K12MATH012: Precalculus I

Unit 2: Linear Functions   Math students often ask, “How can I use this? What’s it for?” The answer is “modeling.” Mathematics is designed to help model real-world phenomena and trends. Modeling allows us to apply mathematics to situations such as predicting population growth, balancing costs and profits, developing inventions, or clarifying the shape of terrain where a road must go. As you learn mathematics, it is important to acquire skill in creating functions to describe our world. The power of mathematics is in its application to real life.
As you delve into specific functions in this and additional units, you will not only learn mathematical processes but also about using functions to model situations. You begin with a familiar form - linear. Linear functions are well named because their graphs are lines. This unit begins by describing linear functions and their graphs. It applies those concepts to modeling as you learn to fit a line to a set of data points and measure how close to being linear those points actually are. The unit closes by exploring absolute value functions and their graphs. 

Unit 2 Time Advisory
Completing this unit should take approximately 12 hours and 40 minutes.

☐    Subunit 2.1: 1 hour and 55 minutes

        ☐    Subunit 2.1.1: 25 minutes

        ☐    Subunit 2.1.2: 1 hour and 30 minutes

☐    Subunit 2.2: 3 hours

        ☐    Subunit 2.2.1: 50 minutes

        ☐    Subunit 2.2.2: 45 minutes

        ☐    Subunit 2.2.3: 1 hour and 25 minutes

☐    Subunit 2.3: 1 hour and 55 minutes

☐    Subunit 2.4: 2 hours and 40 minutes

        ☐    Subunit 2.4.1: 35 minutes

        ☐    Subunit 2.4.2: 30 minutes

        ☐    Subunit 2.4.3: 1 hour and 35 minutes

☐    Subunit 2.5: 3 hours and 10 minutes

        ☐    Subunit 2.5.1: 25 minutes

        ☐    Subunit 2.5.2: 30 minutes

        ☐    Subunit 2.5.3: 2 hours and 15 minutes

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Determine the slope of a linear function.
  - Calculate the rate of change of a linear function.
  - Create graphs of linear functions.
  - Create equations to model given graphs of linear functions.
  - Fit linear models to real-world data.
  - Evaluate a correlation coefficient.
  - Solve absolute value equations.
  - Graph absolute value functions.

Standards Addressed (Common Core): - CCSS.Math.Content.6-NS.C.7.c - CCSS.Math.Content.8-SP.A.1 - CCSS.Math.Content.HSA-SSE.A.1.a - CCSS.Math.Content.HSA-CED.A.1 - CCSS.Math.Content.HSA-CED.A.3 - CCSS.Math.Content.HSA-CED.A.4 - CCSS.Math.Content.HSA-REI.B.3 - CCSS.Math.Content.HSA-REI.D.10 - CCSS.Math.Content.HSA-REI.D.11 - CCSS.Math.Content.HSF-IF.A.1 - CCSS.Math.Content.HSF-IF.A.2 - CCSS.Math.Content.HSF-IF.B.4 - CCSS.Math.Content.HSF-IF.B.6 - CCSS.Math.Content.HSF-IF.C.7
- CCSS Math.Content.HSF-IF.C.8 - CCSS-Math.Content.HSF-BF.A.1 - CCSS.Math.Content.HSF-BF.B.3 - CCSS.Math.Content.HSF-LE.A.2 - CCSS.Math.Content.HSS-ID.B.6 - CCSS.Math.Content.HSS-ID.C.7 - CCSS.Math.Content.HSS-ID.C.8 - CCSS.ELA-Literacy.RST.11-12.2 - CCSS.ELA-Literacy.RST.11-12.3 - CCSS.ELA-Literacy.RST.11-12.4 - CCSS.ELA-Literacy.RST.11-12.6 - CCSS.ELA-Literacy.RST.11-12.9 - CCSS.ELA-Literacy.RST.11-12.10 - CCSS.ELA-Literacy.WHST.11-12.2.d - CCSS.ELA-Literacy.WHST.11-12.4

2.1 Linear Functions   Some of the concepts in this subunit will be familiar and part of the material will be review. Focus carefully; there are seeds of ideas in this unit that will grow into bigger ideas as they are applied to more complex functions in the later parts of the course. This subunit defines linear functions and investigates how to find the rate of change of a function.

