Lesson 10: Comparing Functions and their features
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1 Lesson 10: Comparing Functions and their features Standards: MAFS.912.F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MAFS.912.F-LE.1.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
2 Bellwork:
3 Cut out all of the graphs on the two worksheets. We are going to compare and contrast some features of the graphs and learn some new vocabulary. 1. Find graphs F, K, O, and U. What do they all have in common? 2. Find graphs D, F, M, T, and V. What do they have in common? New Vocabulary: A Line of Symmetry is a line that can be drawn through a graph such that each side of the line is a mirror reflection of the other.
4 3. Use the Vertical line test to sort the graphs into two groups: functions and non functions. List your results below: Functions Non Functions: A function is increasing when the Y variable increases as the X variable increases. (it goes up from left to right). A function is decreasing when the Y variable decreases as the X variable increases (it goes down from left to right). If the Y variable remains constant over the entire domain, then that function is called a constant function (a horizontal line). 4. Sort the functions into one of the following groups: Increasing Decreasing Constant Combination
5 You should have seven functions that are either increasing, decreasing, or constant. Write the following equations on the seven functions. These are the equations that determine the graphs: G. y = x 1 x H. y = 5 2 K. y = 2 x L. y = 2 3 x + 5 O. y = x + 3 P. y = ( 1 2 )x U. y = 2 Sort these graphs into two groups based on the EQUATIONS representing the functions. Group 1 Group 2
6 Your two groups represent two function families The first function family are LINEAR functions. The equations are in the format y = mx + b where m and b are numbers and can be zero (if so the y will appear missing). The second function family is called EXPONENTIAL functions. The equations are in the format y = ab x where a and b are real numbers and b is greater than 0 but not equal to 1. Let s look at those equations again and see how they meet the above definitions:
7 A function has an absolute minimum if there is a point that has a y-coordinate that is less than the y-coordinates of every other point on the graph. In other words, there is a LOWEST point. A function has an absolute maximum if there is a point that has a y- coordinate that is greater than the y-coordinates of every other point on the graph. In other words, there is a HIGHEST point. If the graph goes forever up it would not have an absolute maximum point and if it goes forever down, it would not have an absolute minimum point. We are going to work with the graphs that were in the combination category from an earlier slide. Sort these graphs into three groups: those that have an absolute minimum, those that have an absolute maximum, and those that have no absolute minimum or maximum point. Absolute minimum Absolute maximum No minimum or maximum
8 These are the equations of the graphs with either an absolute maximum or an absolute minimum. Write these equations on the graphs they belong to: T. y = x 2 +8x =12 V. y = x 3 2 M. y = x 2 D. y = x I. y = x F. y = 3x 2 +4 where x is an integer B. y = 1 2 x2 + 2x Q. y = 2 x Consider the above graphs only. Sort the graphs into two groups based on the EQUATIONS. Group 1 Group 2
9 The family of QUADRATIC functions includes functions of the form y = ax 2 + bx + c where a, b, and c are real numbers and a is not zero. The family of Linear ABSOLUTE VALUE functions includes functions of the form y = a x + b + c where a, b, and c are real numbers, and a is not zero. Now look at graphs A, S, and C. These graphs are called LINEAR PIECEWISE functions. They are functions that equation changes for different parts of the domain. The equations are very complicated and we will not write them down right now. For now, you just need to be able to recognize that the shape of the graph changes abruptly from one piece of the x-axis to the other.
10 Now we will sort all of our graphs in to their families on graphic organizers and list all of their definitions, equations, and graphical behavior including increasing or decreasing, maximums and minimums, and curves/lines.
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