The secant and cosecant functions are just the reciprocals of the cosine and sine functions,
so we can apply the same type of analysis as we did to the cotangent function to get
their graphs.
Since the graphs are simpler this time, we won't go through as many steps as we did for
the cotangent function.
Here are the graphs of the cosine and sine functions.
The zeros of the cosine and sine functions turn into asymptotes for the secant and cosecant
functions.
There are no asymptotes for the cosine and sine functions, so we have no zeros for the
secant and cosecant functions.
Also the peaks and valleys of the sine and cosine function are at height plus or minus
1, so these points stay fixed.
When we look at the reciprocals we'll mark those points and gray out the cosine and sine
functions in preparation for graphing the secant and cosecant functions.
Recall that we determine whether the reciprocal function is approaching positive or negative
infinity by the signs of the original functions.
This process will give us the graphs of the secant and cosecant functions.
While we could use key values for the secant and cosecant functions to graph transformations,
it turns out that we have a helpful heuristic based on the sine and cosine graphs that is
a little bit easier to use.
The basic idea is that (the cosine and secant functions) and (the sine and cosecant functions)
transform together.
As the zeros of the cosine and sine functions move around, so do the asymptotes of the secant
and cosecant functions.
As the cosine and sine functions are stretched and shifted vertically the secant and cosecant
functions move with them.
This is best understood through an example.
Example: Graph y = 2 sec(2x + pi) + 1.
To graph this function, we start by graphing the corresponding cosine function,
y = 2 cos(2x + pi) + 1.
If you need to review these steps, you're encouraged to watch the previous videos, which
go through these steps a little more slowly.
Just as before, we determine the fundamental period and the x coordinates of our key values.
We can then substitute the y coordinates into the formula to get the transformed y values.
This gives us all the information we need to graph the cosine function.
To get the secant graph, we first draw the asymptotes through the points where the midline
crosses the cosine function.
The peaks and valleys of the cosine function become reference for the secant function.
And now we use what we know about the shape of the secant function to sketch the graph.
It is important to understand that this works because the two graphs transform together.
We are not actually graphing the cosine function and then inverting it to get the secant function.
If we did this, the formula and the graph would look quite different.
The inversion step actually happens before the vertical stretching and shifting, so we're
simply using the cosine function as a reference for the secant function.
The same process would work for graphing the transform cosecant function except that you
would graph a sine function instead of a cosine function as your starting point.
Your ability to graph the secant and cosecant functions will greatly depend on your ability
to graph the cosine and sine functions.
This is another example of how mathematics builds new ideas on top of old ones.
If you are struggling with graphing these functions you might want to go back and review
the previous couple sections to make sure that you have a solid foundation to build
on.
For more infomation >> 4.6 - Video 3 - Graphing the Secant and Cosecant Functions - Duration: 3:26.-------------------------------------------
Start a Business Using Social Media Video Posts - Duration: 1:38.
- Hey, Joe Fier here from Evergreen Profits,
reporting to you again, from the back of a car,
with my baby friend.
And also, a volcano, we're in Maui
going up the hill right now, almost there.
But, quick thing, these little short videos
that we're doing,
you're seeing Matt and I do this quite often now,
they're about minute long, sometimes longer.
We're posting them in all social media channels,
as much as we can, mainly Facebook and Instagram.
And I recommend you guys doing the same.
And you can go to evergreenprofits.com,
and look up Roland Frasier.
He actually has this bad ass strategy
where he did videos similar to this,
and he raised about, I believe it was four,
three or four million dollars on a limited budget
comparatively to what he made.
And basically you can create a business
by doing videos like this.
So go over there to Evergreen Profits,
and look up Roland Frasier, and you are going to love it.
It just got released.
And that's about it, so keep watching these videos
and experiment for yourself.
Alright, bye.
-------------------------------------------
VIDEO: K9 learns to walk on hot pavement in dog booties - Duration: 0:40.
For more infomation >> VIDEO: K9 learns to walk on hot pavement in dog booties - Duration: 0:40. -------------------------------------------
Learn Colors with Train | Learning Videos | Color Cartoons for Kids - Duration: 1:50.
Old MacDonald had a farm,E-I-E-I-O. And on his farm he had some chicks,E-I-E-I-O.
With a chick chick here, And a chick chick there, Here a chick, there a chick, Everywhere a chick chick,
Old MacDonald had a farm,E-I-E-I-O.
Old MacDonald had a farm. E-I-E-I-O And on his farm he had some dogs. E-I-E-I-O
With a Woof Woof here, And a Woof Woof there. Here a Woof, There a Woof, Everywhere a Woof Woof!
Old MacDonald had a farm. E-I-E-I-O
Old MacDonald had a farm. E-I-E-I-O And on his farm he had some cows. E-I-E-I-O
With a Moo Moo here, And a Moo Moo there. Here A Moo, There a Moo, Everywhere a Moo Moo!
Old MacDonald had a farm. E-I-E-I-O
Old MacDonald had a farm. E-I-E-I-O And on his farm he had some ducks. E-I-E-I-O
With a Quack Quack here, And a Quack Quack there, Here a Quack, There a Quack, Everywhere a Quack Quack!
Old MacDonald had a farm, E-I-E-I-O
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