Chapter 2: Introduction to Functions
|
|
- Abel Tyler
- 6 years ago
- Views:
Transcription
1 Chapter 2: Introduction to Functions Lesson 1: Introduction to Functions Lesson 2: Function Notation Lesson 3: Composition of Functions Lesson 4: Domain and Range Lesson 5: Restricted Domain Lesson 6: One-to-One Functions Lesson 7: Inverse Functions Lesson 8: Ke Features of Functions This assignment is a teacher-modified version of Algebra 2 Common Core Copright (c) 2016 emath Instruction, LLC used b permission.
2 Chapter 2 Lesson 1: Introductions to Functions Introduction to Functions: Definition: A function is an rule that assigns eactl one output value (-value) for each input value (-value). These rules can be epressed in different was, the most common being equations, graphs, and tables of values. We call the input variable independent and output variable dependent. Relations and Functions: A relation is a relationship between sets of information. It is a set of ordered pairs. Recall: Domain: the set of all. ( variable) Range: the set of all. ( variable) A function is a specific tpe of relation. In order for a relation to be a function there must be onl and eactl one that corresponds to a given. Basicall, this means that elements of the domain never repeat. Is it a Function? One wa to test if a relation is a function is b using the vertical line test. This can be used when the relation is graphed. The vertical line test states that if a vertical line is drawn anwhere on the graph so that it hits the graph in more than one spot, then the graph is NOT a function. Eercise #1: Let s tr some (a) Is the graph of a parabola, f() = 2 a function?
3 (b) What about if we have a parabola on the side? Determine if each of the following graphs are functions.
4 Relationship B Relationship A Eercise #2: One of the following graphs shows a relationship where is a function of and one does not. (a) Draw the vertical line whose equation is = 3 on both graphs. (b) Give all output values for each graph at an input of 3. Relationship A: Relationship B: (c) Eplain which of these relationships is a function and wh.
5 Eercise #3: The graph of the function = is shown below. (a) State this function s -intercept. (b) Between what two consecutive integers does the larger -intercept lie? (c) Draw the horizontal line = -2 on this graph. (d) Using these two graphs, find all values of that solve the equation below: = -2
6 Eercise #4: Determine if the following equations represent a function. Justif our answer. (a) 2 + ( - 1) 2 = 4 (b) = (c) = 3 (d) = 3 + 3
7 INTRODUCTION TO FUNCTIONS CC ALGEBRA II HOMEWORK LESSON 1 FLUENCY 1. Determine for each of the following graphed relationships whether is a function of using the Vertical Line Test. (a) (b) (c) (d) (e) (f)
8 2. Given the relation {(8,2), (3,6), (7,5), (k,4)}, which value of k will result in the relation not being a function? (1) 1 (2) 2 (3) 3 (4) 4 3.) Which relation is not a function? (1) ( 2) = 4 (3) + = 4 (2) = 4 (4) = 4 4.) Which graph represents a function?
9 APPLICATIONS 5.) Evan is walking home from the museum. He starts 38 blocks from home and walks 2 blocks each minute. Evan s distance from home is a function of the number of minutes he has been walking. (a) Which variable is independent and which variable is dependent in this scenario? (b) Fill in the table below for a variet of time values. Time, t, in minutes Distance from home, D, in blocks (c) Determine an equation relating the distance, D, that Evan is from home as a function of the number of minutes, t, that he has been walking. (d) Determine the number of minutes, t, that it takes for Evan to reach home.
10 REASONING 6.) In one of the following tables, the variable is a function of the variable. Eplain which relationship is a function and wh the other is not Relationship # Relationship #2
11 Chapter 2 Lesson 2: Function Notation Function Notation: One method of defining a function is b naming the function, indicating the variable inside of the parentheses, and then defining a rule. It is important not to mistake function notation with multiplication. Recall that function rules commonl come in one of three forms: (1) equations, (2) graphs, and (3) tables. Eercise #1: Evaluate each of the following given the function definitions and input values. (a) f() = 5 2 (b) g() = f(3) = g(3) = f(-2) = g(0) = (c) h() = 2 h(3) = h(-2) =
12 Eercise #2: Find each of the following. Answers must be in simplest form. (a) Given: f() = 2-4, find: (1) f(-3) (2) f( 1) (b) Given: g() = , find: (1) g(6) (2) g(2) Eercise #3: (a) Given f() = 5-1, find the value of when f() = 29.
13 (b) Given f() = 2 + 4, find the value(s) of when f() = 20. Eercise #4: Boiling water at 212 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature, T, is a function of the number of hours, h. h (hours) T h F (a) Evaluate T(2) and T(6). (b) For what value of h is T(h) = 76? (c) Between what two consecutive hours will T(h) = 100?
14 Eercise #5: The function = f() is defined b the graph shown below. Answer the following questions based on this graph. (a) Evaluate f(-1), f(1) and f(5). (b) Evaluate f(0). What special feature on a graph does f(0) alwas correspond to? (c) What values of solve the equation f() = 0? What special features on a graph does the set of - values that solve f() = 0 correspond to? (d) Between what two consecutive integers does the largest solution to f() = 3 lie?
15 Eercise #6: For a function = g() it is known that g(-2) = 7. Which of the following points must lie on the graph of g()? (1) (7, -2) (3) (0, 7) (2) (-2, 7) (4) (-2, 0)
16 Chapter 2 Lesson 2: Function Notation Homework 1.) Given: f() = , g() = 2 + 1, and h() = 3 9 evaluate the following: (a) g(4) (b) h(-6) (c) f(-5) 2.) Given the following functions, simplif in terms of : f() = 2 3, g() = 2 + 4, h() = (a) h(2 3) (b) g(3 1) (c) f(9 2) 3.) If f() =, what is the value of f(-10)? (1) (2) (3) (4)
17 4.) Based on the graph of the function = g() shown below, answer the following questions. (a) Evaluate g(-2), g(0), g(3) and g(7). (b) What values of solve the equation g() = 0 (c) Graph the horizontal line = 2 on the grid above and label. (d) How man values of solve the equation g()=2?