2.1.1 Linear Functions   Suppose the local fitness center charges $80 to join and $35 every month that you are a member. The relationship between cost and the number of months you are a member is linear and can be described as C(n) = 80 + 35 n. This function would predict your total fitness expense until the gym raises prices. The subunit begins by defining linear functions and their characteristics and then investigates how to find the rate of change.

2.1.2 Rate of Change   One of the important descriptors of a function is the rate of change. Linear functions have a constant rate of change. Although there is an easy way to “see” the rate of change of the algebraic form of a linear function, it is important to focus on how to determine the rate of change algebraically. This process will help you understand rate of change and will be important when you apply it to more complex functions in subsequent units of the text. This subunit will define rate of change and apply it to linear functions.

2.2 Graphs of Linear Functions   As you explore linear functions further, you will apply ideas from Unit 1 to these familiar graphs. In the previous subunit, you learned about transformations of toolkit functions. This subunit applies those ideas as it delves into graphs of lines and explores relationships in parallel and perpendicular lines. It closes by focusing on the intersections of lines.

2.2.1 Graphing Linear Functions   This subunit begins by reviewing the topics of slopes and intercepts. It extends the exploration by applying graph transformations as we focus on shifts and compressions. The subunit closes by identifying the special types of equations that describe horizontal and vertical lines.

2.2.2 Parallel and Perpendicular Lines   The relationship among parallel lines is very specific. This subunit helps you see the connections among algebraic descriptions of functions that create parallel lines. Similarly it looks at connections among descriptions of functions creating perpendicular lines. 

  • Activity: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions” Link: Notre Dame Open Courseware’s Calculus Applet: “Transformations of Functions” (HTML)
    Instructions: Click on the link above to see more complex transformations. Below the graph, find where f(x) is identified and change the function to f(x) = a*x+b. Change the g function to g(x) = a*x + c. Focus on the left-hand graph, and ignore the one on the right. Move the sliding bars to see the transformations as you alter values of a, b, c, andx. Try to determine which of the bars controls whether the functions are parallel.
    After working with these two functions, rename the g function to be g(x) = -(1/a)x + c. Move the sliding bars to see the transformations as you alter values of a, b, c, and x. Try to determine which of the bars controls whether the functions are perpendicular.
    Completing this activity should take approximately 10 minutes.
    Standards Addressed (Common Core):

    Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 License.

  • Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions (PDF)
    Instructions: Read pages 117 - 119 of the text, focusing on the relationships between parallel lines and between perpendicular lines. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 119. Answers are on page 121.
    Completing this activity should take approximately 15 minutes.
    Standards Addressed (Common Core):

    Terms of Use: This resource is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License.

  • Activity: Getting the Slant on Parallel and Perpendicular Lines Instructions: Address an explanation to a math student who is about to begin this course as you write two paragraphs in answer to the following: Why do parallel lines have the same slope? With perpendicular lines, why would one slope always be positive and the other always be negative? Have someone read your paragraphs and help you polish them to clarity.
    Completing this activity should take approximately 20 minutes.
    Standards Addressed (Common Core):

2.2.3 Intersections of Lines   The connections between graphs and algebraic forms come together in this subunit. A graph gives the big picture of the behavior of functions, but it is the algebraic form that gives specifics. It is impossible from a small graph to tell if a point is at 2 or 2.00146, but the algebra can pin things down. This subunit uses graphs to illustrate the relation between two distinct linear functions and the place at which they have points in common.

