High School Trigonometry/Basic Functions
This chapter will introduce you to a particular family of functions, the trigonometric functions, which are the basis for this book. In this first lesson, we will review the basic characteristics of functions in general: what a function is, what the graph of a function looks like, and the characteristics of several families of functions. While this lesson will not define trigonometric functions, we will consider one of their basic characteristics, and some important applications of these functions.
Learning Objectives[edit]
 Determine if a relation is a function.
 State the domain and range of a function.
 Categorize a function according to a function family.
 Identify key characteristics of functions, including the concept of a periodic function.
The Basics of Functions[edit]
Consider two situations shown in the boxes below:
Age  18  17  18  18  17 
Height  65"  64"  67"  68"  66" 
In the first situation, let the variable x represent the speed of your car, and let y represent the number of miles it can travel using one gallon of gasoline. If you travel at x miles per hour, you will go y = 30 − .5 (x − 55) miles on one gallon of gasoline. For example, if you travel at 60 mph, you will travel 30 − .5(60 − 55) = 27.5 miles on one gallon of gasoline. Notice that you can use your speed to "predict" how far you can travel with one gallon of gasoline.
Now consider the second situation. Can you use the data to "predict" height, given the age of a student?
This is not the case in the second situation. For example, if a student is 18 years old, there are several heights that the student could be.
Both situations are relations. A relation is simply a relationship between two sets of numbers or data. For example, in the second situation, we created a relationship between students' ages and heights, just by writing each student's information as an ordered pair. In the first situation, there is a relationship between the car's speed and how efficiently it can use one gallon of gasoline. The first example is different from the second because it represents a function: every x is paired with only one y. Some relations are mathematically important. For example, circles and ellipses are graphical representations of important relations between x and y coordinates, but there is not a unique y coordinate for each x coordinate. Because of the unique y for each x, functions play an important role in mathematics and the science.
We can represent functions in many ways. Some of the most common ways to represent functions include: sets of ordered pairs, equations, and graphs. The figure below shows a function depicted in each of these representations:

Table 1.1: Examples of functions Representation Example Set of ordered pairs (1,3), (2,6), (3,9), (4,12) (a subset of the ordered pairs for this function) Equation y = 3x Graph
In contrast, the relations shown in the figure below are not functions:

