Team Sine Welcome to team SINE. You guys will become the toughest, strongest and smartest sine students in the class. However, it is also your goal to make the rest of your home team experts on the graph of y = sinx too, so it is very important that you work through this learning activity, understand the WHY s behind the answers and think about how you are going to teach your home team. Please be sure to READ all the text presented on this paper there will be facts, questions and places for you to write answers. Everything is designed for you to discover and understand the WHY s. Feel free to check any fill in the blanks with the teacher. So let s have it here we go! Goal #1: Use the TI-83 to create a graph The graphing calculator can be a little touchy when trying to create the sine wave. You have to change certain SETTINGS, understand the @ and how to use the $ button. Let s check it out. Step 1: Be sure your M menu is set to DEGREES Step 2: Using the! screen simply type sin(x). Clear out any other equations in this screen and double check that none of the Plot1 Plot2 Plot 3 are turned on. Step 3: In order for trig values (degrees and unit circle values) to graph appropriately, it is necessary to use a different # than we are accustomed too. So simply select # then scroll to 7:ZTrig. Now you should be looking at a few waves of the sine wave!! Box 1: A few important comments before we proceed When you selected ZTrig, the @ changed dramatically go ahead and select @. Do see how the graph stretches from nearly to in the x and from to in the y? Also notice that the x scale markers are now worth. This will help you to better understand the values you are seeing on the graph. 90 180 360 360 Goal #2: Create a pencil sketch labeling significant values Its one thing to sketch the sine wave on blank paper (to just get it done) but its another thing to THINK about the numbers / points / graph as you create it; when we do this, we are really getting comfortable with the graph and its features. Eventually you need to have this basic graph memorized. So let s try it for the first time. Step 1: Before you actually draw any sine waves on paper, we the y intercept: need to note some significant values / points. Take a two or three positive x intercepts: moment to $ your calculator graph and identify a max point with a positive x: a min point with a positive x:
Step 2: Now use your home team notes paper to carefully sketch the sine wave and label the significant points that you found from Step 1 Box 2: A few important comments before we proceed The sine wave you created from last night s homework is actually only ONE wave, whereas the sine wave you are looking at today on your calculator is actually TWO waves. The technical word we use for a wave is PERIOD. The period of a sine wave is the total amount of x required to complete one cycle. Therefore, your homework represents full period, whereas this calculator graph represents full periods. Do you know how many x s are required to make one full period? It s actually better thought of as degrees so how many degrees are required to see one full period? Goal #3: Identify the Amplitude and Period Sine waves are unique because they cycle back and forth, and eventually repeat themselves. In addition, they will only reach so high and / or so low. How low do they go? (bad joke!, but do you know?) Step 1: The amplitude of a graph can be thought of two different ways. (both ways create the same answer) Way #1: Simply the vertical distance from the midline to a max or min (always +) Way #2: Using a formula: (max min) 2 So how does this work out for our sine wave? So how does this work out for our sine wave? midline vertical distance? 1 (-1) 2 So the Amplitude = So the Amplitude = Step 1 continued: It is important to understand BOTH methods, because (trust me) there will be times when one method is better than the next. Now go ahead and write BOTH of these Amplitude Definitions (Way#1 & Way#2) at the top of your home team notes paper; also write your amplitude answer onto your home team notes paper beside the sine wave. Step 2: As was mentioned in Box 2 near the top of this page, the Period of a graph is the total amount of D required for the graph to make a full cycle. (Be sure to write this definition at the top of your home team notes paper.) In order to determine this value, we need a starting point, then a corresponding ending point. Do you realize there could be many different starting/ending points that satisfy a full period? S A T B Point A to Point B is a full period. How many degrees is it? Point S to Point T is a full period. How many degrees is it? Now fill this answer into your Home Team Notes
