -2pi/4
Since |a|=1, the max and min values for cosine will be 1 and -1 respectively.
So now we have points at:
(pi/4, 1)
(3pi/8,0)
(pi/2, -1)
(5pi/8, 0)
(3pi/4,1)
(pi/4, 1)
(pi/8,0)
(0, -1)
(-pi/8, 0)
(-pi/4,1)
Graph the Transformations of the Cosine Function
Draw the graph of cosine, modeling the parent graph, but with the new transformations.
Graph the Transformations of the Secant
Draw vertical asymptotes where cosine is equal to zero and points where cosine has a
minimum and maximum.
Draw the secant graph, starting from the max and min points, away from the cosine graph and towards the asymptotes.
We are done!
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