Functions in Tables 2.0

Transcription

1 Ns Activate Prior Knowledge Function Table Game Topic: Functions Functions in Tables 2.0 Date: Objectives: SWBAT (Identify patterns in Tables) Main Ideas: Assignment: What is a relation? What is a function? What is an input of a function? What is an output of a function? Are all relations functions? Are all functions relations? How can you tell if a relation is a function or not? Rules: Find a simple rule that agrees with each table (find as many as you can) Ways to describe (or relate) each: Input to Output Output to Output Examples: In Words take away 3 from each output to get the next output In Words the output is half of the input Algebraically y = 3x + 1 Combination output = input x 5 Output, Input, n A(n) Possible Answers: Each output is 2 more than the previous output. Output = input 2 y = 2x A(n) = 2n Function Notation Output, Input, n B(n) Output, Input, n C(n) Output, Input, n D(n)

2 NAGS Stand Up, Board Up, Pair Up Upper Level Output, Input, n E(n) Input Output Input Output Input Output Input Output Table: # of years # of rabbits Algebraic: Where r is the rabbit population and y is the number of years. Graph: Verbal: Two rabbits live in the new park. The rabbit population doubles each year.

3 Ns Topic: Creating Functions Creating Functions 2.1 Date: Objectives: SWBAT (Identify patterns in Tables and Create Function from that) Main Ideas: Closed/Explicit Form Assignment: Closed/Explicit Form Definition: In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (+,,, ), and functions (nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit. Example: Evaluate: f(x) = x 2 + 2x + 3 f(4) f( 5) f(0) The table below represents the function f(x) = ax + b. Fill out the outputs for the corresponding inputs and then fill out the column. Input (x) Output f(x) = ax + b 1 st Work Space 1 st Ways to tell if it is a linear function: The difference from one output to another output is constant.meaning it is the same throughout the whole table (per input) The highest degree (power) of the variable is 1 in the polynomial you created from the Closed-Form Linear Input, n Output C(n) st : C(0): 4-2 Closed Form:

4 In (x) Out f(x) = ax 2 + bx + c 1 st 2 nd 0 Extending the Differences Work Space What you should notice? The f(0) is the c. The 2 is twice as much as the leading coefficient (a). The of f(1) and f(0) is equal to a + b. The table below represents the function, f(x) = 3x 2 + 4x + 5. Find the outputs for the corresponding inputs. Find the and 2. Then relate the things you find with the a leading coefficient, b middle term coefficient, and c the constant. 2 nd Quadratic Input (x) Output f(x) = 3x 2 + 4x + 5 Steps: #1 - Find the f(0) value this is your c constant. 1 st 2 nd Work Space #2 - Find the and 2.the 2 is twice your a leading coefficient. 2 by 2 #3 - Use your between f(1) and f(0) this is your a + b, use this to find your b middle term coefficient #4 - Write your function in the form, f(x) = ax 2 + bx + c, with all the missing pieces

5 Creating Functions 2.1 Identify if it is linear or quadratic then write the closed-form that represents the table. We, Two, You In Out In Out In Out Group Activity Person #1 In Out Person #2 In Out Person #3 In Out Person #4 In Out Exponential Tables Create a table using the exponential function f(x) = 3 x Use the Domain {0, 1, 2, 3, 4, 5} In Out 1 st 2 nd ratio f(x + 1) f(x) f(x) means this output f(x + 1) means the next output

6 Closed Form of an Exponential Function f(x) = a b x Step #1: If there is no constant difference 1 st, 2 nd, and beyond, find the ratio of the all the consecutive inputs ( f(x+1) ) and see if there is a constant ratio. f(x) *this will be your base factor (b) - if it is constant Step #2: Find f(0): this will be your (a) Step #3: Put it all together in f(x) = a b x form. x f(x) x f(x) x -2-1 f(x) x -2-1 f(x) Your Turn x f(x) x f(x)

