Pre-AP Algebra 2 Lesson 1-1 Basics of Functions

Transcription

1 Lesson 1-1 Basics of Functions Objectives: The students will be able to represent functions verball, numericall, smbolicall, and graphicall. The students will be able to determine if a relation is a function or not, with and without the vertical line test. The students will be able to evaluate functions written in function notation. Materials: Little Black Book; Do Now worksheet; pairwork; hw #1-1 Time 15 min Class Introductions Pass out sllabus, discuss classroom rules, etc Activit Introduce the Little Black Book 15 min Do Now Students work on Functions as Lenses handout. Show a completed picture of the projected image on the overhead. 30 min Eplore Have students create a Fraer Model of function. Use this opportunit to go over group roles (reporter, recorder, materials minder, timekeeper). Put a list of the different was relations/functions can be represented on the board. Each group must choose at least representation to use for their eample/non-eample. Set 1 of those for each group so that all are accounted for. o Use the eamples to talk about domain, range, what makes the eamples functions versus the noneamples o Guiding Questions: What is the difference in a relation and a function? What else do ou know about functions? Does this graph represent a function? How can ou tell? What would make a graph not be a function? Concepts to make sure are discussed as ou are going over the students fraer models A relation is a mapping (or pairing) of input values with output values. The input values are called the domain, represented b the independent variable. The output values are called the range, represented b the dependent variable A relation can be represented in several different was: o Verbal description; Mapping; Set of ordered pairs; Graph; Equation; Table A function is a special tpe of relation where each input is paired with eactl one output. o ever time ou put the same thing in, ou get the same thing out. Direct Instruction Functions are ver important in algebra and we will be using them a lot for the rest of the course. We can use function notation to help represent and work with functions. Can ou eplain what f() means? What kind of functions are = ½ 5 and = - + 1? Functions can be given names and represented with a letter. Let s call the first function f and the second one g. The can be written as f() = ½ 5 and g() = This is read f of of f at not f times. The f() replaces the. It alwas gives ou the value of the output. f() means the output of function f when the input is. Find f(1), f(-), g(0), g(3). Function Machine: Paint a visual picture using a function machine, which converts one number, the input, into another number, the output, b a rule in such a manner that each input has onl one output. Define the input as the independent variable, and the output as the dependent variable, and the rule as the equation or relationship which acts upon the input to produce one output. input: = 5 rule: = + 3 output: = 8 0 min Pairwork - students will tr problems to show understanding of functions and representational fluenc check often for misconceptions Homework #1-1: Basics of functions homework.

2 Lesson 1-1 DO NOW Functions as Lenses One good wa to think about a function in math is to picture it as a lens. Lenses are used to bend and shift light waves, usuall to make images bigger or smaller, or to move them to a new location (think: glasses, magnifing glass, microscope, telescope, etc.). Think of the image on the left as the original, or input, image, and the image on the right as the projected, or output, image. The original image lies on an -ais, and the projected image lies on a -ais. On the left (original image), draw a stick figure with the feet at = 0, the knees at = 1, the waist at =, the chest at = 3 and the top of the head at = 4. Assume that each tick mark equals 1 space. Suppose that the lens takes input, and then squares ever value before output (i.e. = ). Draw a picture of the projected image. In order to do this, draw arrows from each of the initial five -values to where the are projected on the. Use these arrows as a guide for drawing the projection. Lens (Function) = Original (Input) Projection (Output) What do ou notice about the projection? Wh did this happen?

3 Lesson 1-1 Pairwork Part1: Functions Is each relation a function? If not, eplain wh it isn t ½ ¼ ½ ½ ¼ Apple Ton Banana Melvin Carrot Aaron Durian Gar Endive Ton Melvin Aaron Gar Apple Banana Carrot Durian Endive

4 Lesson 1-1 Pairwork Part : Function Notation Practice 1 For problems 1-18, let f( ), g( ) 5, and h( ) 8 1) f(5) = ) f(10) = 3) f(a) = 4) f( 5) = 5) g(3) = 6) g(-1) = 7) g(b) = 8) g( 6) = 9) h(-10) = 10) h(10) = 11) h( ) = 1) h( + 8) = 13) f(5) + g(3) = 14) g(-1) h(-10) = 15) f (4) g( 8) = h(3) 16) h(6) = 17) f(-) = 18) g(-) =.

5 Lesson 1-1 Pairwork Part 3: Representational Fluenc (It means: being able to easil move between different representations of the same concept.) One wa to represent a function is with an in-out table. Here are tables that represent two functions: f: g: ½ Note: The concept is function The representation is - table Find each of the following. If there is not enough information, write CND (cannot be determined). 1) f(-5) = ) g() = 3) f(3) + f(4) = 4) g(0) f(0) = 5) [g(-3)] f(-5) = 6) g(8) = Another wa to represent a function is with a graph. Here are graphs that represent two functions: Note: The concept is function The representation is graph f g Find each of the following: 1) f(0) = ) g(0) = 3) g(-) = 4) f(-4) = 5) g(8) = 6) g(11) = 7) f(-1) + g(-8) = 8) f(-0) g(1.73) = 9) f (6) g(7) =

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7 Lesson 1-1 Homework Check for Understanding Can ou complete these problems correctl b ourself Is each relation a function? If not, eplain wh it isn t For problems 5 8, let ( ) ( ) ( ) 5. ( ) ( ) 6. ( ) ( ) 7. ( ) ( ) 8. ( ) Spiral What do ou remember from Algebra 1? (these are skills we will need in Algebra ) Solve the following equations for (Show all work on a separate page) 1. ( ) ( 4) ( ) 4. 5( ) 3 7 3

8 Lesson 1-1 Homework 8. Graph each equation on the same grid a. b. c. 9. Write an equation for each line. a. The line with slope of and -intercept (0, -4) b. The line that passes through the points (-, 6) and (, 14) Solve the sstem of equations: 8

9 Lesson 1-1 Pairwork Ke Part1: Functions 1. Yes. Yes 3. No. The input value 5 has more than one output 4. Yes. Although 4 is repeated in the chart. It has the same output. 5. Yes 6. No. The input, Gar, is associated with more than one output 7. Yes 8. No. The graph fails the vertical line test 9. No. The graph fails the vertical line test 10. Yes Part : Function Notation Practice 1) f(5) = ) f(10) = 3) f(a) = 4) f( 5) = ( ) 5) g(3) = 6) g(-1) = 7) g(b) = 8) g( 6) = 9) h(-10) = 10) h(10) = 11) h( ) = 1) h( + 8) = 13) f(5) + g(3) = 14) g(-1) h(-10) = 15) f (4) g( 8) = 16) h(3) h(6) = 17) f(-) = ( ) 18) g(-) = Part 3: Representational Fluenc: Tables 1) - ) CND 3) 3½ 4) 1 5) 6) 0 Graphs 1) 1 ) -3 3) 0 4)1 5) 5 6) 7 7) 6 8) -3 9) 1/4