We want to determine what the graph of an exponential function y = a x looks like for all values of a such that 0 < a < 1
|
|
|
- Alaina Stanley
- 5 years ago
- Views:
Transcription
1 Section 5 2B: Graphs of Decreasing Eponential Functions We want to determine what the graph of an eponential function y = a looks like for all values of a such that 0 < a < We will select a value of a such that 0 > a > and eamine several ordered pairs The patterns we find for a = /2 will be true for any value of a such that 0 < a < and y values for the right side of the graph if = 0 0 y =! if = y = 2! if = 2 2 y = 4! if = 3 3 y = 8! if = 4 4 y = 6! if = 5 y = 32 5 As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line. and y values for the left side of the graph if = 6 if = 5 if = 4 if = 3 if = 2 if = 6 y = 64! 5 y = 32! 4 y =6! 3 y = 8! 2 y = 4! y = 2 As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left
2 The table below shows several values for pairs of (, y) for y = (/2) y Domain Range: can be any real number so the domain of the function is all real numbers. The range for y are positive numbers greater than 0. This means that the entire graph of the function will be above the ais. 2. The y intercept is at (0, ) 3. Left end of the graph: As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left. 4. Right end of the graph: As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line.
3 The Graph of a Decreasing Eponential Function The graph of y = a for all values of a such that 0 < a < is shown below. y = a for all values of a such that 0 > a > is called an decreasing eponential function. Domain Range: can be any real number so the domain of the function is all real numbers. The range for y are positive numbers greater than 0 or y > 0. This means that the entire graph of the function will be above the ais. 2. The y intercept is at (0, ) 3. The ais is a horizontal asymptote. We use a dotted line to show the asymptotic line. Every increasing logarithmic function has a horizontal asymptote. 4. Left end of the graph: As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left. 5. Right end of the graph: As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line.
4 There are 6 different transformation that change the position of the graph, asymptote and y intercept. The graph of y = (/2) and be moved RiGHT, moved LEFT, moved UP, moved Down, flipped about the AXIS or flipped about the y AXIS. These transformations are effected by the addition, subtraction or multiplication of various part of the equation. Translation :! Translation 2:! Translation 3: y = (/2) 3! y = (/2) + 3! y = (/2) + 3 subtracting 3 from! adding 3 to! Adding 3 at the end!! moves the graph RIGHT 3! moves the graph LEFT 3! moves the graph UP 3!! The y intercept is found by! The y intercept is found by! The y intercept is found by! letting = 0 and finding y! letting = 0 and finding y! letting = 0 and finding y y = (/2) 3! y = (/2) + 3! y = (/2) + 3 y = (/ 2) (0 3)! y = (/ 2) 0 + 3! y = (/2) y = (/ 2) 3! y = (/ 2) + 3! y = + 3! y = 8! y = / 8! y = 4
5 Translation 4:! Translation 5:! Translation 6: y = (/2) 3! y = (/2)! y = (/2) subtracting 3 at the end! negating y flips the graph! negating flips the graph moves the graph DOWN 3! about the ais! about the y ais!! The y intercept is found by! The y intercept is found by! The y intercept is found by! letting = 0 and finding y! letting = 0 and finding y! letting = 0 and finding y y = (/2) 3! y = (/2)! y = (/2) y = (/ 2) 0 3! y = (/ 2) 0 =! y = (/ 2) 0 y = 3! y =! y = 2 0 y = 2!! y =
