How to Graph Trigonometric Functions for Sine and Cosine Amplitudes Midlines Periods Oh My! Kyle O. Linford
Linford 1 For all of my future students, May this text help you understand the power and beauty of mathematics. Mathematics possesses not only truth, but supreme beauty Bertrand Russel Pure mathematics is, in its way, the poetry of logical ideas Albert Einstein
Linford 2 In astronomy, we use it to measure our distance to stars. In geography, it s used to find distances between places and landmarks here on Earth. It can even be found when studying light waves and economics, or even biology, engineering, and architecture. That s right; we re talking about trigonometry. For over 2000 years, this field of mathematics has helped many great minds make remarkable discoveries about the world and the universe. However, for those 2000 years, it has confused a lot of math students. One of the most difficult aspects of this subject for students to understand is how to graph the confusing trigonometric functions; students are often intimidated because the functions can be very long and create these unusual wave-like graphs. Well, this book will help any student feeling anxiety about graphing these functions. Together, we will break the functions down into different pieces to see how each part of it works to create the graphs. In this way, graphing these functions will be a lot easier for us to understand. To begin graphing these trigonometric functions, we will start with our standard trigonometric identities sine and cosine. Remember that each of these identities have unique graphs for how they behave. We will then start manipulating the graphs to make them fit our specific trigonometric functions. To see how we can manipulate these graphs to model the functions that we want, we will take a look at the many different aspects of these graphs to create an easy to understand, step-by-step process for
Linford 3 determining what the graph looks like. However, before we begin, let s look at what the aspects we will be exploring are. Definitions to Know Midline The horizontal line midway between the function s maximum and minimum values. Amplitude The vertical distance between the function s maximum (or minimum) value and the midline. Period The horizontal length of one complete cycle. Vertical Shift The vertical length of how an entire graph was shifted. Horizontal Shift The horizontal length of how an entire graph was shifted. Constant A number not attached to a variable. Absolute Value An operation which takes any number and finds its distance from the number 0.
Linford 4 Finding the Midline The first thing we need to do when graphing a trigonometric function is to identify the midline. Remember that the midline is the horizontal line that splits our function in half; it will be half way between the highest and lowest points of the graph. Looking at whatever function we are given, the midline will be the vertical shift for that function. If we are given some function with sine or cosine and we added a constant c, the midline will be at whatever y-value that point c is. For example, if we have f(x) = sin(x) + 2, the number 2 would be how much the graph of sin(x) was shifted because we moved the entire graph up 2 units, and y = 2 would be our midline. Notice that when y = 2, we cut the graph horizontally in half.
Linford 5 Finding the Amplitude Once we have the midline located, we then want to figure out what our amplitude will be how high and how low our waves are. This will tell us how far away from the midline we should go. Figuring out what the amplitude is from a given function is just as easy as finding the midline. Looking at whatever function we have, the amplitude is the absolute value of the multiple that is applied to the sine or cosine part of the function. For example, if we are givenf(x) = 3 cos(x), the amplitude would be 3 or 3. Similarly, if we havef(x) = 1 sin(x), the amplitude would be 1 or 1. 2 2 2 3 Units Areas to Watch: Now these amplitudes are pretty easy to notice right away, but what if we have something likef(x) = sin(x) or f(x) = cos(x)? What would the amplitude be for these functions because there is no multiple to them? Be careful when you look at these functions; just because no multiple is written does not mean that there isn t one. Whenever we just have a cos(x) or a sin(x), it is implied that they are being multiplied by the number 1, and we just have not written it. For these examples of f(x) = sin(x) and f(x) = cos(x), we can rewrite them to say f(x) = 1 sin(x) and f(x) = 1 cos(x), so the amplitude would just be 1.
Linford 6 Application: Often times, people wonder why we study sine and cosine graphs. They don t realize how often these types of functions appear in everyday life or how we can use them to model trends in data. Meteorologists will often study these trigonometric functions because the yearly weather temperatures follow the trend of a sine graph. For example, weather data from the Department of Mathematics and Statistics at the University of Saskatchewan found the average weekly temperature in a city to be the data in this graph. Looking at this data, we can notice right away that it looks like a sine function. Then we can use our knowledge of these functions to determine what the midline of the function is and observe its amplitude. Looking at the graph, it appears that the midline should be at about y = 42 with an amplitude of about 30. This helps to tell us that the year s highest weekly temperature was about 72 F and the lowest was about 12 F. Now this at first does not seem too interesting of a discovery. Why would I care to find the midline of a function that might model this data and know its amplitude? Can t I just keep a log and see which temperature is the highest and lowest? While we could just keep a log of the data and look to figure this information easily, the graph tells us a lot more necessary information that could impact this city, the state it is in, and that region of the country as a whole. By studying this data and the graph which models it, meteorologists are able to compare yearly temperature ranges to each other. By identifying the amplitude, they can observe unusual spikes in temperature. These
Linford 7 unusual patterns can help them to better understand the environmental impact of things like global warming as well as predict potential dangers like thunderstorms, tornados, and hurricanes. In this way, we can see the immediate importance to understanding these trigonometric functions and the effect they have on preserving our safety.
