First we were still talking about Sinusodial Functions today, finishing up an example or two and we moved on something new. I would like to recap on Sinusoidal Functions and provide a very brief summary about it (just as a review)
- f(x)=asinb(x-c)+d OR f(x)=acosb(x-c)+d
a: amplitude
b: period
c: horizontal shift
d: vertical shift
- Amplitude: The "highest" or "lowest" a graph can go.
- Period: The length of the x-value in one cycle.
- Horizontal shift: Moving the graph to the right or left.
- Vertical shift: Moving the graph up or down.
Steps into creating equations for Sinusoidal Functions- Find the middle axis (For me I have an artistic mind so I can look at a graph and find the part where it can be cut in half. So I am able to find the middle axis just by looking) The middle axis tells us whether or not the graph moved from the origin aka: 0 (this will be value d)
- Find the amplitude (this will be value a) To find the amplitude, do not go to a graph and look at the highest point or lowest point then say that's the amplitude. It's not correct. Easiest way is to go to the middle axis and count from the middle axis to the highest point the graph is reached.
So if the middle axis is 2 and the highest point is at 6, you would count... 3, 4, 5 and 6. How many numbers was that? 4! So the amplitude is 4. (See, it's not 6!) - Determine the period and then calculate the b-value. Calculating the b-value is not hard if you understand manipulations of an equation. (Below will show you the steps of finding the b-value if you do not know how to transpose/manipulate equations) Sorry guys, the uploader uploaded this so small! You can find a bigger version by clicking here!
- Identify the type of original wave (it's either y=sinx or y=cosx) I would give you guys a great explanation on how to find it but as of right now I am deciphering my own interpretation of how to find it easily for myself!
- Create the first equation using the a, b, c and d values (make sure you guys wrote it down nice and neat somewhere so you can remember!)
- Create the second and third equations including the a,b,c and d values.
- Double check the horizontal shift symbols.
- Double check the symbol of the a value in the equation.
- When creating the two other equations after determining the equation for the sinusoidal function, the two equations have to be the other function that was not the original. So if the original function was cos, then the other two equations are sin. If the original function was sin, then the other two equations are cos.
- The amplitude and period are always the same in all three equations.
- Your goal is to manipulate the type of function you're working with for the second and third equations by making it mask the original equation.
- I noticed a pattern in creating the two extra equations (after determining the first equation) but I am unsure if it is ALWAYS right, but I will share this pattern I saw.
After determining the first equation, to find the second equation you would write the amplitude, period then the function you are working with (if original = cos, then the function you're working with would be sin, vise versa)
Now here's the pattern I saw. For the second equation you would move that function over to the right (or left) a certain amount of times then add the appropriate d-value (the d-value is always the same as the original) This is the second equation, where you will shift the graph using this second equation to mask the original graph wave whatever it is.
For the third equation, all you want to do is flip the graph wave whatever it is to match the original one. To do so you simply add a (-) sign in front of the amplitude then for the horizontal shift, you would go left (if your second equation has a horizontal shift of right) or you would go right (if your second equation has a horizontal shift of left) Then add the d-value. Remember, the a, b and d value are always the same.
Now here's something as well that I do not know if it is always true but is something I noticed. I mentioned earlier about moving a function to the right or left "a certain amount of times" What is that amount of time? Let me try to explain (I'm not a math teacher so I do not know if this is valid but it is something I noticed)
Look at the graph. Most graphs are separated into 4 equal parts of the wave, line or whatever function it is (I really need to know the proper name of it) Remember what Mr. Piatek taught us? Find the period, divide it into two, then divide that number into two etc? Well if you look, the number closest to the 0 that is labeled would indicate how many times we would move. If it's π/2 or π/4, heck 2π/3, that is how many times we would move.
We will move on to what we learned today, which was solving those f(x) type of functions (no graphing involved!! Does anyone want to scream YAY!?) What we should remember in this is some information from circular functions (See? Mr. Piatek wasn't lying when he said every unit will help us in the next unit)
So what is it that we should know from circular functions?
- Knowledge in quadrantal angles, quadrants and special triangle.
- Know how to go from radians to degrees and/or degrees into radians
- Know the CAST rule
- Basic knowledge of pre-calculus (everything we learned so far), or well math overall (there should be concepts that we should know already... like CAST, quadrant rules etc!)
What are quadrantal angles? They are angles that hit these (degrees) on a graph: 0°, 90°, 180°, 270° and 360°. I pray to the lord we all know this and the values of cos(x) and sin(y) when it lays on those measurements/degrees. I already memorized them but if it helps, you can create a little pretty diagram to help you.
Here are your special triangles. Keep in mind that 0°/360° is π, 90° is π/2, 180° is π and 270° is 3π/2. So basically the diagram says at 0°/360°, cos is 1 and sin is 0, at 90° cos is 0 and sin is 1, etc.
