BAM!!! Mr. Tarrou
We are going to do a couple of examples of graphing a trig function, sine and cosine,
that have some transformations in them. Some vertical stretches
or some change in the period which is a horizontal stretch or compression.
They may have vertical shifts.
We are going to do this without the aid of a calculator.
So if your teacher is allowing you to do these graphs with a graphing calculator
you might not like this video very much.
I teach graphing without the use of a calculator to make sure my students understand the unit circle very well
I also want to give them practice adding, subtracting, multiplying, and dividing fractions.
I want them to actually understand this topic as opposed to just tippity tapping on whatever calculator they bought.
This is the general from of our trig functions.
This is either going to be sine or cosine because that is what we are talking about today.
This could also be secant, cosecant, tangent, cotangent. We will be doing those other graphs in a later video.
And, we have got some values in here in different colors that indicate transformations.
These will be the same transformations we learned early in our PreCalculus book, or last year in Algebra 2.
If you are not sure what these transformations are like vertical stretches, horizontal stretch or compression value,
for trig functions B and C are going to work together to make phase shifts which will move the graph left or right,
and vertical shifts.
Then you may want to look up with my search window in the upper right hand corner
and search transformations.
You can see how these apply to some simpler equations like maybe parabolas or what-not.
When we are looking at our trig function, if a is greater than 1 our graph is going to be stretched vertically.
If A is between zero and one there will be a vertical compression.
The value of A is going to be the amplitude of our trig function.
I talk about that in our previous lesson introducing basic sine and cosine functions.
I will point that out once we get an example up here and worked out.
If you leading coefficient is negative that is going to reflect your graph about the x-axis.
Kind of flip it down in a sense.
B and C are going to act together to setup your phase shift.
That is going to be your left or right movement. Think about your parabola.
If you graph y equals x squared and then graph y equals x minus 2 squared, it is going to shift that parabola to the right 2 places.
Well, when we have an addition or subtraction inside of our math function we are going to have those horizontal shifts.
We will be calling them phase shifts in trigonometry.
Now in the textbook I am teaching from, it shows that x can have a coefficient that is not equal to one.
As such, a phase shift is defined as C over B in my book
and that is going to be the horizontal movement.
If you see BX-C, then you will have a shift to the right C/B units.
And if you see BX+C inside your parentheses, then that will be a movement to the left.
Now, maybe your textbook...and my old trig book...would have this value of B in front of a parenthesis
and it would hold the coefficient of X to always be 1.
That way...(B(X-Phase Shift))...you could like inside the parenthesis and see what the phase shift is.
If I factor out a B, then I am going to have a term of C/B in here anyway.
I want to have my notes consistent with the book my students are currently using so my students understand this as much as possible.
If you have a plus or minus outside of you math function, usually at the end, again not inside your actual math function then you have a vertical shift.
This is just like when you first learned about transformations of functions.
Now again we are graphing these without the aid of a calculator.
We need to attempt to graph just the easy points. Not the square root of three over two, or square root of two over two.
I want to graph the easy point of 0, 1, and -1...and that is it.
I am going to say these directions regardless of what trig function we are graphing.
I am going to say these two steps all the time.
When you want to graph a trig function without the aid of a calculator, use your brain.
You start a t-table with the phase shift value and then from there count by one quarter of the period.
As a tip, because you will be doing five points in each of your t-tables,
to make that mental math a little bit easier write your phase shift value, count value, and C with a common denominator.
This way when you work out that t-table you don't have to keep doing those common denominators in your head.
They will already be set up for you.
Now I am talking about the phase shift value, the value of C, and it is not up here but what you are going to count by.
You are going to count by a quarter of the period.
You know your unit circle by now I hope.
And you know as you go around the unit circle at 0, pi over 2, pi, 3pi over 2, and 2pi that there are four places around that unit circle that you have those easy values of 1, 0 , and -1.
Zero degrees where this point is (1,0), pi over 2 or ninety degrees...we are going to do radians...where our point is (0,1)
And then we have where our point is (-1,0) and of course (0,-1) and back to the beginning.
So we are looking for those four points around the unit circle.
Those four easy point. We are going to count by a quarter of the period. That is why we are doing this 1/4 idea
to hit those four easy points around the unit circle.
Lets do an example.
Lets look at the parent function very quickly of y equals the sine of x.
Now this has an amplitude of 1. I erased all my notes because I needed somewhere to write.
The leading coefficient is the amplitude.
The absolute value of a is the amplitude.
Here we have an amplitude that is equal to one.
The period is going to be 2pi divided by B which we find in front of the X. Here B is 1 so it is 2pi.
We have a phase shift...well...there is no plus or minus inside our parenthesis
it is just the sine of x, so there is no phase shift.
The vertical shift is when we have a plus or minus at the end of the equation...not in the function...and here we have none. This is the parent function.
And, we are going to county by one quarter of the period.