Linford 8 Finding the Period Now that we know where our graphs should be centered and how high and low the waves of the graph should reach, we need to figure out the size and location of the waves to finish our graph. To do this, we need to identify what the period of the graph is. Remember that the period is the length of the smallest interval that it takes our graph to repeat its pattern: this is the length of how long it takes us to get back to where we started. Thinking back to the unit circle, it usually takes us 2π units to complete one cycle and end up where we started. However, this is not always the case. Sometimes, the period will be longer or shorter; it is all determined by what is written in the function. While the amplitude and midline were determined outside of the sine and cosine, the period will be based on what is inside of the parenthesis. To calculate the period of a given function, you take 2π units and divide it by the absolute value of the multiple inside of the parenthesis. For example, if we were given f(x) = sin(3x), we could figure out what the period is by taking 2π or 2 π. This suggests 3 3 that we can finish one cycle of this pattern in only 2 π units. 3 2π 3 Units
Linford 9 Areas to Watch: Finding these periods is not that difficult to work from. However, like finding the amplitude, this can be sometimes tricky when no multiple is written inside of the parenthesis. Again, we want to keep in mind that just because no multiple is written does not mean that there isn t one; if no multiple for the variable is written, it is implied that the variable is being multiplied by 1. For example, the function f(x) = sin(x) can be written asf(x) = sin(1 x). To find the period, we would take 2π, which is just 2π. Keep this in mind when working with the different trigonometric functions. 1 2π Units
Linford 10 Application: This understanding of the period can be helpful for musicians and audio engineers when they work with producing and recording music. Often times, people wonder how things like music and math can be related, especially something like trigonometry. Well, one of the amazing areas that trigonometry pops up in our world is with sounds produced by music. Whenever we hear music of some kind, we are listening to sound waves, and these sound waves travel in a pattern that can be modeled by a sine graph. According to the University of Minnesota website Trigonometry in Nature, piano notes each follow unique sine graphs with different periods. For example, the graph of f(x) = sin(880πx) can produced from the note A just above middle C. This function reaches an amplitude of 1 and has a period of about 0.0023. If we were to take this function and put it in a computer program that required a tone of sound to travel in this pattern, we could recreate the sound of this A note. However, if we were to manipulate the period for this function, we would be able to change the sound to another note. For example, if we were to increase the
Linford 11 multiplier to make the function f(x) = sin(1100πx), we would decrease the period and have a graph with more waves than before in the given interval. This is one of the ways in which DJs and producers can use computers to create different sounds and music without having to play notes on an instrument. They can simply use their knowledge of trigonometry to manipulate a standard sine function to create different notes; then they can piece all of these notes together and create a new song.
Linford 12 Orientation While we now know where the graph is centered and how to construct the waves, we want to know which directions the waves are placed. Depending on how the function is written, we can have the waves placed like normal sine and cosine graphs, or we can invert them rather than have the wave go up, it goes down, and rather than going down, it goes up. To decide this, we want to see if the graph is multiplied by a negative number. Think back to when we were finding the amplitude of the function; we took the absolute value of the number that multiplied the sine or cosine. If that number is negative, it will cause the graph to be inverted we will flip the direction of every wave. If that number is positive, we leave the waves in their normal direction. Think of this like you are just waking up. If you feel a little negative, then your day is going to go one way as opposed to the another if did not wake up so negative. Now you won t travel through your day like a sine graph either way, but it is all the more reason to not wake up on the wrong side of the bed! Let s take a closer look at this rule for orientation. If we have the function f(x) = 2 sin(x), we can see that the -2 will cause the graph s orientation to be flipped. If instead we just have f(x) = 2 sin(x), then the orientation will follow like a normal sine graph.
Linford 13 Similarly, the parenthesis inside of the sine or cosine can determine the orientation of the waves. If the variable on the inside of the parenthesis is multiplied by a negative value, the orientation of the graph will change like before, and if it is multiplied by a positive value, the orientation stays the same. Taking the function f(x) = sin( 2x), we can say that the graph will be inverted because the variable x is being multiplied by a negative 2. If this function was just f(x) = sin(2x), then we would leave the waves to be in the regular orientation because we are multiplying the variable x by a positive 2. Remembering this general rule will help us quickly determine the direction of the graph. Areas to Watch: Our rule tells us which direction the graph will be oriented depending on whether the variable or the sine or cosine are multiplied by a negative value. However, what will happen if both of these are multiplied by negative values? For example, what will happen if we were given f(x) = 3 sin( 2x)? In the situation where the variable and the sine are multiplied by negative numbers, these negative values will cancel each other out and keep the orientation for the function like it normally is. Taking a look at the example, if we are given f(x) = 3 sin( 2x), we will not change the orientation. The waves will move just like a normal sine graph. Our rule for changing the orientation will only apply when one of the values are negative, not both of them.