Table 1.2: Examples of nonfunctions Representation Example Set of ordered pairs (4,2), (4,2), (9,3), (9,3) (a subset of the ordered pairs for this function) Equation x = y^{2} Graph
To verify that this relation is not a function, we must show that at least one x value is paired with more than one y value. If you look at the first representation, the set of ordered pairs, you can see that 4 is paired with 2 and with −2. Similarly, 9 is paired with 3 and with −3. Therefore the relation is not a function. If we look at the graph above, we can see that, except for x = 0, the x values of the relation are each paired with two y values. Therefore the above relation is not a function.
One way to quickly determine whether or not a relation is a function is perform the vertical line test, which means that you draw a vertical line through the graph. For example, if we draw the line x = 4 through the graph of x = y^{2}, the line will intersect the graph twice. This means the relation is not a function.
Example 1
Determine if the relation is a function or not. a. (2,4), (3,9), (5,11), (5,12) b. Solution: a. This relation is not a function because 5 is paired with 11 and with 12. If you plotted the points, the line x = 5 would touch 2 points in the relation. b. This relation is a function because every x is paired with only one y. 
Once you are able to determine if a relation is a function, you should then be able to state the set of x values and the set of y values for which a function is defined.
The domain of a function is defined as the set of all x values for which the function is defined. For example, the domain of the function y = 3x is the set of all real numbers, often written as . This means that x can be any real number. Other functions have restricted domains. For example, the domain of the function is the set of all real numbers greater than or equal to zero. The domain of this function is restricted in this way because the square root of a negative number is not a real number. Therefore the domain is restricted to nonnegative values of x so that the function values will be defined.
It is often easy to determine the domain of a function by (1) considering what restrictions there might be and (2) looking at a graph.
Example 2
State the domain of each function. a. y = x^{2} b. c. (2, 4), (3, 9), (5, 11) Solution: a. The domain of this function is the set of all real numbers. There are no restrictions. b. The domain of this function is the set of all real numbers, except x ≠ 0. The domain is restricted this way because a fraction with denominator zero is undefined. c. The domain of this function is the set of x values {2, 3, 5}. 
The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that "y is a function of x". For example, the value of y in the function y = 3x depends on what x value we are considering. If x = 4, we can easily determine that y = 3(4) = 12.
When we are working with a function in the form of an equation, there is a special notation we can use to emphasize the fact that y is a function of x. For example, the equation y = 3x can also be written as f(x) = 3x. It is important to remember that f(x) represents the y values, or the function values, and that the letter f is not a variable. That is, f(x) does not mean that we are multiplying a number f by another number x. Think of a function as a machine that takes in a number, x, and produces another number. In the expression f(x), f is the machine and the parenthesis () are the place where the input, x, is entered into the machine. f(x) is the output that the machine produces with the input x. For example, consider that your machine adds 5 to an input. Then f(3) = 8, or more generally, f(x) = x + 5.
Now that we have considered the domain of a function, we will turn to the range of a function, which is the set of all y values for which a function is defined. Just as we did with domain, we can examine a function and determine its range. Again, it is often helpful to think about what restrictions there might be, and what the graph of the function looks like. Consider for example the function y = x^{2}. The domain of this function is all real numbers, but what about the range?
The range of the function is the set of all real numbers greater than or equal to zero. This is the case because every y value is the square of an x value. If we square positive and negative numbers, the result will always be positive. If x = 0, then y = 0. We can also see the range if we look at a graph of y = x^{2}:
Some functions have sudden jumps. Consider the "rounding" function that takes a number and rounds it to the nearest whole number (rounding up if the number is exactly between two whole numbers). So some values for this function are (2, 2), (1.4, 1), (3.9, 4), (5.5, 6), and (−5.5, −5). The domain of this function is all real numbers, but the range of the function is the integers.
Another function that jumps comes from the way taxis often charge. Suppose a taxi costs $5.00 for the first 2 miles but then $1 for each additional mile or fraction of a mile. Consider the function that has the distance traveled as the input and the cost of the taxi ride as the output. So some values for this function are (1, 5), (1.9, 5), (2.1, 6), (10, 13). The domain of this function is the nonnegative real numbers (since you can't travel a negative distance in a taxi cab). The range of this function is all positive integers greater than or equal to 5: {5, 6, 7, 8, ...}.
Example 3
State the domain and range of the function Solution: For this function, we can choose any x value, except x ≠ 0. Therefore the domain of the function is the set of all real numbers, except x ≠ 0. The range is also restricted to the nonzero real numbers, but for a different reason. Because the numerator of the fraction is 2, the numerator can never equal zero, so the fraction can never equal zero. 
Now that we have defined what it means for a relation to be a function, and we have defined domain and range of a function, we can look at some specific examples of functions and their graphs.
Families of Functions[edit]
The examples we have seen so far have included several different types of functions. From your previous experience working with equations and graphs, you may have already made connections between the forms of the equations of functions, and what the graphs look like. Here we will examine several "families" of functions. A family of functions is a set of functions whose equations have a similar form. The "parent" of the family is the equation in the family with the simplest form. For example, y = x^{2} is a parent to other functions, such as y = 2x^{2} − 5x + 3. The table below summarizes the key aspects of several families of functions.
All of these functions can be used to represent real situations. For example, the linear function y = 3x was used above to represent how much money you would make selling candy bars for $3.00 each. This type of situation is known as direct variation. We say that the amount of money you make varies directly with the number of candy bars you sell. Direct variation between two variables will always be modeled with a linear function of the form y = mx. The slope of the line, m, is the constant of variation. Notice that the y−intercept of the line is 0; that is, the line contains the point (0, 0). This makes sense in terms of the candy selling situation: if you sell 0 candy bars, you make 0 dollars.
Other situations can be modeled with a different kind of linear function. Consider the following situation: a restaurant is having a special: a large cheese pizza costs $8.00, and each topping costs $2.00. The cost of a pizza can be modeled with the function c(x) = 2x + 8, where x is the number of toppings on the pizza. The slope of the line is 2, as each topping adds $2 to the price. The y−intercept is 8: if you do not choose any additional toppings, the pizza costs $8.00.
Quadratic, cubic, and other polynomial functions can be used to model many types of situations Another important family of functions is the rational functions, or quotients of polynomials, such as:
For example, a rational function is used to model inverse variation between two variables. Inverse variation means that the product of two variables is constant: xy = k. If we solve this equation for y, we have , a rational function. The following example shows inverse variation in a real situation:
Example 4
Some days you drive to work, and other days you ride your bike. Yesterday you drove at an average rate of 40 miles per hour, and it took 15 minutes. Today you rode your bike a rate of 20 miles per hour, and it took half an hour. Write an equation that shows the relationship between your speed and the time it takes to get to work. Solution:
First, note that the distance between home and work is 10 miles: We know that in general: Therefore if you drive or ride at a rate of x miles per hour, it will take you y hours to get to work: . 
In general, functions can be used to model real phenomena in many contexts, including different areas of science, business, economics, and more. The type of function that can be used to model a specific situation depends on the key aspects of a function that will match key aspects of the situation. One aspect of many situations is not seen in the function types we have seen so far, but will be seen in the trigonometric functions you will learn about in this chapter. Consider for example, the table below, which shows the average monthly high and low temperatures in the city of Boston, MA, from 1971 to 2000. (Source: rssweather.com)

