ࡱ> -/,5@  Dbjbj22 "FXX."```````t$4t!<X!!!!!!!$9#R%|8!9`5|8!``q!AAA``!A!AAV j`` 0 ,?v !0! T&O& tt````&` (A8!8!ttd7 ttMAT 117 WEEK 4 LESSON PLAN Total estimated time : 145 minutes Objectives: 1. Graphs of Functions Find the domain and range using the graph of a function Vertical Line Test Describe the increasing and decreasing behavior of a function Classify a function as even or odd Identify six common graphs ADVANCE \d6 2. Transformations of Functions Sketch the graph of a function using common graphs and transformations Write the equation of function using common graphs and transformations Motivation: [2 minutes] Why learn about the graphs of functions? One reason is because it helps us recognize the type of relationship that exists between two quantities being considered. Additionally, if were analyzing a set of data, were able to use our knowledge of the different types of graphs of functions to choose the one that best fits the pattern of the data being studied. This allows us to then express the relationship between the two quantities being studied in terms of an equation, which we can then use to estimate, predict, and describe all sorts of things concerning the situation being studied. Individual teachers may want to relate a personal experience with graphs, data, etc. Warm up discussion: [5 minutes] Provide an example of an equation for a function and draw the graph. Prompt students to point out properties of the graph previously learned such as intercepts and symmetry. 1. Graphs of functions Find the Domain and Range using the graph of a function Warm up example or activity: [ 5 minutes] Prompt students to review meaning of domain and range using input/output, independent/dependent variable, and horizontal/vertical axis ideas. On board or overhead, provide graph for students to determine what values constitute the domain and range. Try to stay away from using x and y values because functions are not always in terms of those variables. Formal concept: [ 3 minutes] State the definitions combined with the concept of the graph: the domain is all possible input values on the horizontal axis that give a defined output, which are the range values on the vertical axis. Example: [ 10 minutes] Provide several different graphs and prompt students to find the domain and range. Review the different ways to express sets of numbers, such as set-builder notation, interval notation, etc, and indicate your preference if you have one. Be sure to use examples that include infinite and finite domain/ranges and the corresponding different notation. Some good ones to start with:  EMBED Equation.3   EMBED Equation.3  A step function, such as the greatest integer function. Vertical Line Test Warm up example or activity: [ 10 minutes] Draw a function diagram on the board and ask the students how to connect inputs and outputs so that it does not represent a function. (Hopefully you will have one input in the domain going to at least two outputs in the range.) Show them what this means in terms of two coordinate pairs and plot these on a set of axes. Point out that this wouldnt represent a function because one point is right above the other. Some students will catch on a say vertical line test, if not, point out the concept to them. Remind them of the definition of a function, the output must be unique, or there must be at most one output value for each input. Formal concept: [ 2 minutes] Write a formal definition of the vertical line test on the board. Example: [ 6 minutes] Draw several graphs on the board and apply the VLT. Include tricky examples, like a graph of a step function or one consisting of plotted points. Increasing and decreasing behavior of a function Warm up example or activity: [ 5 minutes] Introduce a function that represents distance D (in miles) from a certain point as a function of time t (in hours). For example, D(t) = 6t t for 0 d" t d" 6. Prompt students to sketch the graph of the function. Ask students to interpret the graph within some context. Prompt students to describe where the function is increasing, decreasing, or neither within same context. Remind students that a functions graph is very useful in determining a functions behavior. This example will hopefully show the students that there is a purpose in looking at this type of behavior. Formal concept: [ 3 minutes] Explain that increasing and decreasing behavior must be expressed in terms of the input variable. Also remind the students to look at the graph from left to right along the horizontal axis. Example: [ 10 minutes] Do a straightforward example (not a word problem) that has more turning points. For instance, f(x) = 2x 6x x + 5. After this initial example, a word problem may be appropriate. Also, introduce a graph that is constant for a small portion of the domain.  y    x Even and odd functions Warm up example or activity: [ 6 minutes] Review y-axis, x-axis, and origin symmetry of graphs, including a discussion of the relationship between the points. For example, if a graph is symmetric about the y-axis and includes the point (3,7), then the point (3,7) must also be on the graph. Prompt students to recall the algebraic method of determining whether a graph has y-axis symmetry, substituting -x for x in the graphs equation, etc., and then make the connection to functions using f(x). Similarly, discuss origin symmetry. A discussion of why we dont consider x-axis symmetry when dealing with functions might be a good review of the definition of function, VLT, etc. Formal concept: [ 2 minutes] State the formal definitions of even and odd that were developed above. Examples: [ 10 minutes] Example 1: Draw several graphs and decide if they are even, odd, or neither. Example 2: Given functions, algebraically determine if they are even, odd, or neither. Ask the students to graph the functions to check and see if their answers are consistent. A good one to start with:  EMBED Equation.3  (you can look at graphs of rationals on the calculator to clear up future questions. ) Identify six common graphs Warm up example or activity: [ 12 minutes] Sketch the graphs prompt students to identify their corresponding functions. Stress the importance of recognizing the general behavior of these graphs and how one is distinguished from another. Have the students state properties previously learned, like increasing/decreasing and even/odd functions as well as domain and range of each. 2.Transformations of functions Vertical and horizontal shifts Warm up example or activity: [10 minutes] Let the students work in groups and explore functions such as g(x)=(x+4) and f(x)=|x| 3 on their calculators. Ask them to write out any rules they find. Formal concept: [ 3 minutes] Define the different types of shifts algebraically and clearly state how they need to write them out in words. For example, f(x) + c means a vertical shift up c units for some c > 0. Example: [ 5 minutes] Combine various shifts into the same problem and have the students state all transformations. Then sketch the graph in terms of the common graph for each. f(x)=(x 2) + 5. Vertical and horizontal stretches and compressions Warm up example or activity: [ 10 minutes] Again, let the students work in groups and explore various functions like f(x) = 4x and g(x) = (3x). Formal concept: [ 3 minutes] Define the different types of stretches and compressions algebraically and clearly state how they need to write them out in words. For example, cf(x) means a vertical stretch by a factor of c units for some c > 1 and a vertical compression for 0 < c < 1. Example: [ 5 minutes] Combine stretches and compressions with shifts, state all transformations, and then graph in terms of the common graph. Use examples such as h(x)=  EMBED Equation.2  or g(x) = x + 4. Reflections about the x and y axes Warm up example or activity: [ 10 minutes] With the students working in groups, let them explore reflections. Include an even functions so they will see what happens with a reflection about the y-axis. Ask the students why we dont talk about reflections about the x-axis and make a connection to even and odd functions. Formal concept: [ 3 minutes] Define the difference between the two types of reflections. For example, f(x) will cause the graph of f(x) to reflect about the y-axis. Example: [ 5 minutes] Combine reflections with stretches, compressions and shifts, state all transformations, and graph in terms of the common graph. Be sure to mention that the order the transformations are applied to the original graph can make a difference. 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