Goal #4: Identify the Domain and Range The domain and range define the specific x and y values that appear on a trig graph. It is helpful to know these values so we can prepare input values and anticipate output values. Depending on the problem, this can be tricky, but let s go for it! Here s a quick review (copy these definitions onto your home team notes) Domain the x values represented by the graph Range the y values represented by the graph all possible input values all possible output values Step 1: Let s find the domain by thinking about the left and right behavior of the sine wave notice how the graph continues to travel left and right, all the while cycling up and down. If you $ over the graph notice how you encounter a bunch different of angles (x values) negative angles, zero, positive angles on and on. So how would you describe the domain? All numbers. Step 2: The range is a different story; we now need to think about the up and down behavior of the sine wave are you noticing which two y values the graph cycles between? Why is it that no matter what the input, the output will always between these two numbers? Think UNIT CIRCLE! ( I love the Unit Circle ) So are you ready to write the range? You will need to use inequalities < y < Make sure you fill these Domain and Range answers into your Home Team Notes. Get them checked just to be sure they are right. Fill in these blanks Prepare to teach the SINE WAVE to your home team Its time to get ready to return to your home team but wait! You need to also be ready to TEACH this material to your group. You want them to discover the same things you did! You want them to really experience the SINE WAVE, don t you?... here s a few guidelines to make this work Do s 1. Give them time to type the equation into their calculator and get it to work. 2. Make sure they are drawing and writing the correct sketches and answers onto their home team notes paper. 3. To the best of your ability ask them questions that lead them to the answer(s). Don t s. 1. Just show them your paper and say HERE, READ THIS! 2. Just show them your paper and tell them to copy it. 3. Just tell them all the answers along the way. Keep in mind that your other team members may have discovered some of the same things that you did (even though they studied different graphs) so you may find that you don t need to cover every last detail. Simply ask your group if they already understand the concept and move to the next.
Team Cosine Welcome to team COSINE. You guys will become the toughest, strongest and smartest cosine students in the class. However, it is also your goal to make the rest of your home team experts on the graph of y = cosx too, so it is very important that you work through this learning activity, understand the WHY s behind the answers and think about how you are going to teach your home team. Please be sure to READ all the text presented on this paper there will be facts, questions and places for you to write answers. Everything is designed for you to discover and understand the WHY s. Feel free to check any fill in the blanks with the teacher. So let s have it here we go! Goal #1: Use the TI-83 to create a graph The graphing calculator can be a little touchy when trying to create the cosine wave. You have to change certain SETTINGS, understand the @ and how to use the $ button. Let s check it out. Step 1: Be sure your M menu is set to DEGREES Step 2: Using the! screen simply type cos(x). Clear out any other equations in this screen and double check that none of the Plot1 Plot2 Plot 3 are turned on. Step 3: In order for trig values (degrees and unit circle values) to graph appropriately, it is necessary to use a different # than we are accustomed too. So simply select # then scroll to 7:ZTrig. Now you should be looking at a few waves of the cosine wave!! Box 1: A few important comments before we proceed When you selected ZTrig, the @ changed dramatically go ahead and select @. Do see how the graph stretches from nearly to in the x and from to in the y? Also notice that the x scale markers are now worth. This will help you to better understand the values you are seeing on the graph. 90 180 360 360 Goal #2: Create a pencil sketch labeling significant values Its one thing to sketch the cosine wave on blank paper (to just get it done) but its another thing to THINK about the numbers / points / graph as you create it; when we do this, we are really getting comfortable with the graph and its features. Eventually you need to have this basic graph memorized. So let s try it for the first time. Step 1: Before you actually draw any cosine waves on paper, the y intercept: we need to note some significant values / points. Take two or three positive x intercepts: a moment to $ your calculator graph and identify a max point with a positive x: a min point with a positive x:
