Identify Non-linear Functions from Data Student Probe Identify which data sets display linear, exponential, or quadratic behavior. x -1 0 1 2 3 y -3-4 -3 0 5 x -2 0 2 4 6 y 9 4-1 -6-11 x -1 0 1 2 3 y ¼ 1 4 16 64 Answer: Quadratic, Linear, Exponential Lesson Description Students are asked to identify the types of nonlinear functions represented in data tables. Students will be expected to distinguish between linear, quadratic and exponential data. Rationale Representations are the means by which mathematical patterns are recorded and analyzed. Different representations support different ways of thinking about mathematical objects. Students should be able to move fluidly between graphical, symbolic, and tabular representations of functions. Preparation None At a Glance What: Students will recognize function types given a table of values Common Core State Standard: CC.9-12.F.LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Matched Arkansas Standard: AR.9-12.LF.TM.1.1 (LF.1.TM.1) Identify a linear relationship represented by a table, by a graph, and by symbolic forms; AR.9-12.EF.TM.2.1 (EF.2.TM.1) Identify exponential growth or decay by creating tables, graphs, and mathematical models; AR.9-12.QEF.AII.3.3 (QEF.3.AII.3) Analyze and solve quadratic equations with and without appropriate technology by: -- graphing Mathematical Practices: Reason abstractly and quantitatively. Model with mathematics. Look for and make use of structure Who: students who have difficulty recognizing non-linear functions given a table of values Grade Level: Algebra I Prerequisite Vocabulary: constant, domain, range, linear, exponential, quadratic, parabola Prerequisite Skills: plotting ordered pairs, graphing Delivery Format: Individual, pairs, small group, whole group Lesson Length: 15-20 minutes Materials, Resources, Technology: Graph paper, graphing calculator (optional) Student Worksheets: None
Lesson The teacher says or Expect students to say or do If students do not, then 1. Look at this table of data. x 0 5 10 15 20 y 800 400 200 100 50 How are the values in the domain (the x values) changing? Is it a constant change? 2. How are the values in the range (the y values) changing? Increasing by 5. This change is constant. Decreasing by one-half. The x values are 0, 5, 10, 15, 20, How are they changing? Is it always changing by 5? Do you see a relationship between 800, 400, 200, 100, 50? 3. If they are decreasing by 1 2, do you subtract, divide, or multiply? So they have a common factor of 1 2. Multiply by 1 2 or divide by 2 4. Now that we have determined that the domain is in regular intervals and the range decreases by a common factor of 1, what type of function do 2 the values describe? 5. Plot the points on a graph to determine the shape of the graph. Exponential (decay) Is it linear? Is it quadratic? Is it exponential? 6. What do you notice about the graph? Answers may vary, but listen for: As the x values increase, the y values decrease quickly and then seem to start to level off Is it increasing or decreasing? Rapidly or slowly?
The teacher says or Expect students to say or do If students do not, then 7. Since it forms this shape, we call it exponential. 8. Look at this table of data. x 0 5 10 15 20 y 3 6 9 12 15 Look for the patterns formed by the x values and the y values. What do notice about them? 9. Plot these points and describe the graph. The graph is a straight line so this is a linear function. The x values are increasing at a constant rate. The y values are increasing at a constant rate. Are the x values increasing or decreasing? At a constant or changing rate? Are the y values increasing or decreasing? At a constant or changing rate? 10. Look at this table of data. x 4 5 6 7 8 y 4 1 0 1 4 Look for the patterns formed by the x values and the y values. What do you notice about them? 11. Plot these points and describe the graph. The graph is U-shaped (parabolic) so the function is quadratic. The x values change at a constant rate. The y values decrease and then increase. How do the x values change? How do the y values change?
The teacher says or Expect students to say or do If students do not, then 12. Suppose you rent a gaming device for a flat fee of $10 plus $4 per day. Create a table of values for the cost for the first 5 days. Day 1 2 3 4 5 Cost ($) 14 18 22 26 30 What is the cost for one day? If it is $4 for each day, what is the cost for 2 days? 13. Graph this function. 14. What type of function is this? How do you know? 15. Stephen had a bag containing 160 jelly beans. He ate one half of the jelly beans in the bag each day. Create a table of values for this situation. Your table should show the day and the number of jelly beans remaining in the bag. Linear The graph is a straight line. Day 1 2 3 4 5 Beans 80 40 20 10 5 How many did he eat the first day? How many did he eat the second day? 16. Graph this function. 17. What type of function is this? How do you know? Exponential It is decreasing by a common factor of 1 2.
Teacher Notes: 1. The general formulas for the three types of functions are listed below. Linear: y = mx + b Quadratic: y = ax 2 + bx + c Exponential: y = a x 2. Graphing calculators may be used with this lesson. Variations The teacher may elect to develop a card sort game with three sets of cards. One set of cards would contain tables of values, another set of cards would contain the graphs of those values, and the third set of cards would contain contextual situations. Students would match the tables to the graphs and the contextual situations and then classify them as linear, quadratic, or exponential. Formative Assessment Identify which data sets display linear, exponential, or quadratic behavior. 1. y 1 3 9 27 81 2. y 6 2 0 2 6 3. y 2 5 8 11 14 Answers: 1. Exponential 2. Quadratic 3. Linear References Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on May 12, 2011.