7 Ns Explicit vs. Recursive Forms 2.2 Topic: Explicit vs. Recursive Forms Date: Objectives: SWBAT (Represent Tables using Explicit and Recursive Forms) Main Ideas: Assignment: Explicit: A formula that allows direct computation of any term in a sequence or table Explicit Form vs. Recursive Form Recursive Definition Recursive: A formula that requires computation of the previous terms (outputs) in order to find the value of a n Little Differences Closed-Form/Explicit Form: -is an equation (in function notation) that describes the pattern (or relationship) from input to output -lets you find any output for any input by direct calculation Input to Output Domain: All real # s Range: All real # s (for linear functions) Linear: f(x) = x + f(0) Exponential: f(x) = f(0) ratio x Example: Find: Recursive: -is an equation that describes the pattern from output to output -to find any value using recursive definition you must have the previous output value to find the next output value Output to Output Domain: All real # s 0 or the first input in table Range: Depends on first output value and if it increases or decreases from there Linear: f(x) = { f(0) f(x 1) + Exponential: f(x) = { f(0) f(x 1) ratio f(n) = { 3 if n = 0 f(n 1) + 5 if n > 0 if x = 0 if x > 0 if x = 0 if x > 0 f(0) f(1) f(2) f(18)

8 Write the closed-form and recursive definitions of the table below. State the domain and range of each function for each definition. Input Output Closed-Form: Input Output Explicit: Recursive: Recursive: Examples 4 11 Input Output Closed-Form: Input Output Explicit: Recursive: Recursive: f(x) = { 5 f(x 1) + 2x if x = 0 if x > 0 Building Tables f(x) = { 1 f(x 1) 2 f(x) = { 3 f(x 1) + 2 if x = 0 if x > 0 if x = 0 if x > 0

9 Ns Introduction to Sequences 2.3 Topic: Introduction to Sequences Date: Objectives: SWBAT (Identify the difference in arithmetic and geometric sequences) Main Ideas: Assignment: Finite Sequence: -Contains a finite number of terms 1, 1, 2, 3, 5, 8 1, 2, 4, 5, 6, 7, 8} finite sequences 1, 1, 1, 1 Finite vs. Infinite Infinite Sequence: -Contains an infinite number of terms A general sequence is shown at the right. The first term in the sequence is a 1, the second term is a 2, the third term is a 3, and the nth term, also called the general term of the sequence, is a n. Frequently, a sequence has a definite pattern that can be expressed by a formula. 1, 3, 5, 7, 9 1, 1 2, 1 4, 1 8 1, 1, 2, 3, 5, 8 } infinite sequences a 1, a 2, a 3, a 4., a n, a n = 3n a 1, a 2, a 3,, a n Building Sequences Write the first three terms of the sequence given by the formula below: a n = 2n 1 Write the first three terms of the sequence given by the formula below: a n = n(n + 1) Upper Level Write the first three terms of the sequence given by the formula below: 1 a n = n(n + 2)

10 Evaluate a Series or Sequence An indicated sum of the terms of a sequence is called a series. To indicate wanting to find the sum of a sequence in compact for, you represent it in summation notation, or sigma notation. The sigma notation to the right indicates it is wanted to find the sum of the first four terms of the sequence in terms of n using the formula a n = 2n. 1,3,5,7 Sequence Series Greek letter SIGMA 4 2n n=1 Your Turn 3 (2i 1) i=1 6 1 n 2 n=3 Upper Level 5 nx n=1 5 x i i=1 Summary What is the difference between a finite sequence and an infinite sequence?

11 Ns Arithmetic/Geometric 2.4 Topic: Arithmetic/Geometric Sequences Date: Objectives: SWBAT (Identify the difference in Arithmetic and Geometric Sequences) Main Ideas: Assignment: Arithmetic Sequences: -is made up successive terms that have a common difference ( or d). EX: 10, 13, 16, 19 Arithmetic vs. Geometric Identify Building a Sequence Recursive Form of an Arithmetic Sequence: a n n th term a 1 first term in sequence d common difference n is any natural number Comes from this: a n = a 1 + d(n 1) y y 1 = m(x x 1 ) point slope form Geometric Sequences: -is a sequence of numbers such that each term is found by multiplying the previous term by a fixed, nonzero constant called the common ratio. Recursive Form of a Geometric Sequence: EX: 1, 3, 9, 27, 81, 243 a n = a 1 r (n 1) a n n th term a 1 first term in sequence r common ratio n is any natural number Identify whether the given sequence represents an arithmetic or geometric sequence. If they do, then identify the common difference or common ratio. 24, 35, 46, 18, 6, 2, 4, 4, 12, 3 Building the first four terms in a sequence with the given Recursive Form of the sequence. a n = (n 1) a n = 36 ( 1 n 1 3 ) a n = (n 1) 4

