CK-12 Trigonometry Read online

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  Chapter 1: Trigonometry and Right Angles

  Basic Functions

  Learning objectives

  A student will be able to:

  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.

  Introduction

  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.

  The basics of functions

  Consider two situations shown in the boxes below:

  Situation 1: Your car can travel on one gallon of gasoline at . For every mile per hour faster you drive, the car travels half a mile less per gallon of gasoline.

  Situation 2: You collect data from several students in your class on their ages and their heights in inches:

  In the first situation, let the variable represent the speed of your car, and let represent the number of miles it can travel using one gallon of gasoline. If you travel at per hour, you will go on one gallon of gasoline. For example, if you travel at , you will travel 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 is paired with only one . Some relations are mathematically important. For example, circles and ellipses are graphical representations of important relations between and coordinates, but there is not a unique coordinate for each coordinate. Because of the unique for each , 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:

  Representation Example

  Set of ordered pairs (a subset of the ordered pairs for this function)

  Equation

  Graph

  In contrast, the relation shown in the figure below is not a function:

  Representation Example

  Set of ordered pairs (A subset of the ordered pairs for this relation)

  Equation

  Graph

  To verify that this relation is not a function, we must show that at least one value is paired with more than one value. If you look at the first representation, the set of ordered pairs, you can see that is paired with and with . Similarly, is paired with and with . Therefore the relation is not a function. If we look at the graph above, we can see that, except for , the values of the relation are each paired with two 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 through the graph of , 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.

  b.

  Solution:

  a.

  This relation is not a function because is paired with and with . If you plotted the pints, the line would touch points in the relation.

  b. This relation is a function because every is paired with only one .

  Once you are able to determine if a relation is a function, you should then be able to state the set of values and the set of values for which a function is defined.

  The domain of a function is defined as the set of all values for which the function is defined. For example, the domain of the function is the set of all real numbers, often written as . This means that 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 non-negative values of 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.

  b.

  c.

  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 . The domain is restricted this way because a fraction with denominator zero is undefined.

  c.

  The domain of this function is the set of values

  The variable is often referred to as the independent variable, while the variable is referred to as the dependent variable. We talk about and this way because the values of a function depend on what the values are. That is why we also say that “ is a function of .” For example, the value of in the function depends on what value we are considering. If , we can easily determine that . Returning to the situation in the introduction, we can say that the amount of money you take in depends on the number of candy bars you have sold.

  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 is a function of . For example, the equation can also be written as . It is important to remember that represents the values, or the function values, and that the letter is not a variable. That is, does not mean that we are multiplying a number by another number . I like to think of a function as a machine that takes in a number, , and produces another number. In the expression , is the machine and the parenthesis are the place where the input, , is entered into the machine. is the output that the machine produces with the input . For example, consider that your machine adds to an input. Then , or more generally, .

  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 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 ex
ample the function . 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 value is the square of an value. If we square positive and negative numbers, the result will always be positive. If , then . We can also see the range if we look at a graph of :

  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 and . 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 for the first but then 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 . The domain of this function is the non-negative 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 : .

  Example 3: State the domain and range of the function

  Solution: The domain and range of the function

  For this function, we can choose any value, except . Therefore the domain of the function is the set of all real numbers, except .

  The range is also restricted to the non-zero real numbers, but for a different reason. Because the numerator of the fraction is , 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

  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, is a parent to other functions, such as . The table below summarizes the key aspects of several families of functions.

  Family Parent(s) Key aspects Example

  Linear

  The slope of the line is the coefficient , and the intercept is the constant .

  The graph of a linear function is a straight line, which is often identified in terms of its slope and its intercept.

  Quadratic

  The vertex is the point . The graph is symmetric across the line .

  These functions have a highest exponent of . The graph is a parabola, which has a vertex that is either a global maximum or minimum of the graph.

  Cubic

  These functions have a highest exponent of . The ends of the graph have opposite behavior. Cubic graphs either have a local maximum and minimum, like the one in the graph to the right, or no local maximums or minimums, like

  Exponential , , etc

  As approaches , the function values approach the axis .

  Exponential functions have a variable as an exponent. The graph has a horizontal asymptote.

  Rational , etc

  As approaches , the values approach

  As approaches , the values approach (the axis).

  These are functions that contain fractions with polynomials in the numerator and denominator. The graphs have a horizontal and a vertical asymptote.

  All of these functions can be used to represent real situations. For example, the linear function was used above to represent how much money you would make selling candy bars for 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 . The slope of the line, , is the constant of variation. Notice that the intercept of the line is ; that is, the line contains the point . This makes sense in terms of the candy selling situation: if you sell candy bars, you make 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 , and each topping costs . The cost of a pizza can be modeled with the function , where is the number of toppings on the pizza. The slope of the line is , as each topping adds to the price. The intercept is : if you do not choose any additional toppings, the pizza costs .

  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: . If we solve this equation for , 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 per hour, and it took . Today you rode your bike a rate of 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: , where is your speed in miles per hour and is the time it takes to get to work in hours.

  First, note that the distance between home and work is :

  We know that in general:

  Therefore if you drive or ride at a rate of per hour, it will take you 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)

  Month Low High

  Jan

  Feb

  Mar

  Apr

  May

  Jun

  Jul

  Aug

  Sept

  Oct

  Nov

  Dec

  The graph below shows the average low temperatures.

  Notice that the graph includes a full year of data, and then ends with December, the 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 , . The function will repeat after one year. What does represent? What is the temperature expected to be based on this data?

  Because the months cycle each year, 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 We can therefore predict that the average low temperature in January in Boston will be about 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 refe
rred to as periodic. A function is periodic if there exists some value such that for all 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

  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

  a. Evaluate the function for , , and .

  b. Describe the end behavior of the function

  c. Approximate all 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:

  The button is right next to the green [ALPHA] button (on the TI-83 model)

  To raise to the 3rd power, press [MATH].

  To raise to the second power, press the button, which is in the left column.

  Be careful with negatives: the blue “” button on the right side is for subtraction. The button on the bottom that says “” is for negative numbers.

  Once you have entered the equation, press [ZOOM 6]. This will take you to the “standard” window: you can see both and from to . (Note that if you scroll down to option , you have to press enter. However, if you just enter the number , you will be taken to the graph. )