In this discussion, we are going to graph the parent graph of the secant function,
discuss how to read the graph to come up with an equation, and I will
explain how to use the standard equation to come up with the graph.
Here is an outline of this article:
- Graph the Parent Graph of the Secant Function
- No Max and min, thus, no Amp
- Observing where Cosine and Secant Coincide
- Find the Period given the Graph
- Find the Phase Shift Given the Graph
- Find the Vertical Shift Given the Graph
- Find the x-scale Given the Graph
- The Standard Equation of the Secant Function
- Find the vertical Stretch/Compress Given the Standard Equation
- Find the Period and Horizontal Stretch/Compress Given the Standard Equation
- Find the Phase Shift Given the Standard Equation
- Find the Vertical Shift Given the Standard Equation
- Find the x-scale Given the Standard Equation
It is assumed that you already understand the unit circle before reading this article.. It is also assumed that you know how to graph the cosine function.
Thus, we begin with what you already know.
Recall the unit circle:
We have terminal points in the form (x,y), and secant takes the value 1/cosine, which is 1/x.
The easiest way to graph the secant function is to first graph the cosine function. So, let us look at the graph of cosine.
Beginning at zero and going counterclockwise,
cos(0) = 1
cos(pi/2) = 0
cos(pi) = -1
cos(3pi/2) = 0
cos(2pi)=1
Starting from zero and going clockwise,
cos(0) = 1
cos(-pi/2) = 0
cos(-pi) = -1
cos(-3pi/2) = 0
cos(-2pi)=1
This pattern continues to the right and to the left.
Now we know that secant is the reciprocal of cosine.
Thus, we take the reciprocal of the cosine values.
Everywhere that cosine is equal to 0, secant is undefined because 1/0 is undefined.
So, we put vertical asymptotes everywhere cosine is equal to zero.
We also see that everywhere cosine is equal to 1 and -1, secant is equal to 1 and -1
Respectively because 1/1=1 and 1/-1= -1.
So, we put points everywhere cosine is equal to 1 and -1.
We draw the graph of secant away from the cosine graph and towards the asymptotes.
The pattern continues to the right and left.
Now, let us gather some information from the graph.
No Max, No Min, No Amplitude.
Since the range of the secant function is from negative infinity to -1, then from 1 to
infinity, there are no max and min values (like sine and cosine has max and min
values). Thus, there is no amplitude.
Observing where Cosine and Secant Coincide
Note that the max and min values of the cosine graph coincide with the secant graph.
The Period Given the Graph
Now, let us discuss the period.
The period of secant is 2pi because the graph repeats itself every 2pi unites. Observing the graph, from -pi/2 to pi/2, the secant graph points up, and from pi/2 to 3pi/2, the secant graph points down; then the graph repeats.
The Phase Shift Given the Graph
The phase shift, or the horizontal shift, determines whether the graph will shift to the right or to the left. It is easier to observe the phase shift of the cosine graph. Since the Cosine graph begins at zero and the max value, we see that the phase shift is zero.
The Vertical Shift Given the Graph
The vertical shift determines whether the graph will go up or down. Again, it is easier to observe the vertical shift of cosine. We see that the horizontal midline of cosine begins at y=0. Thus we say that the vertical shift is zero.
The x-scale Given the Graph
The x-scale is the increment value that goes along the x axis. It is easier to observe the x-scale of the cosine graph. From one major point to the next, we have an increment value of pi/2.
For example,
from 0 to pi/2 is a distance of pi/2,
from pi/2 t
o pi is a distance of pi/2,
from pi to 3pi/2 is a distance of pi/2,
from 3pi/2 to 2pi is a distance of pi/2, and so on.
The Standard Equation of Secant
The standard equation of the secant function is of the form:
y = asec[b(x-c)] + d.
If we were to write the original secant function in standard form, we have
y = 1sec[1(x-0)] + 0.
Let us assume that there is no graph, and we are using the standard equation to find the graph.
Note that, when using the standard equation, it is easier to graph in terms of cosine first. Then graph the secant function based on the cosine graph.
Thus, let us rewrite as follows:
y = 1cos[1(x-0)] + 0.
Now let us look at each part in detail.
The Vertical Stretch Given an Equation
|a| = 1: It is known at the vertical stretch/compress.
If |a| is greater than zero, but less than 1, the graph compresses vertically.
If |a| is greater than 1, the graph stretches vertically.
If “a” itself is less than zero, you will graph the function as if the negative sign did not exit, then reflect the x-axis at the end.
The Period and Horizontal Stretch/Compress Given an Equation
|b| = 1: It is known as the horizontal stretch/compress.
If |b| is greater than 0 but less than 1, the graph stretches horizontally.
If |b| is greater than 1, the graph compresses horizontally.
Notice that “b” operates in the opposite manner of “a.”
If “b” itself is less than zero, the graph reflects the y-axis of the original graph.
Or better yet, since cosine is an even function it is symmetric about the y-axis. Thus, we can simply ignore the negative sign altogether.
y = acos[-b(x-c)] + d = acos [b(x-c)] + d
b is also related to the period by the following equation
Period = pi/|b| = pi/|1| =pi
The Phase Shift Given an Equation
c = 0: It is known as the phase shift, or horizontal shift.
This tells us how many units to shift right or left.
If c is greater than zero, we shift right.
If c is less than zero, we shift left.
Note, that in the form
y = acos[b(x-c)] + d , to find c,
set x-c=0 and solve for x to get x = c.
In the form
y = acos[b(x+c)] + d,
set (x+c)=0 and solve for x to get x = -c.
Notice that we flip the signs from the standard equation when finding c.
The Vertical Shift Given an Equation
d=0: It is known as the vertical shift.
This tells us how many units to shift up and down.
If d is greater than 0, we shift up
If d is less than zero, we shift down.
We begin our graph at the phase shift and the max value of cosine..
The x-scale Given an Equation
The x-scale is the increment value along the x-axis.
To find the x-scale, divide the period by 4.
2pi/4=pi/2.
Now begin at c, the add increments of the x-scale.
c=0
0+pi/2=pi/2
pi/2+pi/2=pi
2pi/2+pi/2=3pi/2
3pi/2+pi/2=2pi
Going the other direction, we have
c=0
0-pi/2=-pi/2
-pi/2-pi/2=-pi
-pi-pi/2=-3pi/2
-3pi/2-pi/2=-2pi
Graph the cosine function, using the x-scale, then draw the secant graph based on the cosine graph.
We see that we get the same graph in the beginning.
Now you are ready to graph the secant function with multiple transformations!
Thank you for reading! I hope that I can assist you on your math journey!
Sincerely,
MathAngel369
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