12 Write an equation for the n th term of each sequence, then find the a n for that sequence. Finding the n th term 8, 2, 1 2, (find a 23) 31, 17, 3, (find a 24 ) 1 3, 2 9, 4 27, 9, 2, 5, (find a 15 ) (find a 7) Partial Sum of an Arithmetic Sequence: S n = n ( a 1 + a n ) 2 18 (6k 1) k=1 21 (5m + 6) m=9 Partial Sums Partial Sum of a Geometric Sequence: 10 4(2) k 1 k=3 S n = a 1 a 1 r n, r 1 1 r m 1 m=4 Level The fifth term of a geometric sequence is 1 27 th of the eighth term. If the ninth term is 702, what is the eighth term?

13 Ns Direct vs. Inverse 2.5 Topic: Direct vs. Inverse Variations Date: Objectives: SWBAT (Identify the Difference between Direct and Inverse Functions) Main Ideas: Assignment: Wing-Span vs. Your Height Direct Variation: Equations in the form of y = kx or f(x) = kx or y = k, where k 0 are called direct variation x Graphs of direct variation (y = kx or f(x) = kx) always pass through the origin Direct Variation Graph Example: Types: The slope is positive when k > 0 The slope is negative when k < 0

14 Suppose y varies directly as x, and y = 9 when x = 3. Write a direct variation equation that relates x and y. Suppose y varies directly as x, and y = 15 when x = 5. Write a direct variation equation that relates x and y. Varies Directly Use the direct variation equation to find x when y = 15. Suppose y varies directly as x, and y = 15 when x = 5. Use the direct variation equation to find x when y = 45. Use the direct variation equation to find x when y = -24. Which of the following formulas represent a direct variation? a) A = s 2 b) E = MC 2 Inverse Variation Suppose you have $300 dollars and you must divide it evenly amongst participants in the school garbage pick-up day. c) D = rt d) A = P(1 + rt) An inverse variation can be represented by the equation y = k or xy = k. x Inverse Variation Graph Example:

15 Direct vs. Inverse 2.5 Assume that y varies inversely as x. If y = 5 when x = 3, write an inverse variation equation that relates x and y. Writing Equations Assume that y varies inversely as x. If y = 3 when x = 8, determine a correct inverse variation equation that relates x and y. If (x 1, y 1 ) and (x 2, y 2 ) are solutions of an inverse variation, then x 1 y 1 = k and x 2 y 2 = k. Product Rule If you set them equal to each other, you get: x 1 y 1 = x 2 y 2 This is called the product rule for inverse variation. x y Assume that y varies inversely as x. If y = 5 when x = 12, find x when y = 15. More Examples If y varies inversely as x and y = 6 when x = 40, find x when y = 30.

16 Operations w/functions 2.6 & 2.7 Topic: Operations with Polynomial Functions Date: Objectives: SWBAT (Perform Operations on Polynomial Functions) Main Ideas: Assignment: Operation Definition Example Let f(x) = 2x and g(x) = x + 5 Operations with Functions Addition (f + g)(x) = f(x) + g(x) 2x + ( x + 5) = x + 5 Subtraction (f g)(x) = f(x) g(x) 2x ( x + 5) = 3x 5 Multiplication (f g)(x) = f(x) g(x) 2x( x + 5) = 2x x Division ( f f(x) 2x ) (x) =, g(x) 0 g g(x) Examples: Given f(x) = 3x 2 + 7x and g(x) = 2x 2 x 1, find (f + g)(x). Given f(x) = 3x 2 + 7x and g(x) = 2x 2 x 1, find (f g)(x)., x 5 x + 5 Given f(x) = 2x 2 + 5x + 2 and g(x) = 3x 2 + 3x 4, find (f + g)(x). Your Turn