18 APPLICATIONS 5.) Ian invested $2500 in an investment vehicle that is guaranteed to earn 4% interest compounded earl. The amount of mone, A, in his account as a function of the number of ears, t, since creating the account is given b the equation At t. (a) Evaluate A 0 and A 10. (b) What do the two values that ou found in part (a) represent?
19 Chapter 2 Lesson 3: Composite Functions Since functions convert the value of an input variable into the value of an output variable, it stands to reason that this output could then be used as an input to a second function. This process is known as composition of functions, in other words, combining the action or rules of two functions. There are two notations that are used to indicate composition of two functions. Two Was of Writing Composite Functions: (1) (2) ***Alwas work from the inside out or start with the function closest to the *** Eercise #1: Given f() = 2 5 and g() = 2 + 3, find values for each of the following. (a) f(g(1)) = (b) g(f(2)) = (c) g(g(0)) = (d) (f o g)(-2)
20 (e) (g o f)(3) = (f) (f o f)(-1) = Eercise #2: Given f() = and g() = 4-5, find values for each of the following. Answers must be in simplest form. (a) g(f( + 1)) (b) (f o g)(2) Eercise #3: If and, which epression is equivalent to? (1) (2) (3) (4)
21 Eercise #4: The graphs below are of the functions = f() and = g(). Evaluate each of the following questions based on these two graphs. f g (a) g(f(2)) = (b) f(g(-1)) = (c) g(g(1)) = (d) (g o f)(-2) = (e) (f o g)(0) = (f) (f o f)(0) = Eercise #5: If f() = 2 and g() = - 5 then f(g()) = (1) (3) 2-5 (2) 2-25 (4)
22 Chapter 2 Lesson 3: Composite Functions Homework FLUENCY 1. Given f() = 3 4 and g() = evaluate: (a) f(g(0)) (b) (g o f)(6) (c) f(f(3)) 2. Given h() = and g() = evaluate: (a) h(g(18)) (b) g(h(4)) (c) (g o g)(11) 3. The graphs of = h() and = k() are shown below. Evaluate the following based on these two graphs. h k (a) h(k(-2)) (b) (k o h)(0) (c) h(h(-2)) (d) (k o k)(-2)
23 4. If g() = 3 5and h() = 2 4 then (g o h)() =? (1) 6-17 (3) 5-9 (2) 6-14 (4) If f() = and g() = + 4 then f(g()) = (1) (3) (2) (4) APPLICATIONS 6. Scientists modeled the intensit of the sun, I, as a function of the number of hours since 6:00 a.m., h, using the function. The then model the temperature of the soil, T, as a function of the intensit using the function T(I) = of the soil at 2:00 p.m.?. Which of the following is closest to the temperature (1) 54 (3) 67 (2) 84 (4) 38
24 REASONING 7. Consider the functions f() = and g() =. Calculate the following. (a) g(f(15)) (b) g(f(-3)) (c) g(f()) (d) What appears to alwas be true when ou compose these two functions?
25 Chapter 2 Lesson 4: Domain & Range of Functions Because functions convert values of inputs into values of outputs, it is natural to talk about the sets that represent these inputs and outputs. The set of inputs that result in an output is called the domain (-values) of the function. The set of outputs is called the range (-values). This works for an relation. To find the domain we need to look from the left to the right. To determine the range we need to look down and up.
26 Eercise #1: Given the function {(0,4), (1,5), (2,6), (4,7), (6,4)}, use roster notation to give the domain and range. Roster Form: Eercise #2: State the range of the function f(n) = 2n + 1 if its domain is the set {1, 3, 5}. Show the domain and range in the mapping diagram below.
27 Eercise #3: The function = g() is completel defined b the graph shown below. Answer the following questions based on this graph. (a) Determine the minimum and maimum -values represented on this graph. (b) Determine the minimum and maimum -values represented on this graph. (c) State the domain and range of this function using set builder notation. Set Builder Notation:
28 Some functions, defined with graphs or equations, have domains and ranges that stretch out to infinit. Eercise #4: The function f() = is graphed on the grid below. Which of the following represents its domain and range written in interval notation? (1) Domain [-2, 4] (3) Domain (-, ) Range [-4, 6] Range [-4, ) (2) Domain [-2, 4] (4) Domain (-2, 4) Range (-4, ) Range (-4, 6) f Interval Notation: Eercise #5: Determine the domain and range of the graphs below.
29 Eercise #6: Determine the domain of the function. (1) -1 < < 6 (2) -1 < < 6 (3) -2 < < 5 (4) -2 < < 5 Eercise #7: Which graph illustrates a quadratic relation whose domain is all real numbers?
30 Chapter 2 Lesson 4: Domain & Range of Functions Homework 1.) A function is given b the set of ordered pairs {(2, 5), (4, 9), (6, 13), (8, 17)}. Write its domain and range in roster form. Domain: Range: 2.) The function h() = maps the domain given b the set {-2, -1, 0, 1, 2}. Which of the following sets represents the range of h()? (1) {0, 6, 10, 12} (3) {5, 6, 9} (2) {5, 6, 7} (4) {1, 4, 5, 6, 9} 3.) The graph below represents the function = f(). State the domain and range of this function.
31 4.) What are the domain and the range of the function shown in the graph below? (1) { > -4}; { > 2} (2) { > -4}; { > 2} (3) { > 2}; { > -4} (4) { > 2}; { > -4} 5.) What is the range of f() = 3 + 2? (1) { > 3} (3) { real numbers} (2) { > 2} (4) { real numbers} 6.) The function = f() is completel defined b the graph shown below. (a) Evaluate f(-4), f(3) and f(6). (b) State the domain and range of this function using interval notation. Domain: Range:
32 APPLICATIONS 7.) A child starts a pigg bank with $2. Each da, the child receives 25 cents at the end of the da and puts it in the bank. If A represents the amount of mone and d stands for the number of das then A(d) = d gives the amount of mone in the bank as a function of das (think about this formula). (a) Evaluate A(1), A(7) and A(30). (b) For what value of d will A(d) = $ (c) Eplain wh the domain does not contain (d) Eplain wh the range does the value d = 2.5. not include the value A = $3.10.
33 Chapter 2 Lesson 5: Restricted Domain Do Now: Eercise #1: Which of the following values of would not be in the domain of the function =? Eplain our answer. (1) = 0 (3) = -3 (2) = 5 (4) = -8 Domain of Radical Functions: Eercise #2: Which of the following values of would not be in the domain of the function =? Eplain our answer. (1) = 1 (3) = 5 (2) = 0 (4) = -5 Domain of Rational Functions:
34 Radical in the Denominator: 3.) Find the domain of the following function:
35 Practice: Find the domain of each of the following functions. 4.) ) f ( ) ) f () ) f () ) 1 f ( ) 9.)
36 Chapter 2 Lesson 5: Restricted Domain Homework FLUENCY 1.) What is the domain of the function? (1) { > 0} (3) { > -4} (2) { > 0} (4) { > -4} 2.) Which of the following values of would not be in the domain of the function defined b f() =? (1) = -3 (3) = 3 (2) = 2 (4) = -2 3.) Determine an values of that do not lie in the domain of the function f() =. Justif our response.
37 4.) Which of the following values of does lie in the domain of the function defined b g() =? (1) = 0 (3) = 3 (2) = 2 (4) = 5 5.) Which of the following would represent the domain of the function =? (1) {: > 3} (3) {: < 3} (2) {: < 3} (4) {: > 3 Find the domain of each of the following functions. 6.) f() = ) f() = ). h() = 5 + 5
38 9.) g() = ) h() = ) g() = ) f() = 2 13.) f() = ) f() =
39 Chapter 2 Lesson 6: One-to-One Functions Functions as rules can be divided into various categories based on shared characteristics. One categor is comprised of functions known as one-to-one. The following eercise will illustrate the difference between a function that is one-to-one and one that is not. Eercise #1: Consider the two simple functions given b the equations f() = 2 and g() = 2. (a) Map the domain {-2, 0, 2} using each function. Fill in the range and show the mapping arrows. Domain of f Range of f Domain of g Range of g (b) What is fundamentall different between these two functions in terms of how the elements of this domain get mapped to the elements of the range? Tip: In a one to one function, each element of the range is paired with element from the domain. Therefore, no repeat.
40 (1) Eercise #2: Of the four tables below, one represents a relationship where is a one-to-one function of. Determine which it is and eplain wh the others are not. (2) (3) (4) Eercise #3: Consider the following four graphs, which show a relationship between the variables and. (1) (2) (3) (4)
41 Eercise #4: Which of the following represents the graph of a one-to-one function? (1) (2) (3) (4)
42 Eercise#5: The distance that a number,, lies from the number 5 on a one-dimensional number line is given b the function D() = 5. Show b eample that D() is not a one-to-one function. Eercise #6: Which function is one-to-one? (1) k() = (2) g() = (3) f() = + 2 (4) j() = Eercise #7: Which function is not one-to-one? (1) {(0,1), (1,2), (2,3), (3,4)} (3) {(0,1), (1,0), (2,3), (3,2)} (2) {(0,0), (1,1), (2,2), (3,3)} (4) {(0,1), (1,0), (2,0), (3,2)}
43 Chapter 2 Lesson 6: One-to-One Functions Homework FLUENCY 1. Which of the following graphs illustrates a one-to-one relationship? (1) (2) (3) (4) 2. Which of the following graphs does not represent that of a one-to-one function? (1) (2) (3) (4)
44 3. In which of the following graphs is each input not paired with a unique output? (1) (2) (3) (4) 4. In which of the following formulas is the variable a one-to-one function of the variable? (Hint tr generating some values either in our head or using TABLES on our calculator.) (1) = 2 (3) = 2 (2) = (4) = 5 5. Which diagram represents a relation that is a one-to-one function? (1) (2) (3) (4)
45 APPLICATIONS 6. A recent newspaper gave temperature data for various das of the week in table format. In which of the tables below is the reported temperature a one-to-one function of the da of the week? (1) (2) (3) (4) Mon 75 Mon 75 Mon 58 Mon 56 Tue 68 Tue 72 Tue 52 Tue 58 Wed 65 Wed 68 Mon 81 Mon 85 Thu 74 Thu 72 Tue 76 Tue 85 REASONING 7.) Consider the function f() = round(), which rounds the input,, to the nearest integer. Is this function one-to-one? Eplain or justif our answer.
46 Chapter 2 Lesson 7: Inverse Functions The inverse of a function, is a relation in which the domain and range of the original function have been echanged. Simpl put, the and have switched places. Inverse functions have their own special notation. It is shown in the bo below. To Find Inverses Algebraicall: 1.) For Ordered Pairs: The inverse of a function is formed b interchanging the and coordinates of each point in the function. 2.) For Equations: (a) Epress f() in terms of and. (b) Switch and to form the inverse. (c) Put back into = form b solving for. (d) Write in inverse notation.
47 Eercise #1: If the point (-3, 5) lies on the graph of = f(), then which of the following points must lie on the graph of its inverse? (1) (3, -5) (3) (5, -3) (2) (-5, 3) (4) (-1/3, 1/5) Eercise #2: Find the equation for the inverse of each function below. (a) f() = (b) g() = ½ 2 8 (c) (d)
48 Eercise #3: f() is a linear function which is graphed below. Use its graph to answer the following questions. (a) Evaluate f -1 (2) and f -1 (-2). (b) Determine the -intercept of f -1 (). (c) On the same set of aes, draw a graph of.
49 Eercise #4: A table of values for the simple quadratic function f() = 2 is given below along with its graph f() a) Graph the inverse b switching the ordered pairs. f 1 ( ) (b) What do ou notice about the graph of this function s inverse?
50 Eercise #5: Given the relation A: {(3,2), (5,3), (6,2), (7,4)}. Which statement is true? (1) Both A and A -1 are functions (2) Neither A nor A -1 is a function. (3) Onl A is a function. (4) Onl A -1 is a function. Eercise #6: Which equation defines a function whose inverse is not a function? (1) = (2) = - (3) = (4) = 2
51 Chapter 2 Lesson 7: Inverse Functions Homework FLUENCY 1. If the point (-7, 5) lies on the graph of = f(), which of the following points must lie on the graph of its inverse? (1) (5, -7) (3) (7, -5) (2) (-1/7, 1/5) (4) (1/7, -1/5) 2. The function = f() has an inverse function = f -1 (). If f(a) = -b then which of the following must be true? (1) f -1 (-b) = -a (3) f -1 (-b) = a (2) f -1 (1/a) = -1/b (4) f -1 (b) = -a 3. The graph of the function = g() is shown below. The value of g -1 (2) is (1) 2.5 (3) 0.4 (2) -4 (4) -1
52 4. Which of the following functions would have an inverse that is also a function? (1) (2) (3) (4) 5. For a one-to-one function it is known that f(0) = 6 and f(8) = 0. Which of the following must be true about the graph of this function s inverse? (1) its -intercept = 6 (3) its -intercept = -6 (2) its -intercept = 8 (4) its -intercept = The function = h() is entirel defined b the graph shown below. (a) Sketch a graph of = h -1 (). Create a table of values if needed. (b) Write the domain and range of = h() and = h -1 () using interval notation. = h() = h -1 () Domain: Domain: Range: Range:
53 APPLICATIONS 7. The function = A(r) = πr 2 is a one-to-one function that uses a circle s radius as an input and gives the circle s area as its output. Selected values of this function are shown in the table below. r Ar (a) Determine the values of A -1 (9π) and A -1 (36π) from using the table. (b) Determine the values of A -1 (100π) and A -1 (225π). (c) The original function = A(r) converted an input, the circle s radius, to an output, the circle s area. What are the inputs and outputs of the inverse function? Input: Output:
54 REASONING 8. The domain and range of a one-to-one function, = f(), are given below in set-builder notation. Give the domain and range of this function s inverse also in set-builder notation. = f() Domain { -3 < < 5} = f -1 () Domain: Range { > -2} Range:
55 Chapter 2 Lesson 8: Ke Features of Functions The graphs of functions have man ke features whose terminolog we will be using all ear. It is important to master this terminolog, most of which ou learned in Common Core Algebra I. Eercise #1: The function = f() is shown graphed to the right. Answer the following questions based on this graph. (a) State the -intercept of the function. f (b) State the -intercepts of the function. What is the alternative name that we give the -intercepts? (c) Over the interval -1 < < 2 is f() increasing or decreasing? How can ou tell?
56 (d) Give the interval over which f() > 0. What is a quick wa of seeing this visuall? (e) State all the -coordinates of the relative maimums and relative minimums. Label each. Relative Minimum Relative Maimum: (f) What are the absolute maimum and minimum values of the function? Where do the occur? (g) State the domain and range of f() using interval notation. Absolute Minimum: Absolute Maimum: (h) If a second function g() is defined b the formula of g?, then what is the -intercept
57 Eercise#2: Consider the function g() = defined over the domain -4 < < 7. (a) Sketch a graph of the function to the right. (b) State the domain interval over which this function is decreasing. (c) State zeroes of the function on this interval. (d) State the interval over which g() < 0. (e) Evaluate g(0) b using the algebraic definition of the function. What point does this correspond to on the graph? (f) Are there an relative maimums or minimums on the graph? If so, which and what are their coordinates?
58 You need to be able to think about functions in all of their forms, including equations, graphs, and tables. Tables can be quick to use, but sometimes hard to understand. Eercise #3: A continuous function f() has a domain of -6 < < 1 with selected values shown below. The function has eactl two zeroes and has eactl two turning points, one at (3, -4) and one at (9, 3) f() (a) State the interval over which f() < 0. (b)state the interval over which f() is increasing
59 We can sketch the graph of functions based on certain characteristics. When sketching graphs, it is helpful to plot the -intercepts and an absolute etrema first. Eercise 4: Draw a graph of = f() that matches the following characteristics. Decreasing on: -10 < < -2 and 6 < < 10 Increasing on: -2 < < 6 Zeros at = -4, 0, and 8; f(-2) = -3 Absolute minimum of -3 and absolute maimum of 5
60 KEY FEATURES OF FUNCTIONS CC ALGEBRA II HOMEWORK LESSON 8 FLUENCY 1. The piecewise linear function f() is shown to the right. Answer the following questions based on its graph. (a) Evaluate each of the following based on the graph: (i) f(4) (ii) f(-3) (b) State the zeroes of f(). (c) Over which of the following intervals is f() alwas increasing? (1) -7 < < -3 (3) -5 < < 5 (2) -3 < < 5 (4) -5 < < 3
61 (d) State the coordinates of the relative maimum and the relative minimum of this function. Relative Maimum: (e) Over which of the following intervals is f() < 0? (1) -7 < < -3 (3) -5 < < 2 (2) 2 < < 7 (4) -5 < < 2 Relative Minimum: (f)a second function g() is defined using the rule g() = 2f() + 5. Evaluate g(0) using this rule. What does this correspond to on the graph of g? (g) A third function h() is defined b the formula h() = 3-3. What is the value of g(h(2))? Show how ou arrived at our answer.
62 2. For the function g() = 9 ( + 1) 2 do the following. (a) Sketch the graph of g on the aes provided. (b) State the zeroes of g. (c) Over what interval is g() decreasing? (d) Over what interval is g() > 0? (e) State the range of g.
63 3. Draw a graph of = f() that matches the following characteristics. Increasing on: -8 < < -4 and -1 < < 5 Decreasing on: -4 < < -1 f(-8) = -5 and zeroes at = -6, -2 and 3 Absolute maimum of 7 and absolute minimum of A continuous function has a domain of -7 < < 10 and has selected values shown in the table below. The function has eactl two zeroes and a relative maimum at (-4, 12) and a relative minimum at(5, -6) f() (a) State the interval on which f() is decreasing. (b) State the interval over which f() < 0.
Graphing Quadratics: Vertex and Intercept Form
Algebra : UNIT Graphing Quadratics: Verte and Intercept Form Date: Welcome to our second function famil...the QUADRATIC FUNCTION! f() = (the parent function) What is different between this function and
More informationName Date. In Exercises 1 6, graph the function. Compare the graph to the graph of ( )
Name Date 8. Practice A In Eercises 6, graph the function. Compare the graph to the graph of. g( ) =. h =.5 3. j = 3. g( ) = 3 5. k( ) = 6. n = 0.5 In Eercises 7 9, use a graphing calculator to graph the
More informationChapter 1 Notes, Calculus I with Precalculus 3e Larson/Edwards
Contents 1.1 Functions.............................................. 2 1.2 Analzing Graphs of Functions.................................. 5 1.3 Shifting and Reflecting Graphs..................................
More information20 Calculus and Structures
0 Calculus and Structures CHAPTER FUNCTIONS Calculus and Structures Copright LESSON FUNCTIONS. FUNCTIONS A function f is a relationship between an input and an output and a set of instructions as to how
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More informationUnit 4: Part 1 Graphing Quadratic Functions
Name: Block: Unit : Part 1 Graphing Quadratic Functions Da 1 Graphing in Verte Form & Intro to Quadratic Regression Da Graphing in Intercept Form Da 3 Da Da 5 Da Graphing in Standard Form Review Graphing
More informationProperties of Quadrilaterals
MIAP Chapter 6: Linear functions Master 6.1a Activate Prior Learning: Properties of Quadrilaterals A quadrilateral is a polgon with 4 sides. A trapezoid is a quadrilateral that has eactl one pair of parallel
More informationIntegrated Algebra A Notes/Homework Packet 9
Name Date Integrated Algebra A Notes/Homework Packet 9 Lesson Homework Graph Using the Calculator HW # Graph with Slope/Intercept Method HW # Graph with Slope/Intercept Method-Continued HW #3 Review Put
More informationChapter 5. Radicals. Lesson 1: More Exponent Practice. Lesson 2: Square Root Functions. Lesson 3: Solving Radical Equations
Chapter 5 Radicals Lesson 1: More Exponent Practice Lesson 2: Square Root Functions Lesson 3: Solving Radical Equations Lesson 4: Simplifying Radicals Lesson 5: Simplifying Cube Roots This assignment is
More informationF8-18 Finding the y-intercept from Ordered Pairs
F8-8 Finding the -intercept from Ordered Pairs Pages 5 Standards: 8.F.A., 8.F.B. Goals: Students will find the -intercept of a line from a set of ordered pairs. Prior Knowledge Required: Can add, subtract,
More informationLesson #17 Function Introduction
Lesson #17 Function Introduction A.A.37 A.A.40 A.A.41 Define a relation and function Write functions in functional notation Use functional notation to evaluate functions for given values in the domain
More informationEssential Question How many turning points can the graph of a polynomial function have?
.8 Analzing Graphs of Polnomial Functions Essential Question How man turning points can the graph of a polnomial function have? A turning point of the graph of a polnomial function is a point on the graph
More informationAlgebra 1B Assignments Chapter 6: Linear Equations (All graphs must be drawn on GRAPH PAPER!)
Name Score Algebra 1B Assignments Chapter 6: Linear Equations (All graphs must be drawn on GRAPH PAPER!) Review Review Worksheet: Rational Numbers and Distributive Propert Worksheet: Solving Equations
More information3.2 Polynomial Functions of Higher Degree
71_00.qp 1/7/06 1: PM Page 6 Section. Polnomial Functions of Higher Degree 6. Polnomial Functions of Higher Degree What ou should learn Graphs of Polnomial Functions You should be able to sketch accurate
More information3.1 Functions. The relation {(2, 7), (3, 8), (3, 9), (4, 10)} is not a function because, when x is 3, y can equal 8 or 9.
3. Functions Cubic packages with edge lengths of cm, 7 cm, and 8 cm have volumes of 3 or cm 3, 7 3 or 33 cm 3, and 8 3 or 5 cm 3. These values can be written as a relation, which is a set of ordered pairs,
More informationRelations and Functions
Relations and Functions. RELATION Mathematical Concepts Any pair of elements (, y) is called an ordered pair where is the first component (abscissa) and y is the second component (ordinate). Relations
More informationGraphing square root functions. What would be the base graph for the square root function? What is the table of values?
Unit 3 (Chapter 2) Radical Functions (Square Root Functions Sketch graphs of radical functions b appling translations, stretches and reflections to the graph of Analze transformations to identif the of
More informationQuadratic Inequalities
TEKS FCUS - Quadratic Inequalities VCABULARY TEKS ()(H) Solve quadratic inequalities. TEKS ()(E) Create and use representations to organize, record, and communicate mathematical ideas. Representation a
More informationFour Ways to Represent a Function: We can describe a specific function in the following four ways: * verbally (by a description in words);
MA19, Activit 23: What is a Function? (Section 3.1, pp. 214-22) Date: Toda s Goal: Assignments: Perhaps the most useful mathematical idea for modeling the real world is the concept of a function. We eplore
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More informationEXAMPLE A {(1, 2), (2, 4), (3, 6), (4, 8)}
Name class date Understanding Relations and Functions A relation shows how one set of things is related to, or corresponds to, another set. For instance, the equation A 5 s shows how the area of a square
More information5.2. Exploring Quotients of Polynomial Functions. EXPLORE the Math. Each row shows the graphs of two polynomial functions.
YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software Eploring Quotients of Polnomial Functions EXPLORE the Math Each row shows the graphs of two polnomial functions.
More informationTransformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations
Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting,
More information3.5 Rational Functions
0 Chapter Polnomial and Rational Functions Rational Functions For a rational function, find the domain and graph the function, identifing all of the asmptotes Solve applied problems involving rational
More informationLESSON 3.1 INTRODUCTION TO GRAPHING
LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered
More informationAttributes and Transformations of f(x) = e x VOCABULARY
- Attributes and Transformations of f() = e TEKS FOCUS TEKS ()(A) Determine the effects on the ke attributes on the graphs of f() = b and f() = log b () where b is,, and e when f() is replaced b af(),
More informationPutting the V in Absolute Value Defining Absolute Value Functions and Transformations
1 Putting the V in Absolute Value Defining Absolute Value Functions and Transformations Warm Up The graph of f() 5 is shown. Graph each transformation. 1. g() 5 f() 1 5 2. h() 5 2? f() 2 3 Learning Goals
More informationUnit 4 Part 1: Graphing Quadratic Functions. Day 1: Vertex Form Day 2: Intercept Form Day 3: Standard Form Day 4: Review Day 5: Quiz
Name: Block: Unit 4 Part 1: Graphing Quadratic Functions Da 1: Verte Form Da 2: Intercept Form Da 3: Standard Form Da 4: Review Da 5: Quiz 1 Quadratic Functions Da1: Introducing.. the QUADRATIC function
More informationTIPS4RM: MHF4U: Unit 1 Polynomial Functions
TIPSRM: MHFU: Unit Polnomial Functions 008 .5.: Polnomial Concept Attainment Activit Compare and contrast the eamples and non-eamples of polnomial functions below. Through reasoning, identif attributes
More informationRational Functions with Removable Discontinuities
Rational Functions with Removable Discontinuities 1. a) Simplif the rational epression and state an values of where the epression is b) Using the simplified epression in part (a), predict the shape for
More informationTransformations of y = x 2
Transformations of = Parent Parabola Lesson 11-1 Learning Targets: Describe translations of the parent function f() =. Given a translation of the function f() =, write the equation of the function. SUGGESTED
More informationEnd of Chapter Test. b. What are the roots of this equation? 8 1 x x 5 0
End of Chapter Test Name Date 1. A woodworker makes different sizes of wooden blocks in the shapes of cones. The narrowest block the worker makes has a radius r 8 centimeters and a height h centimeters.
More informationUnit 4 Test REVIEW: Polynomial Functions
Name Algebra II Date Period Unit 4 Test REVIEW: Polnomial Functions 1. Given a polnomial of the form: = a n + b n 1 + c n 2 + + d 2 + e + f a. What are the maimum number of zeros for this polnomial? b.
More information4.4 Absolute Value Equations. What is the absolute value of a number? Example 1 Simplify a) 6 b) 4 c) 7 3. Example 2 Solve x = 2
4.4 Absolute Value Equations What is the absolute value of a number? Eample Simplif a) 6 b) 4 c) 7 3 Eample Solve = Steps for solving an absolute value equation: ) Get the absolute value b itself on one
More informationRELATIONS AND FUNCTIONS
CHAPTER RELATINS AND FUNCTINS Long-distance truck drivers keep ver careful watch on the length of time and the number of miles that the drive each da.the know that this relationship is given b the formula
More informationDerivatives 3: The Derivative as a Function
Derivatives : The Derivative as a Function 77 Derivatives : The Derivative as a Function Model : Graph of a Function 9 8 7 6 5 g() - - - 5 6 7 8 9 0 5 6 7 8 9 0 5 - - -5-6 -7 Construct Your Understanding
More informationInvestigation Free Fall
Investigation Free Fall Name Period Date You will need: a motion sensor, a small pillow or other soft object What function models the height of an object falling due to the force of gravit? Use a motion
More informationFunction Notation. Essential Question How can you use function notation to represent a function?
. Function Notation Essential Question How can ou use function notation to represent a function? The notation f(), called function notation, is another name for. This notation is read as the value of f
More informationReady To Go On? Skills Intervention 4-1 Graphing Relationships
Read To Go On? Skills Intervention -1 Graphing Relationships Find these vocabular words in Lesson -1 and the Multilingual Glossar. Vocabular continuous graph discrete graph Relating Graphs to Situations
More informationCHECK Your Understanding
CHECK Your Understanding. State the domain and range of each relation. Then determine whether the relation is a function, and justif our answer.. a) e) 5(, ), (, 9), (, 7), (, 5), (, ) 5 5 f) 55. State
More informationPROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS
Topic 21: Problem solving with eponential functions 323 PROBLEM SOLVING WITH EXPONENTIAL FUNCTIONS Lesson 21.1 Finding function rules from graphs 21.1 OPENER 1. Plot the points from the table onto the
More informationFunctions Project Core Precalculus Extra Credit Project
Name: Period: Date Due: 10/10/1 (for A das) and 10/11/1(for B das) Date Turned In: Functions Project Core Precalculus Etra Credit Project Instructions and Definitions: This project ma be used during the
More informationSTRAND G: Relations, Functions and Graphs
UNIT G Using Graphs to Solve Equations: Tet STRAND G: Relations, Functions and Graphs G Using Graphs to Solve Equations Tet Contents * * Section G. Solution of Simultaneous Equations b Graphs G. Graphs
More informationUsing a Table of Values to Sketch the Graph of a Polynomial Function
A point where the graph changes from decreasing to increasing is called a local minimum point. The -value of this point is less than those of neighbouring points. An inspection of the graphs of polnomial
More informationAppendix A.6 Functions
A. Functions 539 RELATIONS: DOMAIN AND RANGE Appendi A. Functions A relation is a set of ordered pairs. A relation can be a simple set of just a few ordered pairs, such as {(0, ), (1, 3), (, )}, or it
More information1-1. Functions. Lesson 1-1. What You ll Learn. Active Vocabulary. Scan Lesson 1-1. Write two things that you already know about functions.
1-1 Functions What You ll Learn Scan Lesson 1- Write two things that ou alread know about functions. Lesson 1-1 Active Vocabular New Vocabular Write the definition net to each term. domain dependent variable
More informationDeveloped in Consultation with Tennessee Educators
Developed in Consultation with Tennessee Educators Table of Contents Letter to the Student........................................ Test-Taking Checklist........................................ Tennessee
More informationLesson 8.1 Exercises, pages
Lesson 8.1 Eercises, pages 1 9 A. Complete each table of values. a) -3 - -1 1 3 3 11 8 5-1 - -7 3 11 8 5 1 7 To complete the table for 3, take the absolute value of each value of 3. b) - -3 - -1 1 3 3
More informationFunctions: The domain and range
Mathematics Learning Centre Functions: The domain and range Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Functions In these notes
More informationLesson 2.1 Exercises, pages 90 96
Lesson.1 Eercises, pages 9 96 A. a) Complete the table of values. 1 1 1 1 1. 1 b) For each function in part a, sketch its graph then state its domain and range. For : the domain is ; and the range is.
More informationAlgebra II Notes Radical Functions Unit Applying Radical Functions. Math Background
Appling Radical Functions Math Background Previousl, ou Found inverses of linear and quadratic functions Worked with composition of functions and used them to verif inverses Graphed linear and quadratic
More information3.4 Graphing Functions
Name Class Date 3. Graphing Functions Essential Question: How do ou graph functions? Eplore Graphing Functions Using a Given Domain Resource Locker Recall that the domain of a function is the set of input
More information2.8 Distance and Midpoint Formulas; Circles
Section.8 Distance and Midpoint Formulas; Circles 9 Eercises 89 90 are based on the following cartoon. B.C. b permission of Johnn Hart and Creators Sndicate, Inc. 89. Assuming that there is no such thing
More information3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately.
Math 65 Weekl Activit 1 (50 points) Name: Simplif the following epressions. Make sure to use the = smbol appropriatel. Due (1) (a) - 4 (b) ( - ) 4 () 8 + 5 6 () 1 5 5 Evaluate the epressions when = - and
More informationCK-12 PreCalculus Concepts 1
Chapter Functions and Graphs Answer Ke. Functions Families. - - - - - - - -. - - - - - - - - CK- PreCalculus Concepts Chapter Functions and Graphs Answer Ke. - - - - - - - -. - - - - - - - - 5. - - - -
More information1.1 Horizontal & Vertical Translations
Unit II Transformations of Functions. Horizontal & Vertical Translations Goal: Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related
More informationACTIVITY: Graphing a Linear Equation. 2 x x + 1?
. Graphing Linear Equations How can ou draw its graph? How can ou recognize a linear equation? ACTIVITY: Graphing a Linear Equation Work with a partner. a. Use the equation = + to complete the table. (Choose
More informationMath 1050 Lab Activity: Graphing Transformations
Math 00 Lab Activit: Graphing Transformations Name: We'll focus on quadratic functions to eplore graphing transformations. A quadratic function is a second degree polnomial function. There are two common
More informationSection 4.2 Graphing Lines
Section. Graphing Lines Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif collinear points. The order of operations (1.) Graph the line
More informationFair Game Review. Chapter 2. and y = 5. Evaluate the expression when x = xy 2. 4x. Evaluate the expression when a = 9 and b = 4.
Name Date Chapter Fair Game Review Evaluate the epression when = and =.... 0 +. 8( ) Evaluate the epression when a = 9 and b =.. ab. a ( b + ) 7. b b 7 8. 7b + ( ab ) 9. You go to the movies with five
More informationUnit 2: Function Transformation Chapter 1
Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph.
More information5.2 Graphing Polynomial Functions
Locker LESSON 5. Graphing Polnomial Functions Common Core Math Standards The student is epected to: F.IF.7c Graph polnomial functions, identifing zeros when suitable factorizations are available, and showing
More information5.2 Graphing Polynomial Functions
Name Class Date 5.2 Graphing Polnomial Functions Essential Question: How do ou sketch the graph of a polnomial function in intercept form? Eplore 1 Investigating the End Behavior of the Graphs of Simple
More information1.5 LIMITS. The Limit of a Function
60040_005.qd /5/05 :0 PM Page 49 SECTION.5 Limits 49.5 LIMITS Find its of functions graphicall and numericall. Use the properties of its to evaluate its of functions. Use different analtic techniques to
More information2.2 Absolute Value Functions
. Absolute Value Functions 7. Absolute Value Functions There are a few was to describe what is meant b the absolute value of a real number. You ma have been taught that is the distance from the real number
More information= = The number system. Module. Glossary Math Tools... 33
- > + > < - %. < + a = - = = b in. F - - Module The number sstem Lesson Rational and Irrational Numbers........ 8.NS. Lesson ompare and Order Numbers......... 8 8.NS., 8.NS. Lesson Estimate the Value of
More informationOnline Homework Hints and Help Extra Practice
Evaluate: Homework and Practice Use a graphing calculator to graph the polnomial function. Then use the graph to determine the function s domain, range, and end behavior. (Use interval notation for the
More informationReady To Go On? Skills Intervention 3-1 Using Graphs and Tables to Solve Linear Systems
Read To Go On? Skills Intervention 3-1 Using Graphs and Tables to Solve Linear Sstems Find these vocabular words in Lesson 3-1 and the Multilingual Glossar. Vocabular sstem of equations linear sstem consistent
More informationConnecticut Common Core Algebra 1 Curriculum. Professional Development Materials. Unit 4 Linear Functions
Connecticut Common Core Algebra Curriculum Professional Development Materials Unit 4 Linear Functions Contents Activit 4.. What Makes a Function Linear? Activit 4.3. What is Slope? Activit 4.3. Horizontal
More informationPerform Function Operations and Composition
TEKS 6.3 a.3, 2A.1.A Perform Function Operations and Composition Before You performed operations with algebraic epressions. Now You will perform operations with functions. Why? So you can model biological
More informationConnecting the Dots Making Connections Between Arithmetic Sequences and Linear Functions
Connecting the Dots Making Connections Between Arithmetic Sequences and Linear Functions Warm Up Use what ou know about arithmetic sequences to complete each task.. Write the first 5 terms of the sequence
More informationLaurie s Notes. Overview of Section 6.3
Overview of Section.3 Introduction In this lesson, eponential equations are defined. Students distinguish between linear and eponential equations, helping to focus on the definition of each. A linear function
More informationACTIVITY: Forming the Entire Coordinate Plane
.5 The Coordinate Plane How can ou graph and locate points that contain negative numbers in a coordinate plane? You have alread graphed points and polgons in one part of the coordinate plane. In Activit,
More informationPre-Algebra Notes Unit 8: Graphs and Functions
Pre-Algebra Notes Unit 8: Graphs and Functions The Coordinate Plane A coordinate plane is formed b the intersection of a horizontal number line called the -ais and a vertical number line called the -ais.
More informationSection 2.2: Absolute Value Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a
Section.: Absolute Value Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
More information1.1. Parent Functions and Transformations Essential Question What are the characteristics of some of the basic parent functions?
1.1 Parent Functions and Transformations Essential Question What are the characteristics of some of the basic parent functions? Identifing Basic Parent Functions JUSTIFYING CONCLUSIONS To be proficient
More informationVocabulary. Term Page Definition Clarifying Example. dependent variable. domain. function. independent variable. parent function.
CHAPTER 1 Vocabular The table contains important vocabular terms from Chapter 1. As ou work through the chapter, fill in the page number, definition, and a clarifing eample. dependent variable Term Page
More informationWhy? Identify Functions A function is a relationship between input and output. In a 1 function, there is exactly one output for each input.
Functions Stopping Distance of a Passenger Car Then You solved equations with elements from a replacement set. (Lesson -5) Now Determine whether a relation is a function. Find function values. Wh? The
More informationGraph Linear Equations
Lesson 4. Objectives Graph linear equations. Identif the slope and -intercept of linear equations. Graphing Linear Equations Suppose a baker s cookie recipe calls for a miture of nuts, raisins, and dried
More information6. f(x) = x f(x) = x f(x) = x f(x) = 3 x. 10. f(x) = x + 3
Section 9.1 The Square Root Function 879 9.1 Eercises In Eercises 1-, complete each of the following tasks. i. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. ii. Complete
More informationPartial Fraction Decomposition
Section 7. Partial Fractions 53 Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the eamples that follow. Note
More informationChapter 3. Exponential and Logarithmic Functions. Selected Applications
Chapter Eponential and Logarithmic Functions. Eponential Functions and Their Graphs. Logarithmic Functions and Their Graphs. Properties of Logarithms. Solving Eponential and Logarithmic Equations.5 Eponential
More information3.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions
76 CHAPTER Graphs and Functions Find the equation of each line. Write the equation in the form = a, = b, or = m + b. For Eercises through 7, write the equation in the form f = m + b.. Through (, 6) and
More informationRotate. A bicycle wheel can rotate clockwise or counterclockwise. ACTIVITY: Three Basic Ways to Move Things
. Rotations object in a plane? What are the three basic was to move an Rotate A biccle wheel can rotate clockwise or counterclockwise. 0 0 0 9 9 9 8 8 8 7 6 7 6 7 6 ACTIVITY: Three Basic Was to Move Things
More information9.1 Exercises. Section 9.1 The Square Root Function 879. In Exercises 1-10, complete each of the following tasks.
Section 9. The Square Root Function 879 9. Eercises In Eercises -, complete each of the following tasks. i. Set up a coordinate sstem on a sheet of graph paper. Label and scale each ais. ii. Complete the
More informationGRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS
GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS 1.1 DIFFERENT TYPES AND SHAPES OF GRAPHS: A graph can be drawn to represent are equation connecting two variables. There are different tpes of equations which
More informationSection 9.3: Functions and their Graphs
Section 9.: Functions and their Graphs Graphs provide a wa of displaing, interpreting, and analzing data in a visual format. In man problems, we will consider two variables. Therefore, we will need to
More information9. f(x) = x f(x) = x g(x) = 2x g(x) = 5 2x. 13. h(x) = 1 3x. 14. h(x) = 2x f(x) = x x. 16.
Section 4.2 Absolute Value 367 4.2 Eercises For each of the functions in Eercises 1-8, as in Eamples 7 and 8 in the narrative, mark the critical value on a number line, then mark the sign of the epression
More information12.4 The Ellipse. Standard Form of an Ellipse Centered at (0, 0) (0, b) (0, -b) center
. The Ellipse The net one of our conic sections we would like to discuss is the ellipse. We will start b looking at the ellipse centered at the origin and then move it awa from the origin. Standard Form
More informationGraphing Radical Functions
17 LESSON Graphing Radical Functions Basic Graphs of Radical Functions UNDERSTAND The parent radical function, 5, is shown. 5 0 0 1 1 9 0 10 The function takes the principal, or positive, square root of.
More informationA Picture Is Worth a Thousand Words
Lesson 1.1 Skills Practice 1 Name Date A Picture Is Worth a Thousand Words Understanding Quantities and Their Relationships Vocabular Write a definition for each term in our own words. 1. independent quantit
More informationTransforming Polynomial Functions
5-9 Transforming Polnomial Functions Content Standards F.BF.3 Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative) find
More informationPolynomial Functions I
Name Student ID Number Group Name Group Members Polnomial Functions I 1. Sketch mm() =, nn() = 3, ss() =, and tt() = 5 on the set of aes below. Label each function on the graph. 15 5 3 1 1 3 5 15 Defn:
More informationIt s Not Complex Just Its Solutions Are Complex!
It s Not Comple Just Its Solutions Are Comple! Solving Quadratics with Comple Solutions 15.5 Learning Goals In this lesson, ou will: Calculate comple roots of quadratic equations and comple zeros of quadratic
More informationSection 1.6: Graphs of Functions, from College Algebra: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative
Section.6: Graphs of Functions, from College Algebra: Corrected Edition b Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license.
More informationTHE INVERSE GRAPH. Finding the equation of the inverse. What is a function? LESSON
LESSON THE INVERSE GRAPH The reflection of a graph in the line = will be the graph of its inverse. f() f () The line = is drawn as the dotted line. Imagine folding the page along the dotted line, the two
More information23.1 Perpendicular Bisectors of Triangles
Name lass Date 3.1 Perpendicular isectors of Triangles Essential Question: How can ou use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle? Resource Locker
More informationUNIT P1: PURE MATHEMATICS 1 QUADRATICS
QUADRATICS Candidates should able to: carr out the process of completing the square for a quadratic polnomial, and use this form, e.g. to locate the vertex of the graph of or to sketch the graph; find
More informationLINEAR PROGRAMMING. Straight line graphs LESSON
LINEAR PROGRAMMING Traditionall we appl our knowledge of Linear Programming to help us solve real world problems (which is referred to as modelling). Linear Programming is often linked to the field of
More informationProblem 1: The relationship of height, in cm. and basketball players, names is a relation:
Chapter - Functions and Graphs Chapter.1 - Functions, Relations and Ordered Pairs Relations A relation is a set of ordered pairs. Domain of a relation is the set consisting of all the first elements of
More information