So, we are counting by one quarter of the period of 2pi
which is 2 time 1 is 2pi
over 4 which reduces down to pi/2.
Now where did I just say the easy points were?
Since our period is a standard 2pi,
we have 0, pi over 2, pi, 3pi over 2, and 2pi
Counting by this value is giving me my easy values of 0,1, and -1.
Lets take a look at this graph.
Excuse me, we need a t-table.
So we are going to set up our t-table.
X and Y equals sine of x and unlike our previous video about basic sine and cosine
we are not going to plot 12 points, just four...excuse me five.
Start with the phase shift.
Now count by one quarter of the period.
So, 0, pi over 2, 2pi over 2, 3pi over 2, and 4pi over 2.
That is 5 x values and that will be enough to get the first period of your graph set up.
Period by the way is the horizontal distance it takes for the graph to repeat.
Once you get your first period, you can just set that stamp and the pattern will just keep cycling over again and again.
As you rotate around that unit circle of course we get repetitive values and that sets up our idea of period
when you graph a function.
So, the sine of zero is zero. The sine of pi over 2 is 1.
This sine of 2pi over 2 or pi, when you rotate pi radians, the sine value of y over r is back to 0.
When you rotate 3pi over 2, the y value or sine value is negative one.
And of course 4pi over 2 is 2pi, so all the way around the unit back to the beginning, the sine value there is zero.
I need some room to write so hopefully you copied these directions down.
I don't need to seem them, I know what they are:)
Lets graph y equals sine of x.
I just made a mistake.
When you graph these without the aid of a graphing calculator,
you should first draw just the x axis. Don't forget the x axis is domain and we do plug angle measures into trig functions.
Our x values are not representing x values on the unit circle,
that is not a graph of sine or cosine that is the graph of a circle.
When we set up this graph the x axis, the domain, it is theta, it is what we plug into the trig function, it is an angle measure.
Use these x values along the t-table to set up your number line, the x axis.
If you do not happen upon zero
count backward or forward until you get to zero.
If you skip over zero, like maybe you have an x value of pi over 4 and the next one is negative 3pi over 4,
that is fine. Just keep your spaces even then drop your y axis down where the x value would be zero.
You do not want a space this big to represent pi over 8 and then have the same interval represent...say...pi over 2.
Equal intervals representing different values would be incorrect. So set up that x axis
count forward and backward until you get to zero, and if you skip over zero that is ok.
You will not have a zero if you would include it in your t-table. Then just slide your y axis down through zero or between your negative and positive x values.
So what do we have.
(0,0) ... (pi over 2, 1)
At pi sine is zero
At 3pi over 2 we are at negative one.
At 2pi we are back to zero.
We have graphed one period of the sine function.
If I want two periods,
now that I have my counts along the x axis, I can just continue that count
And extend my values to the right or the left.
Because my one period is set up, I know the pattern for the rest of the graph.
I am going to keep counting backwards
by my one quarter of the period. Or I can just look at the x axis.
2pi, 3pi over 2, 2pi over 2, pi over 2, 0, negative pi over 2,
negative 2 pi over 2
negative 3 pi over 2
and finally negative 2 pi.
Look at my patter. 0, -1, 0, 1, 0 So what will then number be?
-1
Back to zero
Back to 1
and back to zero. Let me highlight this period in a different color.
See how this looks like a stamp that just repeated,
that is the period.
This horizontal distance it take the graph to repeat, a trig function graph to repeat is the period.
Look at the overall height. We go up to 1 and down to negative 1.
I would say the overall height is 2 units.
1 minus negative 1 is 2.
And the amplitude, that I had written up here a second ago, is half of the overall height.
So if you take 1 minus negative 1 is 2, and half of 2 is equal to 1. There is your amplitude which is equal to 1.
Now lets do another example because you are probably thinking why did you spend 5 minutes graphing the parent function.
Lets now graph an example like y equals sine of 2x
and show you what the value of B does when it is greater than 1.
This is going to have an amplitude equal to one.
The period is equal to 2 pi over 2.
That is supposed to be pi.
The phase shift is going to be zero because there is no plus or minus inside the sine function.
The vertical shift is equal to zero.
And we have a count value that is equal to pi over 4.
I can tell by the alarm in my pocket that I only have one minute left.
I do not have the time to set up a t-table. I am going to do another couple of examples in another video.
But, notice what this period is...it is pi.
So that B value that is greater than one is going to cause this cycle
to not take a distance of 2pi, but this up and down cycle to take a horizontal distance of just pi.
When you have a B that is bigger than 1, it is going to compress the graph.
It is not going to stretch like a vertical transformation.
Again when that B value is greater than one it is going to take that graph and squeeze it together horizontally.
I am going to do two more examples in my next video.
Thanks for listening to me blabber on about how this works and why:)
Check it out if you need it. I am Mr. Tarrou GO DO YOUR HOMEWORK:D