Table 1.4: Average monthly high/low temperatures for Boston, MA Month Low High January 22.1°F 36.5°F February 24.2°F 38.7°F March 31.5°F 46.3°F April 40.5°F 56.1°F May 50.2°F 66.7°F June 59.4°F 76.6°F July 65.5°F 82.2°F August 64.5°F 80.1°F September 56.8°F 72.5°F October 46.4°F 61.8°F November 37.9°F 51.8°F December 27.8°F 41.7°F
The graph below shows the average low temperatures.
Notice that the graph includes a full year of data, and then ends with December, the 12^{th} month. It is possible that the curve suggested by this graph can be approximated by a function in one of the families of functions we've discussed. Not all natural phenomena can be modeled with mathematical functions, but many can.
Suppose this data was representative of Boston weather in general. We could make a function whose input is the time in months from the present and whose output is the average temperature expected. For example f(1) = 22.1, f(5) = 50.2. The function will repeat after one year. What does 13 represent? What is the temperature expected to be based on this data?
Because the months cycle each year, 13 represents January of the next year, and in general, we can predict the weather in January given our knowledge of the usual climate in a location. For example, January is the coldest month of the year in the city of Boston. For the years shown in the table, the average low temperature was about 22 degrees. We can therefore predict that the average low temperature in January in Boston will be about 22 degrees. We could use such a function to compare current weather to past weather and test for climate changes over time.
Because the months of the year and the weather patterns are cyclical in nature, we need to model this situation with a function that is also cyclical in nature. Such functions are referred to as periodic. A function is periodic if there exists some value p such that f(x + p) = f(x) for all x in the domain of the function. The trigonometric functions you will learn about in this chapter are one type of periodic function, and we can use certain trigonometric functions to model the weather data shown above. We will return to this topic at the end of this lesson, but now we will look at the graphs of functions.
Graphing Functions and Technological Tools[edit]
While there are techniques you can use to efficiently graph many functions by hand, using a graphing calculator allows you to quickly graph any function, and to identify key aspects of the function. The following two examples will show you how to use a TI graphing calculator to explore a function.
Example 5
Graph the function y = x^{3} − 3x^{2} + 1 a. Evaluate the function for x = 0, x = 2, and x = −2. b. Describe the end behavior of the function. c. Approximate all x−intercepts. d. Approximate any local maxima and minima. Solution: To graph this function, press [Y=], and clear any equations already entered. In Y1, enter the equation. If you have never entered an equation before, here are some tips:
Once you have entered the equation, press [ZOOM] [6]. This will take you to the "standard" window: you can see both x and y from −10 to 10. (Note that if you scroll down to option 6, you have to press enter. However, if you just enter the number 6, you will be taken to the graph.)
Now that you are in tracing mode, you can enter any x value, and the calculator will tell you the y value. For example, if you press [2] [ENTER], you will see the cursor move to the point (2,−3) and at the bottom of the screen, you will see x = 2 and y = −3. If you press [()] [2] [ENTER], you will see x = −2 and y = −19 at the bottom of the screen. Notice that you cannot see the point on the graph. To see that point, we need to change the window. Press [WINDOW] and scroll down to Ymin. Change the −10 to −25. Then press [GRAPH]. Now press [TRACE] [()] [2] [ENTER]. You should see the point (−2,−19).

Example 6
You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You want the enclosed land to be a rectangle. a. Write a function to model the area of the plot as a function of the width of the plot. b. Graph the function using a graphing calculator. c. What size rectangle should you make with the fence in order to maximize the area of the rectangular enclosure? d. Explain the significance of the x−intercepts. Solution: The plot of land will look like the picture below:
The graph of A(x) is shown here on the interval [0, 100].