Linford 14 Finding Horizontal Shifts We are almost there! Having everything else we need to graph the function, we just need to see if there is a horizontal shift being applied to the graph. This is often times the last thing we will do when determining what the graph will look like. Similar to the vertical shift where we took the trigonometric graph and moved the entire thing up or down on the y-axis, we can also take the entire graph and shift it horizontally left or right on the x-axis. However, unlike the vertical shift, which we found based off of the midline outside of the parenthesis, the horizontal shift will be found like the period was inside of the parenthesis. The horizontal shift in the graph is determined by whatever is being added or subtracted to the variable in the parenthesis. If our parenthesis is of the form (x + c) or (x c) where c is a constant and x is a variable, then we can easily say that the shift is determined by whatever c is; the c tells us how far to move everything. However, the addition or subtract determines what direction we shift the graph along the x-axis. The rule for this is pretty simple to follow: if we are adding the constant to the variable, then we shift to the left; if we are subtracting the constant from the variable, we shift to the right. To view this, let s explore the trigonometric function f(x) = sin(x + 2). For this function, the horizontal shift is determined by what is being added to the variable x. In this case, the horizontal shift is 2, so we take our entire graph and move it 2 units to the left. Similarly, if we were given f(x) = sin(x 2), we would take the entire graph and shift it to the right 2 units.
Linford 15 It is important to keep in mind that not all trigonometric functions will have a horizontal shift. If we are given just a variable multiplied by some number with nothing being added to it, there will be no horizontal shift. To see this, let s examine the function f(x) = sin(3x). We can rewrite this to say f(x) = sin(3x + 0), so it becomes clear that we will not shift the graph to the left or right by anything. In this case, we will not worry about a horizontal shift.
Linford 16 Areas to Watch: Often times, finding the horizontal shift confuses students because it is tangled up with the period. An easy way to make sure that we do not mess up the period or the horizontal shift, is to write whatever is inside of the parenthesis in the form a(x + c), where c is a constant being added to the variable x, and a is just the number multiplying everything. When we have our parenthesis written in this form, we can easily see that c will determine the horizontal shift like before, and the a will be used to determine the period. For example, if we were given the function f(x) = sin(3x + 6), we would want to divide out the 3 to say f(x) = sin[3(x + 3)]. When we write the function like this, we can find the period by taking 2π, and the horizontal shift will just be 3 units to the left. 3 Okay, that example was not too difficult. However, what if we are given something more complicated, like f(x) = sin(3x + 4) where there are no common factors between 3 and 4? Well the answer is just as easy as before; to simplify this down, we would just divide out a 3 again to get f(x) = sin[3(x + 4 )]. Then we can use the 3 to find the period like before and say that the entire graph 3 is shifted horizontally to the left by 4 units. As long as we can write our parenthesis as a(x + c), we are 3 able to see what the period and horizontal shift will be. The same thing can be said when we are using π in the parenthesis. If we had been given the function f(x) = sin(3x + π) instead, we would also divide out a 3 to write f(x) = sin[3 (x + π )]. 3 The best advice I can give to students about finding the period and horizontal shift is to take your time, write the parenthesis in the form a(x + c), and do not be afraid if your answer involves a fraction or π. This is very common when working with these trigonometric functions, so do not worry when you see them. This is a lot of information to absorb at once, so let s practice this technique with some examples.