2.3 Modeling with Linear Functions   Modeling is where mathematics is applied. This subunit focuses on problem solving and writing models to describe life situations. One of the secrets of writing an equation lies in translation. Suppose you want an algebraic description of the following declarative sentence: Doubling the height is the same as adding five to the width. The key to an equation is to find the words in the sentence that represent the equal sign. Everything in front of the words meaning “equal” will be in front of the equal sign; everything after the words meaning “equal” will be after the equal sign. Seldom is it necessary to change the order when translating. The translation is as follows:

       Doubling the height is the same as adding five to the width.
2= 5 + w

2.4 Fitting Linear Models to Data   If you wrote down the outside temperature at noon every day from spring to midsummer, it would tend to get warmer every day, but the data points would not be on an exact straight line. The points might be approximately linear, though. This subunit begins with creating a line on a set of data points to approximate a relation and looks at when it’s appropriate to predict other values from that line. It explores the use of technology to create a line of best fit and then introduces the correlation coefficient, which is a way to measure how well a line describes the relation among a set of data points.

2.4.1 Approximately Linear - Trend Lines   Many relations are only approximately linear, but sometimes we can use lines to estimate them. Suppose a dieter gets weighed every Friday. The weights over time could be approximately linear, but not fall exactly on a line. This subunit places lines on sets of data, identifies the equation of each line, and then uses the equation to identify other values. If, for instance our dieter missed getting weighed one Friday, the equation of the trend line would make it possible to estimate that Friday’s weight later on. 

2.4.2 Interpolation and Extrapolation   Imagine a girl who is 85 centimeters tall at age 2 and 148 centimeters at age 9. We could develop a trend line of height as a function of age: h(a) = 9a + 67. We could use that function to estimate the girl’s height at age 4, which would be h(4) = 9(4) + 67 = 103 centimeters. That would probably be a close approximation. This process is called interpolation.
If, however, we used the function to predict her height at age 70, we would have h(70) = 9(70) + 67 = 697 centimeters. That would mean that this girl would grow to nearly 23 feet tall! Extrapolation is using a trend outside the boundaries of the original input data collected, and it often leads to wild predictions.

2.4.3 Technology and Linear Models   Trend lines like the ones you created in the previous subunit are rough estimates. Statisticians have developed a complex process to find the “line of best fit,” also called the “least squares regression line.” Luckily, technology enables us to find the equation of that line while avoiding the intricate mathematical process. This subunit will develop these regression lines and also introduce the correlation coefficient, which is a value that indicates how close to a line a set of data points actually are.

2.5 Absolute Value Functions   Think of absolute value as distance - that’s why it’s never negative. As you work through the material in this subunit, it is helpful to keep the concept of distance in your mind. You will begin with absolute value and its important features and progress to solving absolute value equations and inequalities.

2.5.1 Defining Absolute Value Functions   A standard absolute value function can be described with a piecewise function of two linear functions. This subunit describes the concept of absolute value and identifies the important features of this type of function.

2.5.2 Solving Absolute Value Equations   If the absolute value of (x - a) is 7, that means that the distance between x and a is seven units. So consider the absolute value of (x - 3): if it’s 7, then the distance between x and 3 is seven, meaning x is seven units away from three - that means that x is at 10 or -4. Sometimes a number line is a useful way to consider absolute value equations. This subunit focuses on an algebraic way to solve absolute value equations and teaches a step-by-step process to address these sometimes confusing equations.

2.5.3 Solving Absolute Value Inequalities   Inequalities look a lot like equations, but they do not use an equal sign (=) as the verb. Inequalities use things like <, >, and <. Keeping the distance idea in mind with absolute value inequalities is very helpful. In the inequality |x – 3| < 2, the distance idea can be useful. Think of this problem as saying the distance between x and 3 is less than 2 units. That means that x can be 2 units away from three, but no farther; thus, x has to be somewhere between 1 and 5, inclusive. If the verb is >, and we have |x – 3| > 2, then we know that x is at least 2 units away from 3, meaning it must be greater than 5 or less than 1. This subunit focuses on algebraic methods to solving inequalities that have absolute value statements within them. If you remember that you can think of absolute value in terms of distance, it may be much easier to solve the problems