Step 2: Now use your home team notes paper to carefully sketch the cosine wave and label the significant points that you found from Step 1 Box 2: A few important comments before we proceed The cosine wave you created from last night s homework is actually only ONE wave, whereas the cosine wave you are looking today at on your calculator is actually TWO waves. The technical word we use for a wave is PERIOD. The period of a cosine wave is the total amount of x required to complete one cycle. Therefore, your homework represents full period, whereas this calculator graph represents full periods. Do you know how many x s are required to make one full period? It s actually better thought of as degrees so how many degrees are required to see one full period? Goal #3: Identify the Amplitude and Period Cosine waves are unique because they cycle back and forth, and eventually repeat themselves. In addition, they will only reach so high and / or so low. How low do they go? (bad joke!, but do you know?) Step 1: The amplitude of a graph can be thought of two different ways. (both ways create the same answer) Way #1: Simply the vertical distance from the midline to a max or min (always +) So how does this work out for our cosine wave? Way #2: Using a formula: (max min) 2 So how does this work out for our cosine wave? midline vertical distance? 1 (-1) 2 So the Amplitude = So the Amplitude = Step 1 continued: It is important to understand BOTH methods, because (trust me) there will be times when one method is better than the next. Now go ahead and write BOTH of these Amplitude Definitions (Way#1 & Way#2) at the top of your home team notes paper; also write your amplitude answer onto your home team notes paper beside the cosine wave. Step 2: As was mentioned in Box 2 near the top of this page, the Period of a graph is the total amount of D required for the graph to make a full cycle. (Be sure to write this definition at the top of your home team notes paper.) In order to determine this value, we need a starting point, then a corresponding ending point. Do you realize there could be many different starting/ending points that satisfy a full period? S A T B Point A to Point B is a full period. How many degrees is it? Point S to Point T is a full period. How many degrees is it? Now fill this answer into your Home Team Notes
Goal #4: Identify the Domain and Range The domain and range define the specific x and y values that appear on a trig graph. It is helpful to know these values so we can prepare input values and anticipate output values. Depending on the problem, this can be tricky, but let s go for it! Here s a quick review (copy these definitions onto your home team notes) Domain the x values represented by the graph Range the y values represented by the graph all possible input values all possible output values Step 1: Let s find the domain by thinking about the left and right behavior of the cosine wave notice how the graph continues to travel left and right, all the while cycling up and down. If you $ over the graph notice how you encounter a bunch of angles (x values) negative angles, zero, positive angles and on and on. So how would you describe the domain? All numbers. Step 2: The range is a different story; we now need to think about the up and down behavior of the cosine wave are you noticing which two y values the graph cycles between? Why is it that no matter what the input, the output will always between these two numbers? Think UNIT CIRCLE! ( I love the Unit Circle ) So are you ready to write the range? You will need to use inequalities < y < Make sure you fill these Domain and Range answers into your Home Team Notes. Get them checked just to be sure they are right. Fill in these blanks Prepare to teach the COSINE WAVE to your home team Its time to get ready to return to your home team but wait! You need to also be ready to TEACH this material to your group. You want them to discover the same things you did! You want them to really experience the COSINE WAVE, don t you?... here s a few guidelines to make this work Do s 1. Give them time to type the equation into their calculator and get it to work. 2. Make sure they are drawing and writing the correct sketches and answers onto their home team notes paper. 3. To the best of your ability ask them questions that lead them to the answer(s). Don t s. 1. Just show them your paper and say HERE, READ THIS! 2. Just show them your paper and tell them to copy it. 3. Just tell them all the answers along the way. Keep in mind that your other team members may have discovered some of the same things that you did (even though they studied different graphs) so you may find that you don t need to cover every last detail. Simply ask your group if they already understand the concept and move to the next.