Introduction to Trigonometric Functions[edit]
Consider again the temperature data from above:
As was noted above, this kind of data needs to be modeled with a function that is periodic. In particular, this kind of data is often modeled by a sinusoid, a graph that oscillates in a particular way, as seen in the graph below.
Every sinusoid repeats its values on a regular interval. If we modeled the weather data with such a graph, the values will repeat every 12 months. Therefore we say that the period of the function is 12.
Notice that the data ranges from about 22 to 65. Also notice that the "wave" centers in between these values, around y = 43. Therefore we say that the amplitude of the wave is about 21, which is the distance from the middle to the top or the bottom of the wave.
Many real phenomena can be modeled with this kind of function.
Lesson Summary[edit]
In this lesson we have reviewed the concept of a function, including major aspects of functions, and different types of functions. We have also used graphing calculators to graph and explore different functions. A key point of this lesson is that we can use functions to model real phenomena. A second key point is that in order to model phenomena that are cyclical in nature, we need to use functions that are periodic. In lesson 4 in this chapter we will define six trigonometric functions. However, because the inputs of these functions are angles, in the next two lessons we will focus on angles. First we will review angles in triangles from Geometry, and then we will consider angles in rotation.
Points to Consider[edit]
 What distinguishes a function from a relation?
 What makes a function periodic?
 What are the pros and cons of using a calculator to graph a function?
Review Questions[edit]
 Determine if each relation is a function:
 A train travels at a constant speed of 95 miles per hour.
 (a) Write an equation that shows the relationship between the number of hours the train has traveled and the distance it has traveled.
 (b) Is this situation direct variation, inverse variation, or neither?
 (c) Use the equation to determine the distance the train has traveled after 3 hours.
 You decide to start a small business making picture frames. You spend $100 on paint and other supplies, as well as $2.00 per wooden frame. You decide to sell each frame for $10.00.
 (a) Write a linear function that models the costs of your business.
 (b) Write a linear function that models the revenue of your business. (Revenue is the amount of money you take in.)
 (c) Write a linear function that models the profits of your business. (The profits can be found by subtracting the costs from the revenue.)
 (d) Use your profit function to determine the minimum number of frames that must be sold to make a profit.
 Consider the function defined by the equation f(x) = x^{2} − x − 3.
 (a) To what family does this function belong?
 (b) State the domain and range of the function.
 (c) Use a graphing calculator to graph the function, to identify the approximate coordinates of the vertex, and the approximate values of the x−intercepts.
 Consider the function
 (a) Use a graphing calculator to graph the function.
 (b) Identify all asymptotes.
 The price of reserving a private party room in a restaurant is $500. The price per person varies inversely with the number of people who attend the party.
 (a) Write an equation that represents the relationship between c, the cost per person, and p the number of people attending.
 (b) Use the equation to find the cost per person if 32 people attend.
 Use a graphing calculator to graph the functions y = x^{3} + x, y = x^{3} + 2x, y = x^{3} − x, and y = x^{3} − 2x. What is the effect of changing the coefficient on the second term?
 The equation p(x) = −.5x^{2} + 90x − 200 represents the profits of a company, where x is the number of units the company sells. Use a graphing calculator to graph the function, and use the graph to answer the questions.
 (a) What is the maximum profit, and how many units must be sold to reach the maximum profit?
 (b) Find the x−intercepts and explain the meaning of these points on the graph in terms of the profits of the company.
 The table below shows the average daylight hours each month in Anchorage, Alaska.
 (a) Use your graphing calculator to plot the data, or graph by hand. Use January = 1.
 (b) What is the period of the data?
 (c) How might the graph look different if the data represented daylight hours where you live?


Table 1.5 Month Average daylight hours January 5.65 February 7.77 March 10.4 April 13.37 May 16.87 June 18.72 July 19.18 August 17.12 September 14.27 October 11.43 November 8.53 December 6.13

Review Answers[edit]

 (a) Not a function
 (b) Is a function
 (c) Not a function

 (a) y = 95x
 (b) The situation is direct variation.
 (c) 285 miles

 (a) C(x) = 2x + 100
 (b) R(x) = 10x
 (c) P(x) = 8x  100
 (d) You must make and sell 13 frames to make a profit.

 (a)
 (b) $15.63

 (a) Maximum profit is $3,850, with 90 units sold.
 (b) 2.25 and 177.75. These are the breakeven points. When 2−3 units are sold, the company has made enough money to make up for initial costs. After selling 177 units, the company is no longer profitable.
Vocabulary[edit]
 dependent variable
 The input variable of a function, usually denoted x.
 domain
 The domain is the set of input values (x) for which a function is defined.
 function
 A relation in which every element of the domain is paired with exactly one element of the range.
 independent variable
 The output variable of a function, usually denoted y.
 periodic function
 Any function that repeats regularly.
 range
 The set of output or function values (y) for a function.
 relation
 A pairing between the items in two sets of numbers or data.
 COODINATES is two points in the graph or function;