Linford 17 Examples: 1. f(x) = sin(6x + 7) For this example, we can see that we do not have the function written in the form that we want, so let s get it to look like a(x + c). To do this we divide out whatever number is attached to the x. For this example, it is a 6. Dividing out a 6, we can rewrite the function as f(x) = sin [6 (x + 7 )]. Now it is in the form that we want, and we can see that the horizontal 6 shift is 7 units to the left. If we wanted to show our work, we would just write what we started 6 with and begin simplifying. f(x) = sin(6x + 7) f(x) = sin [6 ( 6x 6 + 7 6 )] f(x) = sin [6 (x + 7 6 )]
Linford 18 2. f(x) = sin (5x + 5 3 ) Similar to before, we want to rewrite the given function in a way that is easy for us to understand. As always, we divide out what is attached to the x variable, so we divide out a 5 from the terms in the parenthesis. However, this time we are dividing a rational number, 5 3, by 5. Remember, when we divide a rational number by another number, it is the same as the multiplying that rational by the reciprocal of the other number. In this example, we have 5 3 5. This is the same as 5 5 since every whole number can be written as a rational number by 3 1 being put over 1. We then take the dividend and multiply it by the reciprocal of the divisor: 5 1 = 5 = 1. We can now rewrite the function as f(x) = sin [5 (x + 1 )] and see that the 3 5 15 3 3 horizontal shift is 1. If we wanted to show our work, we would just write what we started with 3 and begin simplifying. f(x) = sin (5x + 5 3 ) f(x) = sin [5 ( x 5 + 5 3 5 )] f(x) = sin [5 ( x 5 + 5 3 1 5 )] f(x) = sin [5 ( x 5 + 1 3 )]
Linford 19 Graphing Once we have identified the midline, the amplitude, and the period, we can begin using this information to determine what the graph of our function will look like. Let s work through an example together to see how we can create our graph. 1. f(x) = 2 sin(4x 5) + 1 a. First, we will find the midline. Remember, the midline is just what gets added to the sine portion of our function. In this example, the midline is 1. b. Now, let s find the amplitude to determine how high and low our waves should be. Remember, the amplitude is found by taking the absolute value of the number multiplying the sine portion of the function. In this example, the number multiplying the sine is 2, so let s take 2. This simplifies to just 2. Now we know that the waves will be 2 units away from the midline. c. The next thing to identify is the period, so we know the positioning of our waves. To find the period, we take the absolute value of what is multiplying the variable it is 4 in this problem and divide 2π by it. To find the period of this problem, we would calculate 2π 4. This simplifies to π 2, so our graph will repeat its pattern after π 2 units. d. After the period has been found, the orientation of the waves is the next step for determining the graph. Remembering our rule, to invert the graph, we need to have either the number multiplying the sine be a negative or the number multiplying the variable be negative. For this problem, the 2 and 4 are both positive, so we will not change the normal orientation of a sine graph. e. The final step for creating the graph is to determine if we need to shift the graph horizontally. Let s look inside of the parenthesis at the variable. We can see that we have 4x + 5. Remember, we want to have this in the form of a(x + c), so let s divide out
Linford 20 a 4. We are left with 4(x + 5 4 ). This suggests that we will shift the entire graph 5 4 units, and because it is addition, we want to move the graph to the left. This graph is the final product that we wanted! It is shifted up 1 unit and over 5 units to the left. Its 4 waves stretch up 2 units from the midline, so they touch the line y = 3 and y = -1. Also, following a normal orientation, the pattern repeats itself every π units, so we end up at the same place for the y- 2 direction we started after every π units we go in the x-direction. 2 By exploring this breakdown, we are now better able to work with trigonometric functions and interpret their graphs. All we need to do is take our time and remember what we are checking for in the function. Then we can easily create our graph and interpret our solutions! Maybe know we can better understand what is necessary for studying geography and the stars alike.
Linford 21 Resources Boswell, Laurie, et al. Algebra 2. Boston: Houghton Mifflin Harcourt, 2006. Print. David. Tips from the Pros: Audio Editing. Soundstage Studios. Soundstage Studios, 6 Nov. 2014. Web. 25 Oct. 2015. JPEG File. Example: Amplitude and Period. Khan Academy. Khan Academy, n.d. Web. 8 Oct. 2015. Eye (cyclone). Wikipedia. Wikipedia, 19 Oct. 2015. Web. 25 Oct. 2015. JPEG file. Hurricanes On Land. Zwallpix. Zwallpix, n.d. JPEG file. Lemonick, Michael. Is There a Huge Hole in Outer Space? Time. Time, 27 Aug. 2007. Web. 25 Oct. 2015. JPEG File. O Toole, Tim. Structural Lessons From Hurricanes And Tornadoes Webinar. Mason Contractors Association of America. Mason Contractors Association of America, 24 May 2012. Web. 25 Oct. 2015. JPEG file. Rogness, Jonathon. Trigonometry in Nature: Sinusoidal Waves as Sound. Trigonometry in Nature. University of Minnesota College of Science & Engineering, n.d. Web. 24 Oct. 2015. Using Trig. Functions to Model Real World Problems. Using Trig. Functions to Model Real World Problems. University of Saskatchewan Department of Mathematics & Statistics, n.d. Web. 24 Oct. 2015.
Linford 22 About the Author Kyle Linford was born in 1993 in Plymouth, Michigan, where he attended Eastern Michigan University to study education and mathematics. His research with his professors has involved explorations of the Riemann zeta function; proofs of algebraic, transcendental, and constructible numbers; and developing a computer package using matrix multiplication to represent transformations of the plane. As a Michigan native, Linford has a deep desire to help improve the understanding of mathematics for Michigan students and instill a love for the beauty the subject provides. 2015