3.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions

Size: px
Start display at page:

Download "3.6 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions"

Transcription

1 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 (, ). Vertical line; through -, -0. Horizontal line; through (, 0). Through, -9 and -6, -. Through -, with slope - 6. Slope -; -intercept a0, b 7. Slope ; -intercept 0, - 8. Through a, 0b with slope 9. Through -, -; parallel to - = 0. Through (0, ); perpendicular to - = 0. Through, -; perpendicular to + =. Through -, 0; parallel to + =. Undefined slope; through -,. m = 0; through -,.6 Graphing Piecewise-Defined Functions and Shting and Reflecting Graphs of Functions S Graph Piecewise-Defined Functions. Vertical and Horizontal Shts. Reflect Graphs. Graphing Piecewise-Defined Functions Throughout Chapter, we have graphed functions. There are man special functions. In this objective, we stud functions defined b two or more epressions. The epression used to complete the function varies with and depends upon the value of. Before we actuall graph these piecewise-defined functions, let s practice finding function values. EXAMPLE Evaluate f, f -6, and f0 for the function f = e Then write our results in ordered pair form Solution Take a moment and stud this function. It is a single function defined b two epressions depending on the value of. From above, 0, use f = +. If 7 0, use f = - -. Thus f = - - = - since 7 0 f = - Ordered pairs:, - f -6 = -6 + = -9 since -6 0 f -6 = -9-6, -9 f0 = 0 + = since 0 0 f0 = 0, Evaluate f, f -, and f0 for the function f = e Now, let s graph a piecewise-defined function.

2 Section.6 Graphing Piecewise-Defined Functions and Shting and Reflecting Graphs of Functions 77 EXAMPLE Graph f = e Solution Let s graph each piece. If 0, f = + If 7 0, f = - - f f 0 Closed circle - Values 0 μ - Values 7 0 μ The graph of the first part of f listed will look like a ra with a closed-circle end point at 0,. The graph of the second part of f listed will look like a ra with an open-circle end point. To find the eact location of the open-circle end point, use f = - - and find f0. Since f0 = -0 - = -, we graph the values from the second table and place an open circle at 0, -. Graph of f() Notice that this graph is the graph of a function because it passes the vertical line test. The domain of this function is -, and the range is ( -,. Graph f = e Vertical and Horizontal Shting Review of Common Graphs We now take common graphs and learn how more complicated graphs are actuall formed b shting and reflecting these common graphs. These shts and reflections are called transformations, and it is possible to combine transformations. A knowledge of these transformations will help ou simpl future graphs. Let s begin with a review of the graphs of four common functions. Man of these functions we graphed in earlier sections. First, let s graph the linear function f, or. Ordered pair solutions of this graph consist of ordered pairs whose - and -values are the same. or f - -

3 78 CHAPTER Graphs and Functions Net, let s graph the nonlinear function f or. This equation is not linear because the term does not allow us to write it in the form A + B = C. Its graph is not a line. We begin b finding ordered pair solutions. Because this graph is solved for f, or, we choose -values and find corresponding f, or -values. If = -, then = -, or 9. If = -, then = -, or. If = -, then = -, or. If = 0, then = 0, or 0. If =, then =, or. If =, then =, or. If =, then =, or 9. f or Stud the table for a moment and look for patterns. Notice that the ordered pair solution (0, 0) contains the smallest -value because an other -value squared will give a positive result. This means that the point (0, 0) will be the lowest point on the graph. Also notice that all other -values correspond to two dferent -values, for eample, = 9 and - = 9. This means that the graph will be a mirror image of itself across the -ais. Connect the plotted points with a smooth curve to sketch its graph. This curve is given a special name, a parabola. We will stud more about parabolas in later chapters. Net, let s graph another nonlinear function, f 0 0 or 0 0. This is not a linear equation since it cannot be written in the form A + B = C. Its graph is not a line. Because we do not know the shape of this graph, we find man ordered pair solutions. We will choose -values and substitute to find corresponding -values. (, 9) (, 9) (, ) (, ) (, ) Verte (0, 0) f() or (, ) If = -, then = 0-0, or. If = -, then = 0-0, or. If = -, then = 0-0, or. If = 0, then = 0, or 0. If =, then = 0 0, or. If =, then = 0 0, or. If =, then = 0 0, or (, ) (, ) (, ) (, ) (, ) (, ) (0, 0) f() or Again, stud the table of values for a moment and notice an patterns. From the plotted ordered pairs, we see that the graph of this absolute value equation is V-shaped. Finall, a fourth common function, f = or =. For this graph, ou need to recall basic facts about square roots and use our calculator to approimate some square roots to help locate points. Recall also that the square root of a negative number is not a real number, so be careful when finding our domain. Now let s graph the square root function f, or. To graph, we ident the domain, evaluate the function for several values of, plot the resulting points, and connect the points with a smooth curve. Since represents the nonnegative square root of, the domain of this function is the set of all nonnegative numbers, Ú 06, or [0, ). We have approimated on the net page to help us locate the point corresponding to,.

4 Section.6 Graphing Piecewise-Defined Functions and Shting and Reflecting Graphs of Functions 79 If = 0, then = 0, or 0. If =, then =, or. If =, then =, or.7. If =, then =, or. If = 9, then = 9, or. f.7 9 (, ) (, ) (, ) (0, 0) f() or (9, ) Notice that the graph of this function passes the vertical line test, as epected. Below is a summar of our four common graphs. Take a moment and stud these graphs. Your success in the rest of this section depends on our knowledge of these graphs. Common Graphs f() = f() = f() = 0 0 f() = Your knowledge of the slope intercept form, f = m + b, will help ou understand simple shting of transformations such as vertical shts. For eample, what is the dference between the graphs of f = and g = +? g() f = slope, m is 0, 0 g = + slope, m is 0, f() Notice that the graph of g = + is the same as the graph of f =, but moved upward units. This is an eample of a vertical sht and is true for graphs in general.

5 80 CHAPTER Graphs and Functions Vertical Shts (Upward and Downward) Let k be a Positive Number Graph of Same As Moved g = f + k f k units upward g = f - k f k units downward EXAMPLES Without plotting points, sketch the graph of each pair of functions on the same set of aes.. f = and g = +. f = and g = S Without plotting points, sketch the graphs of each pair of functions on the same set of aes.. f = and g = -. f = and g = + A horizontal sht to the left or right ma be slightl more dficult to understand. Let s graph g = 0-0 and compare it with f = 0 0. EXAMPLE Sketch the graphs of f = 0 0 and g = 0-0 on the same set of aes. Solution Stud the table to the left to understand the placement of both graphs. f g (0, 0) f() g() (, 0) Sketch the graphs of f = 0 0 and g = 0-0 on the same set of aes. The graph of g = 0-0 is the same as the graph of f = 0 0, but moved units to the right. This is an eample of a horizontal sht and is true for graphs in general.

6 Section.6 Graphing Piecewise-Defined Functions and Shting and Reflecting Graphs of Functions 8 Horizontal Sht (To the Left or Right) Let h be a Positive Number Graph of Same as Moved g = f - h f h units to the right g = f + h f h units to the left Helpful Hint Notice that f - h corresponds to a sht to the right and f + h corresponds to a sht to the left. Vertical and horizontal shts can be combined. EXAMPLE 6 Sketch the graphs of f = and g = - + on the same set of aes. Solution The graph of g is the same as the graph of f shted units to the right and unit up f() g() ( ) 6 Sketch the graphs of f = 0 0 and g = on the same set of aes. Reflecting Graphs Another tpe of transformation is called a reflection. In this section, we will stud reflections (mirror images) about the -ais onl. For eample, take a moment and stud these two graphs. The graph of g = - can be veried, as usual, b plotting points. f() g() Reflection about the -ais The graph of g = -f is the graph of f reflected about the -ais.

7 8 CHAPTER Graphs and Functions EXAMPLE 7 Sketch the graph of h = Solution The graph of h = is the same as the graph of f = 0 0 reflected about the -ais, then moved three units to the right and two units upward. 7 Sketch the graph of h = h() Vocabular, Readiness & Video Check There are other transformations, such as stretching, that won t be covered in this section. For a review of this transformation, see the Appendi. Match each equation with its graph.. =. =. =. = 0 0 A B C D Martin-Ga Interactive Videos See Video.6 Watch the section lecture video and answer the following questions.. In Eample, onl one piece of the function is defined for the value = -. Wh do we find f ( ) for f ()? 6. For Eamples 8, wh is it helpful to be familiar with common graphs and their basic shapes? 7. Based on the lecture before Eample 9, complete the following statement. The graph of f = has the same shape as the graph of f = + 6 but it is reflected about the..6 Eercise Set Graph each piecewise-defined function. See Eamples and f = e + Ú f = e + Ú 0. f =. f =

8 Section.6 Graphing Piecewise-Defined Functions and Shting and Reflecting Graphs of Functions 8 A B -. g = e g = e f = e 8. f = e Ú - - Ú - MIXED (Sections.,.6) Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. See Eamples and f = e g = e h = e - + Ú - 6. f = e - + Ú f = e - + Ú -. h = e Ú - 0. g = e - Ú f = e - Ú MIXED Sketch the graph of function. See Eamples through f = + 8. f = - 9. f = - 0. f = +. f = -. f = +. f = +. f = -. = - 6. = + 7. f = + 8. f = - 9. f = f = - +. f = - +. f = - +. f = + +. f = + +. f = f = g = h = f = f = + + Sketch the graph of each function. See Eamples through 7.. f = - -. g = - +. h = - +. f = - +. h = g = f = f = - + REVIEW AND PREVIEW Match each equation with its graph. See Section.. 9. = - 0. = -. =. = C D CONCEPT EXTENSIONS. Draw a graph whose domain is -, ] and whose range is [,.. In our own words, describe how to graph a piecewisedefined function.. Graph: f = μ 6. Graph: f = μ Write the domain and range of the following eercises. 7. Eercise 9 8. Eercise 0 9. Eercise 60. Eercise 6 Without graphing, find the domain of each function. 6. f = g = h = f = g = h = Sketch the graph of each piecewise-defined function. Write the domain and range of each function. 67. f = e g = e Ú g = e + - Ú f = e 6 0 Ú 0

8.5 Quadratic Functions and Their Graphs

8.5 Quadratic Functions and Their Graphs CHAPTER 8 Quadratic Equations and Functions 8. Quadratic Functions and Their Graphs S Graph Quadratic Functions of the Form f = + k. Graph Quadratic Functions of the Form f = - h. Graph Quadratic Functions

More information

Graphing Review. Math Tutorial Lab Special Topic

Graphing Review. Math Tutorial Lab Special Topic Graphing Review Math Tutorial Lab Special Topic Common Functions and Their Graphs Linear Functions A function f defined b a linear equation of the form = f() = m + b, where m and b are constants, is called

More information

Graphs and Functions

Graphs and Functions CHAPTER Graphs and Functions. Graphing Equations. Introduction to Functions. Graphing Linear Functions. The Slope of a Line. Equations of Lines Integrated Review Linear Equations in Two Variables.6 Graphing

More information

Derivatives 3: The Derivative as a Function

Derivatives 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 information

Lesson 8.1 Exercises, pages

Lesson 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 information

Transformations of Functions. 1. Shifting, reflecting, and stretching graphs Symmetry of functions and equations

Transformations 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 information

Math 1050 Lab Activity: Graphing Transformations

Math 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 information

9. 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.

9. 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 information

3.5 Equations of Lines

3.5 Equations of Lines Section. Equations of Lines 6. Professional plumbers suggest that a sewer pipe should be sloped 0. inch for ever foot. Find the recommended slope for a sewer pipe. (Source: Rules of Thumb b Tom Parker,

More information

QUADRATIC FUNCTIONS Investigating Quadratic Functions in Vertex Form

QUADRATIC FUNCTIONS Investigating Quadratic Functions in Vertex Form QUADRATIC FUNCTIONS Investigating Quadratic Functions in Verte Form The two forms of a quadratic function that have been eplored previousl are: Factored form: f ( ) a( r)( s) Standard form: f ( ) a b c

More information

Graphing f ( x) = ax 2 + c

Graphing f ( x) = ax 2 + c . Graphing f ( ) = a + c Essential Question How does the value of c affect the graph of f () = a + c? Graphing = a + c Work with a partner. Sketch the graphs of the functions in the same coordinate plane.

More information

SLOPE A MEASURE OF STEEPNESS through 7.1.5

SLOPE A MEASURE OF STEEPNESS through 7.1.5 SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the

More information

Graphing square root functions. What would be the base graph for the square root function? What is the table of values?

Graphing 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 information

p Graph square root functions. VOCABULARY Radical expression Radical function Square root function Parent square root function

p Graph square root functions. VOCABULARY Radical expression Radical function Square root function Parent square root function . Graph Square Root Functions Goal p Graph square root functions. Your Notes VOCABULARY Radical epression Radical function Square root function Parent square root function PARENT FUNCTION FOR SQUARE ROOT

More information

3.7 Graphing Linear Inequalities

3.7 Graphing Linear Inequalities 8 CHAPTER Graphs and Functions.7 Graphing Linear Inequalities S Graph Linear Inequalities. Graph the Intersection or Union of Two Linear Inequalities. Graphing Linear Inequalities Recall that the graph

More information

GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM

GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM FOM 11 T7 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM 1 1 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM I) THE STANDARD FORM OF A QUADRATIC FUNCTION (PARABOLA) IS = a +b +c. To graph a quadratic function

More information

REVIEW, pages

REVIEW, pages REVIEW, pages 69 697 8.. Sketch a graph of each absolute function. Identif the intercepts, domain, and range. a) = ƒ - + ƒ b) = ƒ ( + )( - ) ƒ 8 ( )( ) Draw the graph of. It has -intercept.. Reflect, in

More information

g(x) h(x) f (x) = Examples sin x +1 tan x!

g(x) h(x) f (x) = Examples sin x +1 tan x! Lecture 4-5A: An Introduction to Rational Functions A Rational Function f () is epressed as a fraction with a functiong() in the numerator and a function h() in the denominator. f () = g() h() Eamples

More information

Graphing Quadratics: Vertex and Intercept Form

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 information

TIPS4RM: MHF4U: Unit 1 Polynomial Functions

TIPS4RM: 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 information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE. Eponential Functions. Logarithmic Properties. Graphs of Eponential

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Eponential and Logarithmic Functions Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) Chapter Outline. Eponential Functions. Logarithmic Properties. Graphs of Eponential

More information

Lesson 2.1 Exercises, pages 90 96

Lesson 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 information

Ready to Go On? Skills Intervention 1-1. Exploring Transformations. 2 Holt McDougal Algebra 2. Name Date Class

Ready to Go On? Skills Intervention 1-1. Exploring Transformations. 2 Holt McDougal Algebra 2. Name Date Class Lesson - Read to Go n? Skills Intervention Eploring Transformations Find these vocabular words in the lesson and the Multilingual Glossar. Vocabular transformation translation reflection stretch Translating

More information

Algebra II Quadratic Functions

Algebra II Quadratic Functions 1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations

More information

SLOPE A MEASURE OF STEEPNESS through 2.1.4

SLOPE A MEASURE OF STEEPNESS through 2.1.4 SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written

More information

Polar Functions Polar coordinates

Polar Functions Polar coordinates 548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar

More information

Transformations. which the book introduces in this chapter. If you shift the graph of y 1 x to the left 2 units and up 3 units, the

Transformations. which the book introduces in this chapter. If you shift the graph of y 1 x to the left 2 units and up 3 units, the CHAPTER 8 Transformations Content Summar In Chapter 8, students continue their work with functions, especiall nonlinear functions, through further stud of function graphs. In particular, the consider three

More information

Topic 2 Transformations of Functions

Topic 2 Transformations of Functions Week Topic Transformations of Functions Week Topic Transformations of Functions This topic can be a little trick, especiall when one problem has several transformations. We re going to work through each

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate 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 information

Name Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n =

Name Class Date. subtract 3 from each side. w 5z z 5 2 w p - 9 = = 15 + k = 10m. 10. n = Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations

More information

12.4 The Ellipse. Standard Form of an Ellipse Centered at (0, 0) (0, b) (0, -b) center

12.4 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 information

2.2 Absolute Value Functions

2.2 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

Quadratic Functions In Standard Form In Factored Form In Vertex Form Transforming Graphs. Math Background

Quadratic Functions In Standard Form In Factored Form In Vertex Form Transforming Graphs. Math Background Graphing In Standard Form In Factored Form In Vertex Form Transforming Graphs Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic

More information

Exponential Functions

Exponential Functions 6. Eponential Functions Essential Question What are some of the characteristics of the graph of an eponential function? Eploring an Eponential Function Work with a partner. Cop and complete each table

More information

REMARKS. 8.2 Graphs of Quadratic Functions. A Graph of y = ax 2 + bx + c, where a > 0

REMARKS. 8.2 Graphs of Quadratic Functions. A Graph of y = ax 2 + bx + c, where a > 0 8. Graphs of Quadratic Functions In an earlier section, we have learned that the graph of the linear function = m + b, where the highest power of is 1, is a straight line. What would the shape of the graph

More information

This is called the vertex form of the quadratic equation. To graph the equation

This is called the vertex form of the quadratic equation. To graph the equation Name Period Date: Topic: 7-5 Graphing ( ) Essential Question: What is the vertex of a parabola, and what is its axis of symmetry? Standard: F-IF.7a Objective: Graph linear and quadratic functions and show

More information

2.3 Polynomial Functions of Higher Degree with Modeling

2.3 Polynomial Functions of Higher Degree with Modeling SECTION 2.3 Polnomial Functions of Higher Degree with Modeling 185 2.3 Polnomial Functions of Higher Degree with Modeling What ou ll learn about Graphs of Polnomial Functions End Behavior of Polnomial

More information

Graphing Equations. The Rectangular Coordinate System

Graphing Equations. The Rectangular Coordinate System 3.1 Graphing Equations The Rectangular Coordinate Sstem Ordered pair two numbers associated with a point on a graph. The first number gives the horizontal location of the point. The second gives the vertical

More information

GRAPHS AND GRAPHICAL SOLUTION OF EQUATIONS

GRAPHS 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 information

Transformations of Absolute Value Functions. Compression A compression is a. function a function of the form f(x) = a 0 x - h 0 + k

Transformations of Absolute Value Functions. Compression A compression is a. function a function of the form f(x) = a 0 x - h 0 + k - Transformations of Absolute Value Functions TEKS FOCUS VOCABULARY Compression A compression is a TEKS (6)(C) Analze the effect on the graphs of f() = when f() is replaced b af(), f(b), f( - c), and f()

More information

Sketching graphs of polynomials

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.

More information

Chapter 1 Notes, Calculus I with Precalculus 3e Larson/Edwards

Chapter 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 information

Section 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 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 information

Further Differentiation

Further Differentiation Worksheet 39 Further Differentiation Section Discriminant Recall that the epression a + b + c is called a quadratic, or a polnomial of degree The graph of a quadratic is called a parabola, and looks like

More information

3.2 Polynomial Functions of Higher Degree

3.2 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 information

4.1 Graph Quadratic Functions in

4.1 Graph Quadratic Functions in 4. Graph Quadratic Functions in Standard Form Goal p Graph quadratic functions. Your Notes VOCABULARY Quadratic function Parabola Verte Ais of smmetr Minimum and maimum value PARENT FUNCTION FOR QUADRATIC

More information

Section 9.3: Functions and their Graphs

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

More information

Graphs of quadratics functions are parabolas opening up if a > 0, and down if a < 0. Examples:

Graphs of quadratics functions are parabolas opening up if a > 0, and down if a < 0. Examples: Quadratic Functions ( ) = a + b + c Graphs o quadratics unctions are parabolas opening up i a > 0, and down i a < 0. Eamples: = = + = = 0 MATH 80 Lecture B o 5 Ronald Brent 07 All rights reserved. Notes:

More information

2.3. Horizontal and Vertical Translations of Functions. Investigate

2.3. Horizontal and Vertical Translations of Functions. Investigate .3 Horizontal and Vertical Translations of Functions When a video game developer is designing a game, she might have several objects displaed on the computer screen that move from one place to another

More information

Appendix A.6 Functions

Appendix 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 information

ACTIVITY: Describing an Exponential Function

ACTIVITY: Describing an Exponential Function 6. Eponential Functions eponential function? What are the characteristics of an ACTIVITY: Describing an Eponential Function Work with a partner. The graph below shows estimates of the population of Earth

More information

Lesson 2.4 Exercises, pages

Lesson 2.4 Exercises, pages Lesson. Eercises, pages 13 10 A 3. Sketch the graph of each function. ( - )( + 1) a) = b) = + 1 ( )( 1) 1 (- + )( - ) - ( )( ) 0 0 The function is undefined when: 1 There is a hole at 1. The function can

More information

Essential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1.

Essential Question: How do you graph an exponential function of the form f (x) = ab x? Explore Exploring Graphs of Exponential Functions. 1. Locker LESSON 4.4 Graphing Eponential Functions Common Core Math Standards The student is epected to: F-IF.7e Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric

More information

The Graph Scale-Change Theorem

The Graph Scale-Change Theorem Lesson 3-5 Lesson 3-5 The Graph Scale-Change Theorem Vocabular horizontal and vertical scale change, scale factor size change BIG IDEA The graph of a function can be scaled horizontall, verticall, or in

More information

Calculus & Its Applications Larry J. Goldstein David Lay Nakhle I. Asmar David I. Schneider Thirteenth Edition

Calculus & Its Applications Larry J. Goldstein David Lay Nakhle I. Asmar David I. Schneider Thirteenth Edition Calculus & Its Applications Larr J. Goldstein David La Nakhle I. Asmar David I. Schneider Thirteenth Edition Pearson Education Limited Edinburgh Gate Harlow Esse CM20 2JE England and Associated Companies

More information

5.1 Introduction to the Graphs of Polynomials

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,

More information

Lines and Their Slopes

Lines and Their Slopes 8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation

More information

2.4. A LIBRARY OF PARENT FUNCTIONS

2.4. A LIBRARY OF PARENT FUNCTIONS 2.4. A LIBRARY OF PARENT FUNCTIONS 1 What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal function. Identify and graph step and

More information

Partial Fraction Decomposition

Partial 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 information

What is the relationship between the real roots of a polynomial equation and the x-intercepts of the corresponding polynomial function?

What is the relationship between the real roots of a polynomial equation and the x-intercepts of the corresponding polynomial function? 3.3 Characteristics of Polnomial Functions in Factored Form INVESTIGATE the Math The graphs of the functions f () 5 1 and g() 5 1 are shown.? GOAL Determine the equation of a polnomial function that describes

More information

Example 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs

Example 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have

More information

3x 4y 2. 3y 4. Math 65 Weekly Activity 1 (50 points) Name: Simplify the following expressions. Make sure to use the = symbol appropriately.

3x 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 information

Essential Question How many turning points can the graph of a polynomial function have?