Team Tangent Welcome to team TANGENT. So what do you call an ant at the beach? A Tangent (If this joke was a bomb, don t tell it to anyone else.) Anyhow, you guys will become the toughest, strongest and smartest tangent students in the class. However, it is also your goal to make the rest of your home team experts on the graph of y = tanx too, so it is very important that you work through this learning activity, understand the WHY s behind the answers and think about how you are going to teach your home team. Please be sure to READ all the text presented on this paper there will be facts, questions and places for you to write answers. Everything is designed for you to discover and understand the WHY s. Feel free to check any fill in the blanks with the teacher. So let s have it here we go! Goal #1: Use the TI-83 to create a graph The graphing calculator can be a little touchy when trying to create the tangent wave. You have to change certain SETTINGS, understand the @ and how to use the $ button. Let s check it out. Step 1: Be sure your M menu is set to DEGREES Step 2: Using the! screen simply type tan(x). Clear out any other equations in this screen and double check that none of the Plot1 Plot2 Plot 3 are turned on. Step 3: In order for trig values (degrees and unit circle values) to graph appropriately, it is necessary to use a different # than we are accustomed too. So simply select # then scroll to 7:ZTrig. Now you should be looking at a few waves of the tangent wave!! Box 1: A few important comments before we proceed When you selected ZTrig, the @ changed dramatically go ahead and select @. Do see how the graph stretches from nearly to in the x and from to in the y? Also notice that the scale markers are now worth. This will help you to better understand the values you are seeing on the graph. Special Note: The graph of tangent is unique in that it possess vertical asymptotes. These may or may not show up with YOUR graphing calculator. That really doesn t matter as much as you realizing that they are there. Remember this is happening because of the Unit Circle, y/x, and a zero in the denominator. More on these asymptotes later! 90 180 360 360 asymptotes @ x = 270, 90, 90, 270
Goal #2: Create a pencil sketch labeling significant values Its one thing to sketch the tangent wave on blank paper (to just get it done) but its another thing to THINK about the numbers / points / graph as you create it; when we do this, we are really getting comfortable with the graph and its features. Eventually you need to have this basic graph memorized. So let s try it for the first time. Step 1: Before you actually draw any tangent waves on paper, we the y intercept: need to note some significant values / points. Take a moment to $ your calculator graph and identify a few x intercepts: Box 2: Important comments on asymptotes There are different ways to find the location of these asymptotes. You can simply look at the graph. But you can also $the graph left and right and wait for the y value to disappear. When the y value disappears, this is a sign that there is no value, i.e. you @ at an asymptote. You can also investigate the table by using ` % the x position of a few asymptotes Step 2: Now use your home team notes paper to carefully sketch the tangent wave and label the significant points / asymptotes that you found from Step 1. The asymptotes should be sketched in as dashed vertical lines Box 3: A few important comments before we proceed The tangent waves you created from last night s homework actually represent TWO waves, whereas the tangent waves you are looking at today on your calculator actually represent THREE full waves, plus a HALF and HALF wave, making a total of FOUR WAVES. The technical word we use for a wave is PERIOD. The period of a tangent wave is the total amount of x required to complete one cycle. Therefore, your homework represents full periods, whereas this calculator graph represents full periods. Do you know how many x s are required to make one full period? It s actually better thought of as degrees so how many degrees are required to see one full period? That s right it s actually the total degrees BETWEEN two asymptotes. (More on this next.) Goal #3: Identify the Period Tangent waves are unique because you will always see one full wave (one full period) between two consecutive sets of asymptotes. After that, they will repeat themselves Step 1: As was mentioned in Box 3 above, the period of a graph is the total amount of D required for the graph to make a full cycle. (Be sure to write this definition at the top of your home team notes paper.) In order to determine this value, we need a starting point, then a corresponding ending point. For a tangent graph we can also determine the distance between two consecutive sets of asymptotes. Both methods will give you ONE FULL PERIOD. A B Point A to Point B is a full period. How many degrees is it? Now fill this answer into your Home Team Notes Asymptote S to Asymptotes T is a full period. How many degrees is it? S T