6 The graph of y = (/ 2) 3 compared to the graph of y = (/ 2) y = (/ 2) 3 is equivalent to y = (/ 2) ( 3) Subtracting 3 from the inside a bracket moves the graph 3 units to the RIGHT The graph of y = (/ 2) + 3 compared to the graph of y = (/ 2) y = is equivalent to y = 2 ( + 3) Adding 3 to the inside a bracket moves the graph 3 units to the LEFT
7 The graph of y = (/ 2) + 3 compared to the graph of y = (/ 2) y = (/ 2) + 3 Adding 3 to the (/ 2) at the end of the equation moves the graph UP 3 units The graph of y = (/ 2) 3 compared to the graph of y = (/ 2) Subtracting 3 to the (/ 2) at the end of the equation moves the graph DOWN 3 units
8 The graph of y = (/ 2) compared to the graph of y = (/ 2) Multiplying the (/ 2) by is equivalent to negating the y values (negating the y values flips the graph about the ais) The graph of y = (/ 2) compared to the graph of y = (/ 2) Multiplying the in the eponent by a negative one has the effect of flipping the graph about the y (negating the values flips the graph about the y ais)
Graphs of Increasing Exponential Functions
Section 5 2A: Graphs of Increasing Exponential Functions We want to determine what the graph of an exponential function y = a x looks like for all values of a > We will select a value of a > and examine
Graphs of Increasing Exponential Functions
Section 5 2A: Graphs of Increasing Exponential Functions We want to determine what the graph of an exponential function y = a x looks like for all values of a > We will select a value of a > and examine
Skills Practice Skills Practice for Lesson 7.1
Skills Practice Skills Practice for Lesson.1 Name Date What s the Inverse of an Eponent? Logarithmic Functions as Inverses Vocabulary Write the term that best completes each statement. 1. The of a number
Section 4.2 Graphs of Exponential Functions
238 Chapter 4 Section 4.2 Graphs of Eponential Functions Like with linear functions, the graph of an eponential function is determined by the values for the parameters in the function s formula. To get
f(x) = b x for b > 0 and b 1
7. Introduction to Eponential Functions Name: Recall - Eponents are instructions for repeated multiplication. a. 4 = ()()()() = b. 4 = c.!!!! = Properties of Eponential Functions Parent: Why is the parameter
Lesson #1: Exponential Functions and Their Inverses Day 2
Unit 5: Logarithmic Functions Lesson #1: Exponential Functions and Their Inverses Day 2 Exponential Functions & Their Inverses Exponential Functions are in the form. The inverse of an exponential is a
Section 4.3. Graphing Exponential Functions
Graphing Exponential Functions Graphing Exponential Functions with b > 1 Graph f x = ( ) 2 x Graphing Exponential Functions by hand. List input output pairs (see table) Input increases by 1 and output
Graphs 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
3 = Advanced Math 3 Fall Final Exam Review. Unit 1: If f(x) = x 2 + 3, g(x) = 3x + 1, and h(x) = x + 1, evaluate each.
Advanced Math Fall Final Eam Review Name: Unit 1: If f() +, g() + 1, and h() + 1, evaluate each. 1. f(g()). f(h()). g(- 4) 4. Given ff() + 9, represent its inverse as a (a) graph, (b) chart, and (c) function.
CURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
2.1 Transforming Linear Functions
2.1 Transforming Linear Functions Before we begin looking at transforming linear functions, let s take a moment to review how to graph linear equations using slope intercept form. This will help us because
The 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
Example 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
What is the reasonable domain of this volume function? (c) Can there exist a volume of 0? (d) Estimate a maximum volume for the open box.