Keep in mind that π/6 = 30°, π/4 = 45° and π/3° = 60 (you guys should remember this)
And now lastly, the CAST rule. I hope we all remember this! It's been drilled in our brains since grade 10 pre-calculus and possibly earlier.
Okay so let's move on. I'll work with an example and go step by step so we can figure out what the hell we have to do to solve these equations.First example comes from our in class example:
g(x) = 4Cosπ/2(x-1)+2
Step 1: Identify the x value and plug it into all the x variables/values into the equation.
g(23) = .......... so that number in the bracket tells us that that's what the x value is. So you will subtitute all x values in the equation that is present into 23.
So you will get g(x) = 4Cosπ/2(23-1)+2
Step 2: Order of operations. BEDMAS. B for brackets, so we will solve whatever is in brackets.
Since in the brackets there is (23-1), we will solve that. Obviously 23=1 is 22
g(x) = 4Cosπ/2(22)+2
Step 3: Notice the brackets? I left the brackets because it identifies multiplication. You will multiply (cos)π/2 by 22. Which is easy. It'll be π/2 muliplied by 22 / 1. Then simplify if possible.
g(x) = 4Cos11π+2
Step 4: Notice that there's a Cos11π, that will make things difficult for you so let's change that to a number from radians. No, we will not use a calculator. (Mr. Piatek says we shouldn't use a calculator for this. So let's get into the habit of some mental math and of course, using our past lessons to assist us on this)
There are a few ways to get the absolute value of cos11π. You can make a diagram and count 1π, 2π... all the way to 11π. If we do that, it will end up at π, and in that quadrantal cos would be -1.
g(x) = 4(-1)+2
Note: It's multiplication because whenever you do not see an addition (+) or subtraction (-) sign, generally it means to multiply. Multiplication can be expressed without those signs, with an x, with brackets beside a number without a (+) or (-) separating it or with a ●.
Step 5: Now simply solve (it's easy peasy!)
g(x) = -4+2
g(x) = -2
g(x) = 4Cosπ/2(x-1)+2
Step 1: Identify the x value and plug it into all the x variables/values into the equation.
g(23) = .......... so that number in the bracket tells us that that's what the x value is. So you will subtitute all x values in the equation that is present into 23.
So you will get g(x) = 4Cosπ/2(23-1)+2
Step 2: Order of operations. BEDMAS. B for brackets, so we will solve whatever is in brackets.
Since in the brackets there is (23-1), we will solve that. Obviously 23=1 is 22
g(x) = 4Cosπ/2(22)+2
Step 3: Notice the brackets? I left the brackets because it identifies multiplication. You will multiply (cos)π/2 by 22. Which is easy. It'll be π/2 muliplied by 22 / 1. Then simplify if possible.
g(x) = 4Cos11π+2
Step 4: Notice that there's a Cos11π, that will make things difficult for you so let's change that to a number from radians. No, we will not use a calculator. (Mr. Piatek says we shouldn't use a calculator for this. So let's get into the habit of some mental math and of course, using our past lessons to assist us on this)
There are a few ways to get the absolute value of cos11π. You can make a diagram and count 1π, 2π... all the way to 11π. If we do that, it will end up at π, and in that quadrantal cos would be -1.
g(x) = 4(-1)+2
Note: It's multiplication because whenever you do not see an addition (+) or subtraction (-) sign, generally it means to multiply. Multiplication can be expressed without those signs, with an x, with brackets beside a number without a (+) or (-) separating it or with a ●.
Step 5: Now simply solve (it's easy peasy!)
g(x) = -4+2
g(x) = -2
Note: You can count revolutions to find the absolute value for something (like Cos11π) or you can change that to degrees and look at the diagram in degrees and determine the values like that. Either way would be easy for you. However if it's a huge number like 11π (1980), it might be a bit hard so I personally would count revolutions. If it's a smaller number like 5π/3, then I CAN count revolutions (in my head I can go π/3, 2π/3, 3π/3, 4π/3, 5π/3... ahh it lies in quadrant 4!!) or I can change it into degrees (which would be 300°)
If you guys forgot, here's how to change from radians to degrees:
Took me roughly 2 hours to do all of this. I think it's because I typed a lot! Anyways you guys, here's a quick reminder!
If you guys forgot, here's how to change from radians to degrees:
Took me roughly 2 hours to do all of this. I think it's because I typed a lot! Anyways you guys, here's a quick reminder!Tuesday: Unit 2 (Transformation Test)
Tuesday: Transformation Assignment 2 Due?
March 15: iPhoto assignment due
Don't forget, do the review and worksheet as well as study! I'll be busy studying pre-calculus this long weekend! Hope you guys have a fun weekend!
Tuesday: Transformation Assignment 2 Due?
March 15: iPhoto assignment due
Don't forget, do the review and worksheet as well as study! I'll be busy studying pre-calculus this long weekend! Hope you guys have a fun weekend!
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