17 Given f(x) = 2x 2 + 5x + 2 and g(x) = 3x 2 + 3x 4, find (f g)(x). Given f(x) = 3x 2 2x + 1 and g(x) = x 4, find (f g)(x). Given f(x) = 3x 2 2x + 1 and g(x) = x 4, find ( f ) (x). Make sure to include excluded g values for x. Words: Suppose f and g are functions such that the range of g is a subset of the domain of f. Then the composition function [f g](x) = f[g(x)]. Composition of Functions Simple terms: it means that the outputs of g(x) at those certain inputs will become the inputs of the function f(x). Model: Domain of g Range of g Domain of f Range of f x g(x) f[g(x)] [f o g](x)

18 Operations w/functions 2.6 & 2.7 If f(x) = (2, 6), (9, 4), (7, 7), (0, 1) and g(x) = (7, 0), ( 1, 7), (4, 9), (8, 2), find [f g](x) and [g f](x). Examples Composing the Functions Find [f g](x) and [g f](x) for f(x) = 3x 2 x + 4 and g(x) = 2x 1. If f(x) = {(1, 2), (0, 3), (6, 5), (2, 1)} and g(x) = {(2, 0),( 3, 6), (1, 0), (6, 7)}, find [f g](x) and [g f](x). Your Turn Find [f g](x) and [g f](x) for f(x) = x 2 + 2x + 3 and g(x) = x + 5.

19 Ns Topic: Functions Regression Equations 2.8 Date: Objectives: SWBAT (Identify which equation fits a set of data the best) Main Ideas: Self- Discovery Assignment: You will go to the following website and go through the lessons of A Good Enough Fit, Blame it on the Rain, and Which Model Makes Sense with a partner -Make sure to download the Journals for each Lesson when you get to it filled out save to one of your drives then share with me at shawn.huffman@midlandisd.net Correlation is defined as a mutual relationship or connection between two or more things, or in our case a data set of ordered pairs (x, y). Where x is the independent variable or explanatory variable and y is the dependent variable or response variable. To find the strength of connection between two quantities is to calculate the correlation coefficient (r). And Use the equation as a Line of Best Fit used for predictions. The possible values for the correlation coefficient are as follows: 1 r 1. Linear Regression (y = ax + b) Strong negative linear correlation Steps for Calculator: Hours of Sleep GPA No linear relationship or week linear relationship Hours of Sleep GPA Strong positive linear correlation 1 0 1

20 Quadratic Regression (y = ax 2 + bx + c, a 0) Exponential Regression (y = a b x, a 0) To find the strength of connection between two quantities is to calculate the correlation coefficient (r 2 ). And Use the equation as a Parabola of Best Fit used for predictions. The possible values for the correlation coefficient are as follows: 0 < r 2 1. But you will have to do a little algebra to find the true value of r. Steps for Calculator: The data set shows the number of Americans living in multigenerational households. Number Year (in Millions) a value -the initial value -parameter -also the y-intercept or f(0) of the function f(x) = a b x b value -the growth/decay factor -the base -If b > 1, then this represents an exponential growth and if 0 < b < 1, then it represents an exponential decay. To find the strength of connection between two quantities is to calculate the correlation coefficient (r). And Use the equation as an Exponential of Best Fit used for predictions. The possible values for the correlation coefficient are as follows: 1 r 1.

21 Regression Equations 2.8 Steps for Calculator: The data set shows the percent of the world s population owning a smart phone. Year % of Population Continued Make a sketch of the scatter plot with the most appropriate model from the following criteria. Quadratic, a > 0, r 2 =. 99 Quadratic, a < 0, r 2 =. 78 Linear, a < 0, r =. 85 Depth of Knowledge Linear, a > 0, r =. 65 Exponential, b > 1, r =. 91 Exponential, 0 < b < 1, r =. 99