Goal #4: Identify the Domain and Range The domain and range define the specific x and y values that appear on a trig graph. It is helpful to know these values so we can prepare input values and anticipate output values. Depending on the problem, this can be tricky, but let s go for it! A quick heads up... the first time we do the domain for tangent it may seem a bit strange but we will eventually figure it out Here s a quick review (copy these definitions onto your home team notes) Domain the x values represented by the graph Range the y values represented by the graph all possible input values all possible output values Step 1a: Let s find the domain by thinking about the left and right behavior of the tangent waves notice how the graph continues to travel left and right, over various x (angle) values. However, there are some x (angle) values that the graph is unable to reach. That s right; the graph will never touch these asymptotes. (Remember how we can $ to some of these asymptotes as described in Box 2 on the previous page) So somehow we need to describe a domain that includes all x values EXCEPT for the asymptotes. For starters we could say Domain: All x such that x asymptotes (you can copy this onto your Home Team Notes) Step 1a: However, there is a bit of a fancier way to describe these asymptotes using a formula. This formula, when written correctly will actually output the value of ALL the asymptotes, thereby giving us specific values that are NOT in the domain. Let s see what we can do using these steps The variable k when used in this formula represents any integer. That means k could equal 0, 1, 2 or -1, -2, So go ahead and pick a k value and plug it into the formula. What happens? Let s try k=1. So 180(1) + 90 = 270. Well, well, 270 is the value of an asymptote. So no matter what integer you pick for k, it will always output the location of another asymptote. Guaranteed! 1. Determine the distance between any two consecutive asymptotes. You should have already found this to be 2. Now determine the x location of the first positive asymptotes. Are you finding 90? 3. Finally, carefully using the variable k, write the formula x 180k + 90 distance between 1 st positive So what is the final Domain Answer All x such that x 180k+90 180k+90 (you can copy this onto your Home Team Notes) Step 2: The range is a different story; we now need to think about the up and down behavior of the tangent waves. Even though the waves break at each asymptote, you need to know that they will still travel up up up or down down down beside each asymptote. Therefore, tangent s range will eventually encounter real numbers. Make sure you fill these Domain and Range answers into your Home Team Notes. Get them checked just to be sure they are right. More fun on the back
Prepare to teach the TANGENT WAVE to your home team Its time to get ready to return to your home team but wait! You need to also be ready to TEACH this material to your group. You want them to discover the same things you did! You want them to really experience the TANGENT WAVE. Here s a few guidelines to make this work Do s 1. Give them time to type the equation into their calculator and get it to work. 2. Make sure they are drawing and writing the correct sketches and answers onto their home team notes paper. 3. To the best of your ability ask them questions that lead them to the answer(s). Don t s. 1. Just show them your paper and say HERE, READ THIS! 2. Just show them your paper and tell them to copy it. 3. Just tell them all the answers along the way Keep in mind that your other team members may have discovered some of the same things that you did (even though they studied different graphs) so you may find that you don t need to cover every last detail. Simply ask your group if they already understand the concept and move to the next.
Home Team Notes Name Graphs of Trigonometric Functions Date Period The trigonometric functions that you graphed for homework last night can also be graphed on your calculator. Further, being able to successfully use our calculator to come up with additional insights and answers will prove advantageous to our studies. And lastly, there are certain characteristics of these graphs, such as amplitude, period, domain and range that we want to become familiar with. So let s explore and learn these ideas in a new way today. Instead of there being ONE teacher, today there will be (give or take) 25 teachers. Today you will become an expert on ONE graph then help other students to also understand your graph. When the period is over, you should have successfully worked through this notes paper and be ready to use these ideas for the remainder of the chapter. How this is going to work: 1. Create / join a Home Team made up of exactly three students and give yourself a number between 1 and 3. 2. Student #1 is assigned to SINE Student #2 is assigned to COSINE Student #3 is assigned to TANGENT 3. Now go join your new expert team according to SINE, COSINE or TANGENT (You may want to break these 3 BIG groups into smaller pods) 4. While in these expert groups, learn, discover and understand your graph. Take notes on this paper (and your expert paper), ask questions and ultimately be ready to teach these ideas to your Home Team. 5. Now return to your Home Team and one by one, help your members to learn, discover and understand your graph. Everyone should be filling in the remaining notes on this paper during this last session. Some BIG Ideas and Definitions for Trigonometric Graphs Amplitude Period Domain Range y = sin (x) Amplitude: Period: Domain: Range: Make sure you are labeling points and values on your graph don t just draw a wavy line
y = cos (x) Amplitude: Period: Domain: Range: Make sure you are labeling points and values on your graph don t just draw a wavy line y = tan (x) Amplitude: tan does not have amplitude Period: Domain: Range: Make sure you are labeling points and values on your graph don t just draw a wavy line For homework tonight, try to work out the graph and features for the SECANT function (remember y=secx is y = 1/cosx) y = sec (x) Domain: Amplitude: sec does not have amplitude Range: Period: