1
|
|
- Matthew Paul
- 5 years ago
- Views:
Transcription
1 Zeros&asymptotes Example 1 In an early version of this activity I began with a sequence of simple examples (parabolas and cubics) working gradually up to the main idea. But now I think the best strategy is to start with an example that is rich enough to display the big idea. I would work this with the class with a diagram of the graph on the screen at the front and get them to explain to me why the factored form of the polynomial really does provide the qualitative form of the graph. I arm them with the page from the workbook that allows them to make notes on the story at each root. Following that rather chaotic class discussion, I go through the rather elegant procedure of moving from right to left keeping track of the sign of y as we pass through each root. Consider the polynomial function f(x) = x 7 15x x 5 265x x x The graph f(x) is given at the right. I used technology to draw it. Without technology, that would not have been so easy to do, even armed with calculus. It turns out that f(x) can be factored as f(x) = x(x 1) 2 (x 2)(x 3)(x 4) 2 and thus the zeros of f (where f(x) = 0) are the values x = 0, 1, 2, 3 and 4. The point of this example is that this factored form gives us a lot of information about the graph that we didn t have with the expanded formula. To be precise, the factored form allows us to easily see where y is positive (graph lies above the x-axis), where y is negative (graph lies below the x-axis), and where y is zero (graph intersects the x- axis). Solution The factored form gives f(x) as a product of factors, and thus the sign of y is the product of the signs of these factors. And the sign of each factor is easy to find for any given x. Indeed, a linear term (x a) is positive for x > a and negative for x < a. Thus the only time it changes sign is at x = a. Here s what I think is the neatest way to make the argument. Start with x greater than 4. At this point, all terms are positive so y is positive. Now let x decrease and take account of what happens as we pass through each root. And of course we only have to note what happens to the factor belonging to that root. Well from the form of the equation we see that that as x passes through each root, the sign of y: does not change at x=4 changes at x=3 changes at x=2 does not changes at x=1 changes at x=0. At each root we can record the sign of y in the next interval. This is enough to give us the sign of y in each interval. Given that, we can make a pretty good start at drawing the graph f(x). peter.taylor@queensu.ca 1
2 Example 2 At the right we have the graph of the rational function: x 2 x + 1 Note from both the equation and the diagram that the equation has only one root: x = 2. Note that the sign of y is also clear from the diagram: for x greater than 2, y is positive for x between 1 and 2, y is negative for x less than 1, y is positive (a) Argue that this is also clear from the equation. This time y is a quotient of two terms (rather than a product) but its sign is still determined by the two signs. We use the same argument as before. For x greater than 4 all terms are positive so y is positive. Now let x decrease. The sign of y: changes at x=2 making y negative changes at x= 1 making y positive This is enough to give us the sign of y in each interval. But we have a new kind of behavior at x = 1. For x very close to 1, the denominator x + 1 is very close to zero, whereas the numerator gets close to 3. That causes y to blow up in magnitude and using what we know about the sign of y, we see that it will take a large negative value just above 1 and a large positive value just below 1. That s what causes the vertical asymptote at x = 1. (b) Here s a good question. What happens to y as x gets large, both positive and negative? From the graph, it looks like y increases on the right side and decreases on the left side, but how far? Does it approach a limit? Is it asymptotic to the x-axis? It s hard to get this information from the graph but the equation is not so transparent either as x gets large, both numerator and denominator get large making it hard to see what happens to the quotient. The students can of course use their calculators to get a good idea of the answer, but how would an algebraic argument go? There s a nice way to rewrite the quotient that will deliver the answer divide top and bottom by x: x 2 x + 1 = 1 2/x 1 + 1/x Now what happens to this as x gets large in absolute value? The numerator and the denominator both approach 1. In terms of the graph, the curve approaches a horizontal line of height 1. That s called a horizontal asymptote of the equation. peter.taylor@queensu.ca 2
3 Example 3 (a) Make a reasonable sketch of the graph of the equation x(x 2) (x + 1)(x 1) 2 Pay attention to the asymptotic behavior, occasions when either x or y become large in magnitude. Solution. Note that the equation has roots at x = 0 and x = 2. The equation allows us to find the sign of y at any point. There are four terms in the y-expression and even though only two of these are in the numerator with the remaining two in the denominator, the sign of y is still the product of the signs of the four terms. The linear terms change sign only at their roots while the quadratic term (x 1) 2 never changes sign and is zero only at x = 1. Thus we argue as follows. For x greater than 2 all four terms are positive so y is positive. Now as x decreases, the sign of y: changes at x=2 does not change at x=1 changes at x=0 changes at x= 1 This is enough to give us the sign of y in each interval. The graph is given at the right. Note that the two terms in the denominator have roots at x = 1 and x = 1. This indicates that there will be vertical asymptotes at these two x-values. However at x = 1, there is no change of sign, and the graph goes down the asymptote on both sides. Note that we are not asking for the more detailed sketch we would expect from a calculus student. We are looking for the students to get the sign right and the behavior of the graph at the roots. Using the vertical scale they can try to get the height of the graph about right at different points. (b) What happens to y as x gets large, both positive and negative? Is there a horizontal asymptote? A good way to answer this question is to rewrite the quotient by dividing top and bottom by. One way to do this gives us: x x (x 2 x ) (x + 1) ( x 1 x ) 2 = (1 2/x) (x + 1)(1 1/x) 2 ~ 1 (x + 1)(1) 2 As x gets large (positive or negative) the expression (x+1) in the denominator gets large sending the expression close to zero. In terms of the graph, the x-axis is a horizontal asymptote. peter.taylor@queensu.ca 3
4 Example 4. Here we will work with the parameterized family of graphs x(x a) (x + 1)(x 1) (0 a 2) (a) The endpoint cases a = 0 and a = 2 are sketched below. But before I give these to the students, I ask them to go to the board and come up with rough graphs for these two cases. a = 0: (x + 1)(x 1) a = 2: x(x 2) (x + 1)(x 1) Here s the interesting question: if we allow a to increase gradually from 0 to 2, we will see the shape of the graph change continuously, starting at a = 0 as the graph on the left and winding up at a = 2 as the graph on the right. What does this journey look like? The puzzling behavior here concerns the asymptote at x = 1. At a = 0 it goes down the asymptote on the left side and up the asymptote on the right side. But at a = 2 this behavior is reversed: up on the left side and down on the right side. How can that happen gradually? There must be a point where it happens suddenly but what can the graph look like at such a point? And how can there be sudden changes when the parameter changes continuously? Your job is to understand completely what this continuous transformation looks like. Certainly you could pull Desmos out of the cloud and input the general equation with a slider for a and see exactly how it happens. But before you do that, see if you can figure it out. Draw rough graphs of some key configurations along the way. Provide some discussion to help the reader understand exactly how a curve plunging down to infinitely can suddenly change to plunging up. Or does it? (b) A good place to start the investigation is to check out the halfway point at a = 1. Indeed the mathematical form of the equation suggests that there ought to be a significant shift at that point (do you see why?). Okay, when a = 1: x(x 1) (x + 1)(x 1) = x (x + 1) That s a pretty simple equation and since the (x 1) term in the denominator has been canceled out, the asymptote at x = 1 has disappeared. I ask the students to draw the graph. a = 1: x (x + 1) peter.taylor@queensu.ca 4
5 (c) Okay. For a < 1, the asymptote at x = 1 goes up on the left and goes down on the right. And for a > 1, it s the reverse the asymptote at x = 1 goes down on the left and goes up on the right. And at the transition, a = 1, the asymptote disappears. The question we are interested in is exactly how does this happen? What does the transition look like? I ask the students to draw what they think the graph might look like just before and after a = 1, for example, at a = 0.99 and a = Well here they are: a = 0.99: x(x 0.99) (x + 1)(x 1) a = 1.01: x(x 1.01) (x + 1)(x 1) It s fun using Desmos to see the journey in real time. The two branches seem to lean towards one another in what only can be described as an expectant kiss. At the moment of the kiss, the asymptote disappears! and right away returns as the curves go off in the opposite directions. peter.taylor@queensu.ca 5
6 Example 5. Here we work with the parameterized family of graphs (x 2)2 (x c) 1 c 1 (a) The endpoint cases c = 1 and c = 1 are sketched below. I will shortly give the students these two graphs but before that I ask them to produce rough sketches a good exercise for NPVS (nonpermanent vertical surfaces!). In the right-hand graph the behaviour on the interval [0, 3] has been magnified to better display the roots x = 1 and 2. (x 2)2 (x + 1) (x 2)2 (x 1) Here s the question: if we allow c to increase gradually from 1 to 1, we will see the shape of the graph change continuously. What does this journey look like? Again, the puzzling behavior concerns the asymptote at x = 0, the y-axis. For c = 1 it goes up the asymptote on both sides and for c = 1 it goes down the asymptote on both sides. How can that transition happen gradually? It would seem that there must be a point where there s a sudden shift but what can the graph look like at such a point? Again, before they are permitted to enjoy the luxuries of Desmos, they have to try to figure it out. (b) A good place to start the investigation is to check out the journey s halfway point at c = 0. Indeed the mathematical form of the equation suggests that there ought to be a significant shift at that point, as will get an x-cancelation. Okay, when c = 0: (x 2)2 (x 0) (x 2)2 = x I ask the students to draw the graph. They discover, using their sign analysis, that the vertical asymptote has to go both up and down down on the left side of the y-axis and up on the right side. peter.taylor@queensu.ca 6
7 Okay. Let s consolidate. We are considering the family: (x 2)2 (x c) 1 c 1 The story of the sign of y is quite simple it changes once and only once at x = c: for x > c for x < c y is positive y is negative That tells us, for every value of c, whether the asymptote goes up or down the y-axis. When c is positive, it has to go down the axis and when c is negative, it has to go up the axis. And finally, as we have just seen, when c is zero, it goes up on the right side and down on the left side. (c) Now the question is, what does the graph look like for c very close to zero. How does it prepare for the big up-down shift? I put the students in small groups at the (white or black) board and ask them to draw the graph for c = ±0.1. They don t of course have calculus, but they can still do a pretty good job. Take, for example, c = 0.1. They know that the graph looks like a parabola opening up at x = 2. Then, as x decreases (moving left) it crosses the x-axis from above at x = c = 0.1. So it must have a maximum somewhere between 0.1 and 2. And then as it gets closer to x = 0, it moves rapidly down the y-axis. That gives them a reasonable picture of the graph (which is plotted at the right). This is also a good problem for a grade 12 calculus class as they can show that as c approaches zero, that peak gets higher and higher only to fail at some point and come crashing down. It won t succeed at getting all the way up until c hits zero. The story is much the same on the other side of the y- axis. The curve tries harder and harder to descend but only succeeds when c = 0. Here s an animation from c = 1 to c = 1: and here s one from c = 0.1 to c = 0.1: c > 0 y>0 y>0 c c y<0 y<0 c < 0 graph crosses the x-axis at x = c peter.taylor@queensu.ca 7
8 Exercises. 1. At the right we have the graph of the polynomial (x + 1)(x 3) Note that both the equation and the diagram give us the roots of the equation: x = 1 and x = 3 Note that it is clear from the diagram that: for x greater than 3, y is positive for x between 1 and 3, y is negative for x less than 1, y is positive Show that this is also clear from the form of the equation. Solution. The variable y is a product of two terms and its sign is determined by the signs of those two terms. In fact there s a powerful way to make the argument. Start with x greater than 3 and see what happens to the sign of each factor as x gradually decreases: for x greater than 3, both terms are positive so that y is positive as x decreases the signs both stay positive until x passes through 3 this causes the term (x 3) to change sign but (x+1) stays positive thus y changes sign and becomes negative as x continues to decrease nothing changes until x passes through 1 this causes the term (x+1) to change sign but (x 3) stays negative since only one term changes sign, y changes sign and becomes positive and y stays that way for the rest of the journey. peter.taylor@queensu.ca 8
9 2. At the right we have the graph of the polynomial (x + 1)(x 3) 2 Note that both the equation and the diagram give us the roots of the equation: x = 1 and x = 3 Note that it is clear from the diagram that: for x greater than 3, y is positive for x between 1 and 3, y is positive for x less than 1, y is negative (a) Show that this is also clear from the form of the equation. Solution. Start with x greater than 3 and see what happens to the sign of each factor as x gradually decreases: for x greater than 3, both terms are positive so that y is positive as x decreases the signs both stay positive until the second term becomes zero at x = 3 As x passes through 3 the term (x 3) 2 does not change sign Thus y does not changes sign and stays positive as x continues to decrease nothing changes until x passes through 1 this causes the term (x+1) to change sign and thus y changes sign to negative and stays that way for the rest of the journey. (b) Here s another observation about this example. The piece of the graph around x = 3 looks like a parabola and indeed that s expected from the fact that the (x 3) terms appears as a square. But what parabola is it close to? Well when x is close to 3, the other term (x+1) is close to 4 so that the expression for y will be close to 4(x 3) 2 And at the right I have drawn the graph of this parabola along with the cubic curve. The match is good near x = 3. peter.taylor@queensu.ca 9
10 3. Show algebraically that the slant asymptote for the graph is the line x 3. (x 2)2 (x + 1) Of course to do this exercise the students need to know what it means for a curve to be asymptotic to a line and I ask the class to use their intuition to formulate a definition. Here s what they hopefully (after perhaps some coaching) arrive at: Definition. The graph f(x) is asymptotic to the line mx + b if the distance between the graph and the line approaches zero as x becomes large. That is: f(x) (mx + b) approaches zero as x gets large (large positive or large negative or often both). Solution. We need to show that the expression (x 2) 2 (x + 1) (x 3) approaches zero for large x. This is a good algebraic calculation. We first get a common denominator. = (x 2)2 (x + 1) x = (x2 4x + 4)(x + 1) x = (x ) x = 4 This clearly approaches zero as x get large in magnitude. peter.taylor@queensu.ca 10
11 4. For any given value c find the slant asymptote of the equation (x 2)2 (x c) Solution. There are a couple of ways to go about this. Perhaps the obvious one is to take mx + b to be the unknown asymptote and find the values of m and b that satisfy the asymptote condition. I do that one first. This condition requires that (x 2) 2 (x c) (mx + b) approach zero for large x. Take a common denominator: = (x2 4x + 4)(x c) mx 3 b Expand: = (x3 (4 + c) + (4c + 4)x 4c) mx 3 b Now collect the like terms in x. = (1 m)x3 (4 + c + b) + (4c + 4)x 4c = (1 m)x (4 + c + b) + 4(c + 1) 4c x Now how do we choose the parameters m and b so that this approaches zero for large x? We clearly need to zero out those first two terms. Thus we need m = 1 b = (4 + c) A more direct way is to simplify the equation of the graph: (x 2)2 (x c) = (x2 4x + 4)(x c) = (x3 (4 + c) + (4c + 4)x 4c) = x (4 + c) + 4(c + 1) 4c x Now what do we have to subtract from this to make what s left approach 0 for large x? The answer is clearly x (4 + c). Thus the asymptote is x (4 + c). peter.taylor@queensu.ca 11
12 5. Make a reasonable sketch of the graph of the equation x(x 2) 2 (x + 1)(x 1) 2 Pay attention to the asymptotic behavior, occasions when either x or y become large in magnitude. peter.taylor@queensu.ca 12
3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationGraphing Techniques. Domain (, ) Range (, ) Squaring Function f(x) = x 2 Domain (, ) Range [, ) f( x) = x 2
Graphing Techniques In this chapter, we will take our knowledge of graphs of basic functions and expand our ability to graph polynomial and rational functions using common sense, zeros, y-intercepts, stretching
More informationy= sin( x) y= cos( x)
. The graphs of sin(x) and cos(x). Now I am going to define the two basic trig functions: sin(x) and cos(x). Study the diagram at the right. The circle has radius. The arm OP starts at the positive horizontal
More informationSection Rational Functions and Inequalities. A rational function is a quotient of two polynomials. That is, is a rational function if
Section 6.1 --- Rational Functions and Inequalities A rational function is a quotient of two polynomials. That is, is a rational function if =, where and are polynomials and is not the zero polynomial.
More informationCCNY Math Review Chapter 2: Functions
CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and
More informationSection 3.7 Notes. Rational Functions. is a rational function. The graph of every rational function is smooth (no sharp corners)
Section.7 Notes Rational Functions Introduction Definition A rational function is fraction of two polynomials. For example, f(x) = x x + x 5 Properties of Rational Graphs is a rational function. The graph
More informationNot for reproduction
x=a GRAPHING CALCULATORS AND COMPUTERS (a, d ) y=d (b, d ) (a, c ) y=c (b, c) (a) _, by _, 4 x=b FIGURE 1 The viewing rectangle a, b by c, d _4 4 In this section we assume that you have access to a graphing
More informationAlgebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor
Algebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor In this section, we will look at the hyperbola. A hyperbola is a set of points P in a plane such that the absolute value of the difference between
More information1) Complete problems 1-65 on pages You are encouraged to use the space provided.
Dear Accelerated Pre-Calculus Student (017-018), I am excited to have you enrolled in our class for next year! We will learn a lot of material and do so in a fairly short amount of time. This class will
More information, etc. Let s work with the last one. We can graph a few points determined by this equation.
1. Lines By a line, we simply mean a straight curve. We will always think of lines relative to the cartesian plane. Consider the equation 2x 3y 4 = 0. We can rewrite it in many different ways : 2x 3y =
More informationRational functions, like rational numbers, will involve a fraction. We will discuss rational functions in the form:
Name: Date: Period: Chapter 2: Polynomial and Rational Functions Topic 6: Rational Functions & Their Graphs Rational functions, like rational numbers, will involve a fraction. We will discuss rational
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS In this section, we assume that you have access to a graphing calculator or a computer with graphing software. FUNCTIONS AND MODELS 1.4 Graphing Calculators
More informationx 2 + 3, r 4(x) = x2 1
Math 121 (Lesieutre); 4.2: Rational functions; September 1, 2017 1. What is a rational function? It s a function of the form p(x), where p(x) and q(x) are both polynomials. In other words, q(x) something
More informationCollege Algebra. Fifth Edition. James Stewart Lothar Redlin Saleem Watson
College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson 4 Polynomial and Rational Functions 4.6 Rational Functions Rational Functions A rational function is a function of the form Px (
More informationMath 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
More informationPlanting the Seeds Exploring Cubic Functions
295 Planting the Seeds Exploring Cubic Functions 4.1 LEARNING GOALS In this lesson, you will: Represent cubic functions using words, tables, equations, and graphs. Interpret the key characteristics of
More informationAlgebra 2 Semester 1 (#2221)
Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester
More informationWhy Use Graphs? Test Grade. Time Sleeping (Hrs) Time Sleeping (Hrs) Test Grade
Analyzing Graphs Why Use Graphs? It has once been said that a picture is worth a thousand words. This is very true in science. In science we deal with numbers, some times a great many numbers. These numbers,
More informationf( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving
The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already
More informationPreCalculus Chapter 9 Practice Test Name:
This ellipse has foci 0,, and therefore has a vertical major axis. The standard form for an ellipse with a vertical major axis is: 1 Note: graphs of conic sections for problems 1 to 1 were made with the
More informationExploring Graphs of Power Functions Using the TI-Nspire
Exploring Graphs of Power Functions Using the TI-Nspire I. Exploration Write Up: Title: Investigating Graphs of Parabolas and Power Functions Statement of Mathematical Exploration: In this exploration,
More informationObjectives. Materials
Activity 13 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the
More information1.1 Pearson Modeling and Equation Solving
Date:. Pearson Modeling and Equation Solving Syllabus Objective:. The student will solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical
More informationSupplemental 1.5. Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient
Supplemental 1.5 Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient Interval Notation Many times in this class we will only want to talk about what
More informationToday is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class Back board says your name if you are on my roster. I need parent financial
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More informationDerivatives and Graphs of Functions
Derivatives and Graphs of Functions September 8, 2014 2.2 Second Derivatives, Concavity, and Graphs In the previous section, we discussed how our derivatives can be used to obtain useful information about
More informationCollege Algebra. Gregg Waterman Oregon Institute of Technology
College Algebra Gregg Waterman Oregon Institute of Technology c 2016 Gregg Waterman This work is licensed under the Creative Commons Attribution.0 International license. The essence of the license is that
More informationThe Bisection Method versus Newton s Method in Maple (Classic Version for Windows)
The Bisection Method versus (Classic Version for Windows) Author: Barbara Forrest Contact: baforres@uwaterloo.ca Copyrighted/NOT FOR RESALE version 1.1 Contents 1 Objectives for this Lab i 2 Approximate
More informationAH Properties of Functions.notebook April 19, 2018
Functions Rational functions are of the form where p(x) and q(x) are polynomials. If you can sketch a function without lifting the pencil off the paper, it is continuous. E.g. y = x 2 If there is a break
More informationFamily of Functions Lesson
Family of Functions Lesson Introduction: Show pictures of family members to illustrate that even though family members are different (in most cases) they have very similar characteristics (DNA). Today
More informationLearning Packet. Lesson 6 Exponents and Rational Functions THIS BOX FOR INSTRUCTOR GRADING USE ONLY
Learning Packet Student Name Due Date Class Time/Day Submission Date THIS BOX FOR INSTRUCTOR GRADING USE ONLY Mini-Lesson is complete and information presented is as found on media links (0 5 pts) Comments:
More informationMath Calculus f. Business and Mgmt - Worksheet 9. Solutions for Worksheet 9 - Piecewise Defined Functions and Continuity
Math 220 - Calculus f. Business and Mgmt - Worksheet 9 Solutions for Worksheet 9 - Piecewise Defined Functions and Continuity Evaluating and Graphing Functions Exercise 1: Compose these pairs of functions
More informationA function: A mathematical relationship between two variables (x and y), where every input value (usually x) has one output value (usually y)
SESSION 9: FUNCTIONS KEY CONCEPTS: Definitions & Terminology Graphs of Functions - Straight line - Parabola - Hyperbola - Exponential Sketching graphs Finding Equations Combinations of graphs TERMINOLOGY
More information6.4 Vertex Form of a Quadratic Function
6.4 Vertex Form of a Quadratic Function Recall from 6.1 and 6.2: Standard Form The standard form of a quadratic is: f(x) = ax 2 + bx + c or y = ax 2 + bx + c where a, b, and c are real numbers and a 0.
More information. As x gets really large, the last terms drops off and f(x) ½x
Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be
More informationSample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications
Name Period Date QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications QUAD 4.1 Vertex Form of a Quadratic Function 1 Explore how changing the values of h and
More informationMath 101 Exam 1 Review
Math 101 Exam 1 Review Reminder: Exam 1 will be on Friday, October 14, 011 at 8am. It will cover sections 1.1, 1. and 10.1 10.3 Room Assignments: Room Sections Nesbitt 111 9, 14, 3, 4, 8 Nesbitt 15 0,
More informationPre-Calculus Notes: Chapter 3 The Nature of Graphs
Section Families of Graphs Name: Pre-Calculus Notes: Chapter 3 The Nature of Graphs Family of graphs Parent graph A group of graphs that share similar properties The most basic graph that s transformed
More informationMastery. PRECALCULUS Student Learning Targets
PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions,
More informationMath 1330 Section : Rational Functions Definition: A rational function is a function that can be written in the form f ( x ), where
2.3: Rational Functions P( x ) Definition: A rational function is a function that can be written in the form f ( x ), where Q( x ) and Q are polynomials, consists of all real numbers x such that You will
More informationThe x-intercept can be found by setting y = 0 and solving for x: 16 3, 0
y=-3/4x+4 and y=2 x I need to graph the functions so I can clearly describe the graphs Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points. What is the
More informationExemplar for Internal Achievement Standard. Mathematics and Statistics Level 1
Exemplar for Internal Achievement Standard Mathematics and Statistics Level 1 This exemplar supports assessment against: Achievement Standard (2.2) Apply graphical methods in solving problems An annotated
More informationCore Mathematics 1 Graphs of Functions
Regent College Maths Department Core Mathematics 1 Graphs of Functions Graphs of Functions September 2011 C1 Note Graphs of functions; sketching curves defined by simple equations. Here are some curves
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.1 Quadratic Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic
More informationVoluntary State Curriculum Algebra II
Algebra II Goal 1: Integration into Broader Knowledge The student will develop, analyze, communicate, and apply models to real-world situations using the language of mathematics and appropriate technology.
More informationLesson 10 Rational Functions and Equations
Lesson 10 Rational Functions and Equations Lesson 10 Rational Functions and Equations In this lesson, you will embark on a study of rational functions. Rational functions look different because they are
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my Algebra course that I teach here at Lamar University, although I have to admit that it s been years since I last taught this course. At this point in my career I
More information1.) ( ) Step 1: Factor the numerator and the denominator. Find the domain. is in lowest terms.
GP3-HW11 College Algebra Sketch the graph of each rational function. 1.) Step 1: Factor the numerator and the denominator. Find the domain. { } Step 2: Rewrite in lowest terms. The rational function is
More informationVertical and Horizontal Translations
SECTION 4.3 Vertical and Horizontal Translations Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the vertical translation of a sine or cosine function. Find the horizontal
More informationFinal Exam Review Algebra Semester 1
Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)
More informationSECTION 1.3: BASIC GRAPHS and SYMMETRY
(Section.3: Basic Graphs and Symmetry).3. SECTION.3: BASIC GRAPHS and SYMMETRY LEARNING OBJECTIVES Know how to graph basic functions. Organize categories of basic graphs and recognize common properties,
More information5.5 Newton s Approximation Method
498CHAPTER 5. USING DERIVATIVES TO ANALYZE FUNCTIONS; FURTHER APPLICATIONS 4 3 y = x 4 3 f(x) = x cosx y = cosx 3 3 x = cosx x cosx = 0 Figure 5.: Figure showing the existence of a solution of x = cos
More informationMath 1: Solutions to Written Homework 1 Due Friday, October 3, 2008
Instructions: You are encouraged to work out solutions to these problems in groups! Discuss the problems with your classmates, the tutors and/or the instructors. After working doing so, please write up
More informationSection 1.1 Graphs Graphs
Section 1.1 Graphs 55 1.1 Graphs Much of algebra is concerned with solving equations. Many algebraic techniques have been developed to provide insights into various sorts of equations, and those techniques
More informationChapter 12: Quadratic and Cubic Graphs
Chapter 12: Quadratic and Cubic Graphs Section 12.1 Quadratic Graphs x 2 + 2 a 2 + 2a - 6 r r 2 x 2 5x + 8 2y 2 + 9y + 2 All the above equations contain a squared number. They are therefore called quadratic
More information1.7 Limit of a Function
1.7 Limit of a Function We will discuss the following in this section: 1. Limit Notation 2. Finding a it numerically 3. Right and Left Hand Limits 4. Infinite Limits Consider the following graph Notation:
More informationMath Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:
1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain
More information1 MATH 253 LECTURE NOTES for FRIDAY SEPT. 23,1988: edited March 26, 2013.
1 MATH 253 LECTURE NOTES for FRIDAY SEPT. 23,1988: edited March 26, 2013. TANGENTS Suppose that Apple Computers notices that every time they raise (or lower) the price of a $5,000 Mac II by $100, the number
More informationCollege Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013
College Pre Calculus A Name Period Weekly Review Sheet # 1 Assigned: Monday, 9/9/013 Due: Friday, 9/13/013 YOU MUST SHOW ALL WORK FOR EVERY QUESTION IN THE BOX BELOW AND THEN RECORD YOUR ANSWERS ON THE
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.7 Graphs of Rational Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze and
More informationSection 1.5 Transformation of Functions
6 Chapter 1 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations in order to explain or
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationGraphing with a Graphing Calculator
APPENDIX C Graphing with a Graphing Calculator A graphing calculator is a powerful tool for graphing equations and functions. In this appendix we give general guidelines to follow and common pitfalls to
More informationMATH 1113 Exam 1 Review. Fall 2017
MATH 1113 Exam 1 Review Fall 2017 Topics Covered Section 1.1: Rectangular Coordinate System Section 1.2: Circles Section 1.3: Functions and Relations Section 1.4: Linear Equations in Two Variables and
More informationTHE RECIPROCAL FUNCTION FAMILY AND RATIONAL FUNCTIONS AND THEIR GRAPHS L E S S O N 9-2 A N D L E S S O N 9-3
THE RECIPROCAL FUNCTION FAMILY AND RATIONAL FUNCTIONS AND THEIR GRAPHS L E S S O N 9-2 A N D L E S S O N 9-3 ASSIGNMENT 2/12/15 Section 9-2 (p506) 2, 6, 16, 22, 24, 28, 30, 32 section 9-3 (p513) 1 18 Functions
More informationChapter 9 Review. By Charlie and Amy
Chapter 9 Review By Charlie and Amy 9.1- Inverse and Joint Variation- Explanation There are 3 basic types of variation: direct, indirect, and joint. Direct: y = kx Inverse: y = (k/x) Joint: y=kxz k is
More informationMore Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a
More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing
More information2/22/ Transformations but first 1.3 Recap. Section Objectives: Students will know how to analyze graphs of functions.
1 2 3 4 1.4 Transformations but first 1.3 Recap Section Objectives: Students will know how to analyze graphs of functions. 5 Recap of Important information 1.2 Functions and their Graphs Vertical line
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More information2-3 Graphing Rational Functions
2-3 Graphing Rational Functions Factor What are the end behaviors of the Graph? Sketch a graph How to identify the intercepts, asymptotes and end behavior of a rational function. How to sketch the graph
More informationLimits. f(x) and lim. g(x) g(x)
Limits Limit Laws Suppose c is constant, n is a positive integer, and f() and g() both eist. Then,. [f() + g()] = f() + g() 2. [f() g()] = f() g() [ ] 3. [c f()] = c f() [ ] [ ] 4. [f() g()] = f() g()
More informationNEW CONCEPTS LEARNED IN THIS LESSON INCLUDE: Fundamental Theorem of Algebra
2.5. Graphs of polynomial functions. In the following lesson you will learn to sketch graphs by understanding what controls their behavior. More precise graphs will be developed in the next two lessons
More informationSection 4.4: Parabolas
Objective: Graph parabolas using the vertex, x-intercepts, and y-intercept. Just as the graph of a linear equation y mx b can be drawn, the graph of a quadratic equation y ax bx c can be drawn. The graph
More informationSection 2.2 Graphs of Linear Functions
Section. Graphs of Linear Functions Section. Graphs of Linear Functions When we are working with a new function, it is useful to know as much as we can about the function: its graph, where the function
More informationGraphs and transformations, Mixed Exercise 4
Graphs and transformations, Mixed Exercise 4 a y = x (x ) 0 = x (x ) So x = 0 or x = The curve crosses the x-axis at (, 0) and touches it at (0, 0). y = x x = x( x) As a = is negative, the graph has a
More information3.7 Rational Functions. Copyright Cengage Learning. All rights reserved.
3.7 Rational Functions Copyright Cengage Learning. All rights reserved. Objectives Rational Functions and Asymptotes Transformations of y = 1/x Asymptotes of Rational Functions Graphing Rational Functions
More informationRational Functions Video Lecture. Sections 4.4 and 4.5
Rational Functions Video Lecture Sections 4.4 and 4.5 Course Learning Objectives: 1)Demonstrate an understanding of functional attributes such as domain and range. Determine these attributes for a function
More informationMAC Learning Objectives. Transformation of Graphs. Module 5 Transformation of Graphs. - A Library of Functions - Transformation of Graphs
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
More informationMAC Module 5 Transformation of Graphs. Rev.S08
MAC 1105 Module 5 Transformation of Graphs Learning Objectives Upon completing this module, you should be able to: 1. Recognize the characteristics common to families of functions. 2. Evaluate and graph
More informationDefinitions Associated w/ Angles Notation Visualization Angle Two rays with a common endpoint ABC
Preface to Chapter 5 The following are some definitions that I think will help in the acquisition of the material in the first few chapters that we will be studying. I will not go over these in class and
More information1 Shapes of Power Functions
MA 1165 - Lecture 06 1 Wednesday, 1/28/09 1 Shapes of Power Functions I would like you to be familiar with the shape of the power functions, that is, the functions of the form f(x) = x n, (1) for n = 1,
More informationBuilding Concepts: Moving from Proportional Relationships to Linear Equations
Lesson Overview In this TI-Nspire lesson, students use previous experience with proportional relationships of the form y = kx to consider relationships of the form y = mx and eventually y = mx + b. Proportional
More informationCourse of study- Algebra Introduction: Algebra 1-2 is a course offered in the Mathematics Department. The course will be primarily taken by
Course of study- Algebra 1-2 1. Introduction: Algebra 1-2 is a course offered in the Mathematics Department. The course will be primarily taken by students in Grades 9 and 10, but since all students must
More informationP.5 Rational Expressions
P.5 Rational Expressions I Domain Domain: Rational expressions : Finding domain a. polynomials: b. Radicals: keep it real! i. sqrt(x-2) x>=2 [2, inf) ii. cubert(x-2) all reals since cube rootscan be positive
More informationEquations and Functions, Variables and Expressions
Equations and Functions, Variables and Expressions Equations and functions are ubiquitous components of mathematical language. Success in mathematics beyond basic arithmetic depends on having a solid working
More information2.1 Basics of Functions and Their Graphs
.1 Basics of Functions and Their Graphs Section.1 Notes Page 1 Domain: (input) all the x-values that make the equation defined Defined: There is no division by zero or square roots of negative numbers
More informationSpecific Objectives Students will understand that that the family of equation corresponds with the shape of the graph. Students will be able to create a graph of an equation by plotting points. In lesson
More informationGraphing Rational Functions
Graphing Rational Functions Return to Table of Contents 109 Vocabulary Review x-intercept: The point where a graph intersects with the x-axis and the y-value is zero. y-intercept: The point where a graph
More informationGeometry: Conic Sections
Conic Sections Introduction When a right circular cone is intersected by a plane, as in figure 1 below, a family of four types of curves results. Because of their relationship to the cone, they are called
More information0.4 Family of Functions/Equations
0.4 Family of Functions/Equations By a family of functions, we are referring to a function definition such as f(x) = mx + 2 for m = 2, 1, 1, 0, 1, 1, 2. 2 2 This says, work with all the functions obtained
More informationConic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
More informationSection 1.5 Transformation of Functions
Section 1.5 Transformation of Functions 61 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations
More information3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )
More information1 Vertical and Horizontal
www.ck12.org Chapter 1. Vertical and Horizontal Transformations CHAPTER 1 Vertical and Horizontal Transformations Here you will learn about graphing more complex types of functions easily by applying horizontal
More informationparabola (quadratic) square root semicircle (top, bottom, both) or ellipse or use template greatest integer (step) graphed with int() 10.
Calculator Art Algebra Two DUE Monday MAY 19, 2014 The purpose of this project is for you to program your nspire calculator to draw a picture that incorporates many of the basic graphs that we have studied
More informationIncreasing/Decreasing Behavior
Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second
More informationMath January, Non-rigid transformations. Parent function New function Scale factor
Non-rigid transformations In non-rigid transformations, the shape of a function is modified, either stretched or shrunk. We will call the number which tells us how much it is changed the scale factor,
More informationGraphs of Exponential
Graphs of Exponential Functions By: OpenStaxCollege As we discussed in the previous section, exponential functions are used for many realworld applications such as finance, forensics, computer science,
More information