MA 15800 Lesson 11 Summer 016 E 1: From a rectangular piece of cardboard having dimensions 0 inches by 0 inches, an open bo is to be made by cutting out identical squares of area from each corner and,
Lesson #6: Basic Transformations with the Absolute Value Function
Lesson #6: Basic Transformations with the Absolute Value Function Recall: Piecewise Functions Graph:,, What parent function did this piecewise function create? The Absolute Value Function Algebra II with
9.1 Linear Inequalities in Two Variables Date: 2. Decide whether to use a solid line or dotted line:
9.1 Linear Inequalities in Two Variables Date: Key Ideas: Example Solve the inequality by graphing 3y 2x 6. steps 1. Rearrange the inequality so it s in mx ± b form. Don t forget to flip the inequality
RATIONAL EQUATIONS AND FUNCTIONS
ALGEBRA II CHAPTER 9 NOTES RATIONAL EQUATIONS AND FUNCTIONS Name Algebra II 9. Graphing Simple Rational Functions Day One Today I am graphing simple rational functions. I am successful today when I can
Practice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise
Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,
Announcements: Quiz next Week Worksheet for over the weekend. Turn UP! Maximize Academic Potential
3/7/3 OBJ: SWBAT graph rational functions and recognize eponential functions. Bell Ringer: Start notes for Eponential functions Homework Requests: pg 46 #-9 odds 37, 39, 4, 43 Homework: p86 #-9 odds Read
Student Exploration: General Form of a Rational Function
Name: Date: Student Eploration: General Form of a Rational Function Vocabulary: asymptote, degree of a polynomial, discontinuity, rational function, root Prior Knowledge Questions (Do these BEFORE using
Domain: The domain of f is all real numbers except those values for which Q(x) =0.
Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such
Basic Transformations
Basic Transformations www.math40s.com 90 Part I: Vertical Stretches Vertical Stretches: A vertical stretch is represented by the form y = af(x), where a is the vertical stretch factor. Example 1: Stretch
EXPLORING RATIONAL FUNCTIONS GRAPHICALLY
EXPLORING RATIONAL FUNCTIONS GRAPHICALLY Precalculus Project Objectives: To find patterns in the graphs of rational functions. To construct a rational function using its properties. Required Information:
Graphical Methods Booklet
Graphical Methods Booklet This document outlines the topic of work and the requirements of students working at New Zealand Curriculum level 7. Parabola, vertex form y = x 2 Vertex (0,0) Axis of symmetry
Section 1.2. Graphing Linear Equations
Graphing Linear Equations Definition of Solution, Satisfy, and Solution Set Definition of Solution, Satisfy, and Solution Set Consider the equation y = 2x 5. Let s find y when x = 3. y = 2x 5 Original
Math 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
UNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
Rational number operations can often be simplified by converting mixed numbers to improper fractions Add EXAMPLE:
Rational number operations can often be simplified by converting mixed numbers to improper fractions Add ( 2) EXAMPLE: 2 Multiply 1 Negative fractions can be written with the negative number in the numerator
Rational 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
Section 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
Topic 6: Calculus Integration Volume of Revolution Paper 2
Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x
Pure Math 30: Explained!
www.puremath30.com 30 part i: stretches about other lines Stretches about other lines: Stretches about lines other than the x & y axis are frequently required. Example 1: Stretch the graph horizontally
3.5D Graphing Rational Functions
3.5D Graphing Rational Functions A. Strategy 1. Find all asymptotes (vertical, horizontal, oblique, curvilinear) and holes for the function. 2. Find the and intercepts. 3. Plot the and intercepts, draw
Radical and Rational Function Exam Questions
Radical and Rational Function Exam Questions Name: ANSWERS 2 Multiple Choice 1. Identify the graph of the function x y. x 2. Given the graph of y f x, what is the domain of x f? a. x R b. 2 x 2 c. x 2
Chapter 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
A 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
Section 1.2: Points and Lines
Section 1.2: Points and Lines Objective: Graph points and lines using x and y coordinates. Often, to get an idea of the behavior of an equation we will make a picture that represents the solutions to the
RATIONAL FUNCTIONS Introductory Material from Earl Please read this!
RATIONAL FUNCTIONS Introductory Material from Earl Please read this! In working with rational functions, I tend to split them up into two types: Simple rational functions are of the form or an equivalent
6B Quiz Review Learning Targets ,
6B Quiz Review Learning Targets 5.10 6.3, 6.5-6.6 Key Facts Double transformations when more than one transformation is applied to a graph o You can still use our transformation rules to identify which
Math 121. Graphing Rational Functions Fall 2016
Math 121. Graphing Rational Functions Fall 2016 1. Let x2 85 x 2 70. (a) State the domain of f, and simplify f if possible. (b) Find equations for the vertical asymptotes for the graph of f. (c) For each
HYPERBOLAS BY: Valencia Williams
HYPERBOLAS BY: Valencia Williams A hyperbola is the set of all points (x, y) in a plane such that the absolute value of the difference of the distance from 2 fixed points (foci) is a positive constant.
Omit Present Value on pages ; Example 7.
MAT 171 Precalculus Algebra Trigsted Pilot Test Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions and Equations 5.1 Exponential Functions 5.2 The Natural Exponential
.2 Transformations of Exponential Functions. Math
.2 Transformations of Eponential Functions Math 30-1 1 Vertical Translation Given the graph of f() = 2 g() = 2 + 3 Shifts the graph up if k > 0. The graph of f() moves upward 3 units. (, y) (, y + k) (0,
Algebra 2 Notes Name: Section 8.4 Rational Functions. A function is a function whose rule can be written as a of. 1 x. =. Its graph is a, f x
Algebra Notes Name: Section 8. Rational Functions DAY ONE: A function is a function whose rule can be written as a of two polynomials. The parent rational function is f. Its graph is a, which has two separate
Transformation of curve. a. reflect the portion of the curve that is below the x-axis about the x-axis
Given graph of y f = and sketch:. Linear Transformation cf ( b + a) + d a. translate a along the -ais. f b. scale b along the -ais c. scale c along the y-ais d. translate d along the y-ais Transformation
Section 1.5. Finding Linear Equations
Section 1.5 Finding Linear Equations Using Slope and a Point to Find an Equation of a Line Example Find an equation of a line that has slope m = 3 and contains the point (2, 5). Solution Substitute m =
Math Stuart Jones. 4.3 Curve Sketching
4.3 Curve Sketching In this section, we combine much of what we have talked about with derivatives thus far to draw the graphs of functions. This is useful in many situations to visualize properties of
Section 4.4 Rational Functions and Their Graphs. 1, the line x = 0 (y-axis) is its vertical asymptote.
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, 16 is a rational function.
Section 4.1 Review of Quadratic Functions and Graphs (3 Days)
Integrated Math 3 Name What can you remember before Chapter 4? Section 4.1 Review of Quadratic Functions and Graphs (3 Days) I can determine the vertex of a parabola and generate its graph given a quadratic
Section 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
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
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
Section 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
Important Things to Remember on the SOL
Notes Important Things to Remember on the SOL Evaluating Expressions *To evaluate an expression, replace all of the variables in the given problem with the replacement values and use (order of operations)
2-3 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
Transformations of Exponential Functions
Transformations of Eponential Functions Math Objectives Students will eplore the famil of eponential functions of the form f ( ) c b a and be able to describe the effect of each parameter on the graph
Algebra II Chapter 8 Part 2: Rational Functions
Algebra II Chapter 8 Part 2: Rational Functions Chapter 8 Lesson 4 Multiply and Divide Rational Functions Vocabulary Words to Review: Reciprocal The rules of fractions DO NOT change! *When adding and subtracting,
x 16 d( x) 16 n( x) 36 d( x) zeros: x 2 36 = 0 x 2 = 36 x = ±6 Section Yes. Since 1 is a polynomial (of degree 0), P(x) =
9 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Section -. Yes. Since is a polynomial (of degree 0), P() P( ) is a rational function if P() is a polynomial.. A vertical asymptote is a vertical line a that
Section 2.3 (e-book 4.1 & 4.2) Rational Functions
Section 2.3 (e-book 4.1 & 4.2) Rational Functions Definition 1: The ratio of two polynomials is called a rational function, i.e., a rational function has the form, where both and are polynomials. Remark
I. Function Characteristics
I. Function Characteristics Interval of possible x values for a given function. (Left,Right) Interval of possible y values for a given function. (down, up) What is happening at the far ends of the graph?
MAT 106: Trigonometry Brief Summary of Function Transformations
MAT 106: Trigonometry Brief Summary of Function Transformations The sections below are intended to provide a brief overview and summary of the various types of basic function transformations covered in
WARM UP DESCRIBE THE TRANSFORMATION FROM F(X) TO G(X)
WARM UP DESCRIBE THE TRANSFORMATION FROM F(X) TO G(X) 2 5 5 2 2 2 2 WHAT YOU WILL LEARN HOW TO GRAPH THE PARENT FUNCTIONS OF VARIOUS FUNCTIONS. HOW TO IDENTIFY THE KEY FEATURES OF FUNCTIONS. HOW TO TRANSFORM
State the domain and range of the relation shown in the graph. Is the relation a function? You try: A relation is represented by
1 State the domain and range of the relation shown in the graph. Is the relation a function? 1a A relation is represented by! Remember: A relation is a set of ordered pairs that can be represented by a
2-4 Graphing Rational Functions
2-4 Graphing Rational Functions Factor What are the zeros? What are the end behaviors? How to identify the intercepts, asymptotes, and end behavior of a rational function. How to sketch the graph of a
Math 154 Elementary Algebra. Equations of Lines 4.4
Math Elementary Algebra Caspers Name Date Equations of Lines. For each graph, solve each equation for y (if necessary), then write down the slope and y-intercept.. y x. y x - - - - - - - - - - - - - -
Graphing 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
Graphs and transformations 4G
Graphs and transformations 4G a f(x + ) is a translation by one unit to the left. d A (0, ), B ( ),0, C (, 4), D (, 0) A (, ), B (0, 0), C (, 4), D (5, 0) e f(x) is a stretch with scale factor b f(x) 4
I Doubt It Lesson 6-1 Transforming Functions
I Doubt It Lesson 6- Learning Targets: Graph transformations of functions and write the equations of the transformed functions. Describe the symmetry of the graphs of even and odd functions. SUGGESTED
Exponential and Logarithmic Functions. College Algebra
Exponential and Logarithmic Functions College Algebra Exponential Functions Suppose you inherit $10,000. You decide to invest in in an account paying 3% interest compounded continuously. How can you calculate
Example 1: Use the graph of the function f given below to find the following. a. Find the domain of f and list your answer in interval notation
When working with the graph of a function, the inputs (the elements of the domain) are always the values on the horizontal ais (-ais) and the outputs (the elements of the range) are always the values on
In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM:
In this class, we addressed problem 14 from Chapter 2. So first step, we expressed the problem in STANDARD FORM: Now that we have done that, we want to plot our constraint lines, so we can find our feasible
Algebra 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
Section 7.6 Graphs of the Sine and Cosine Functions
Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.
Exploring Quadratic Graphs
Exploring Quadratic Graphs The general quadratic function is y=ax 2 +bx+c It has one of two basic graphs shapes, as shown below: It is a symmetrical "U"-shape or "hump"-shape, depending on the sign of
Graphs of the Circular Functions. Copyright 2017, 2013, 2009 Pearson Education, Inc.
4 Graphs of the Circular Functions Copyright 2017, 2013, 2009 Pearson Education, Inc. 1 4.3 Graphs of the Tangent and Cotangent Functions Graph of the Tangent Function Graph of the Cotangent Function Techniques
Mastery. 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,
In this section we continue our study of functions. We have already been introduced to
DETAILED SOLUTIONS AND CONCEPTS - GRAPHS OF COMMON FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE
Now each of you should be familiar with inverses from your previous mathematical
5. Inverse Functions TOOTLIFTST: Knowledge of derivatives of basic functions, including power, eponential, logarithmic, trigonometric, and inverse trigonometric functions. Now each of you should be familiar
Graphing 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
ICM ~Unit 4 ~ Day 2. Section 1.2 Domain, Continuity, Discontinuities
ICM ~Unit 4 ~ Day Section 1. Domain, Continuity, Discontinuities Warm Up Day Find the domain, -intercepts and y-intercepts. 1. 3 5. 1 9 3. Factor completely. 6 4 16 3 4. Factor completely. 8 7 Practice
Translation of graphs (2) The exponential function and trigonometric function
Lesson 35 Translation of graphs (2) The exponential function and trigonometric function Learning Outcomes and Assessment Standards Learning Outcome 2: Functions and Algebra Assessment Standard Generate
CHAPTER 5: Exponential and Logarithmic Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
Using Linear Programming for Management Decisions
Using Linear Programming for Management Decisions By Tim Wright Linear programming creates mathematical models from real-world business problems to maximize profits, reduce costs and allocate resources.
Cumulative Review Problems Packet # 1
April 15, 009 Cumulative Review Problems Packet #1 page 1 Cumulative Review Problems Packet # 1 This set of review problems will help you prepare for the cumulative test on Friday, April 17. The test will
Vertical 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
Lesson 20: Every Line is a Graph of a Linear Equation
Student Outcomes Students know that any non vertical line is the graph of a linear equation in the form of, where is a constant. Students write the equation that represents the graph of a line. Lesson
Math 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
Algebra 1. 7 th Standard Complete Graphs. Categories Quadratic (p. 3-9) Exponential (p ) Absolute Value (p ) Linear (p.
Algebra 1 7 th Standard Complete Graphs Categories Quadratic (p. -9) Eponential (p. 10-1) Absolute Value (p. 14-17) Linear (p. 18-9) Summative Assessment Date: Wednesda, November 8 th Page 1 Standard:
2. Graphical Transformations of Functions
2. Graphical Transformations of Functions In this section we will discuss how the graph of a function may be transformed either by shifting, stretching or compressing, or reflection. In this section let
Quadratic Functions. *These are all examples of polynomial functions.
Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real
Notes Lesson 3 4. Positive. Coordinate. lines in the plane can be written in standard form. Horizontal
A, B, C are Notes Lesson 3 4 Standard Form of an Equation: Integers Ax + By = C Sometimes it is preferred that A is Positive All lines in the plane can be written in standard form. Oblique Coordinate Horizontal
Transformation of Functions You should know the graph of the following basic functions: f(x) = x 2. f(x) = x 3
Transformation of Functions You should know the graph of the following basic functions: f(x) = x 2 f(x) = x 3 f(x) = 1 x f(x) = x f(x) = x If we know the graph of a basic function f(x), we can draw the
Sketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
Sections Transformations
MCR3U Sections 1.6 1.8 Transformations Transformations: A change made to a figure or a relation such that it is shifted or changed in shape. Translations, reflections and stretches/compressions are types
c Sa diyya Hendrickson
Transformations c Sa diyya Hendrickson Introduction Overview Vertical and Horizontal Transformations Important Facts to Remember Naming Transformations Reflections Stretches and Compressions The Rebel
Worksheet A GRAPHS OF FUNCTIONS
C GRAPHS F FUNCTINS Worksheet A Sketch and label each pair of graphs on the same set of aes showing the coordinates of any points where the graphs intersect. Write down the equations of any asymptotes.
Goal: Graph rational expressions by hand and identify all important features
Goal: Graph rational expressions by hand and identify all important features Why are we doing this? Rational expressions can be used to model many things in our physical world. Understanding the features
PreCalculus 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
3. parallel: (b) and (c); perpendicular (a) and (b), (a) and (c)
SECTION 1.1 1. Plot the points (0, 4), ( 2, 3), (1.5, 1), and ( 3, 0.5) in the Cartesian plane. 2. Simplify the expression 13 7 2. 3. Use the 3 lines whose equations are given. Which are parallel? Which
5.1 Introduction to the Graphs of Polynomials
Math 3201 5.1 Introduction to the Graphs of Polynomials In Math 1201/2201, we examined three types of polynomial functions: Constant Function - horizontal line such as y = 2 Linear Function - sloped line,
2.6: Rational Functions and Their Graphs
2.6: Rational Functions and Their Graphs Rational Functions are quotients of polynomial functions. The of a rational expression is all real numbers except those that cause the to equal. Example 1 (like