Essential 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 information

Module 2, Section 2 Graphs of Trigonometric Functions

Module 2, Section 2 Graphs of Trigonometric Functions Principles of Mathematics Section, Introduction 5 Module, Section Graphs of Trigonometric Functions Introduction You have studied trigonometric ratios since Grade 9 Mathematics. In this module ou will

More information

Rational functions and graphs. Section 2: Graphs of rational functions

Rational functions and graphs. Section 2: Graphs of rational functions Rational functions and graphs Section : Graphs of rational functions Notes and Eamples These notes contain subsections on Graph sketching Turning points and restrictions on values Graph sketching You can

More information

Lesson/Unit Plan Name: Comparing Linear and Quadratic Functions. Timeframe: 50 minutes + up to 60 minute assessment/extension activity

Lesson/Unit Plan Name: Comparing Linear and Quadratic Functions. Timeframe: 50 minutes + up to 60 minute assessment/extension activity Grade Level/Course: Algebra 1 Lesson/Unit Plan Name: Comparing Linear and Quadratic Functions Rationale/Lesson Abstract: This lesson will enable students to compare the properties of linear and quadratic

More information

Laurie s Notes. Overview of Section 6.3

Laurie 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 information

Appendix F: Systems of Inequalities

Appendix F: Systems of Inequalities Appendi F: Sstems of Inequalities F. Solving Sstems of Inequalities The Graph of an Inequalit What ou should learn The statements < and ⱖ are inequalities in two variables. An ordered pair 共a, b兲 is a

More information

Transformations of Functions. Shifting Graphs. Similarly, you can obtain the graph of. g x x 2 2 f x 2. Vertical and Horizontal Shifts

Transformations of Functions. Shifting Graphs. Similarly, you can obtain the graph of. g x x 2 2 f x 2. Vertical and Horizontal Shifts 0_007.qd /7/05 : AM Page 7 7 Chapter Functions and Their Graphs.7 Transormations o Functions What ou should learn Use vertical and horizontal shits to sketch graphs o unctions. Use relections to sketch

More information

Answers. Investigation 4. ACE Assignment Choices. Applications

Answers. Investigation 4. ACE Assignment Choices. Applications Answers Investigation ACE Assignment Choices Problem. Core Other Connections, ; Etensions ; unassigned choices from previous problems Problem. Core, 7 Other Applications, ; Connections ; Etensions ; unassigned

More information

1.1 Horizontal & Vertical Translations

1.1 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 information

Instructor: Virginia Davis Course: Foundations for College Math (1)

Instructor: Virginia Davis Course: Foundations for College Math (1) 5/19/01 Final Eam Review Ch 10,11-Virginia Davis Student: Date: Instructor: Virginia Davis Course: Foundations for College Math (1) Assignment: Final Eam Review Ch 10,11 1. Simplif b factoring. Assume

More information

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.

GRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),

More information

Problem 1: The relationship of height, in cm. and basketball players, names is a relation:

Problem 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

Domain of Rational Functions

Domain of Rational Functions SECTION 46 RATIONAL FU NCTIONS SKI LLS OBJ ECTIVES Find the domain of a rational function Determine vertical, horizontal, and slant asmptotes of rational functions Graph rational functions CONCE PTUAL

More information

2.4 Polynomial and Rational Functions

2.4 Polynomial and Rational Functions Polnomial Functions Given a linear function f() = m + b, we can add a square term, and get a quadratic function g() = a 2 + f() = a 2 + m + b. We can continue adding terms of higher degrees, e.g. we can

More information

Shifting, Reflecting, and Stretching Graphs

Shifting, Reflecting, and Stretching Graphs Shifting, Reflecting, and Stretching s Shifting s 1 ( ) ( ) This is f ( ) This is f ( ) This is f ( ) What happens to the graph? f ( ) is f () shifted units to the right. f ( ) is f () shifted units to

More information

Section 4.2 Graphs of Exponential Functions

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

More information

Math 20C. Lecture Examples.

Math 20C. Lecture Examples. Math 0C. Lecture Eamples. (8/30/08) Section 14.1, Part 1. Functions of two variables Definition 1 A function f of the two variables and is a rule = f(,) that assigns a number denoted f(,), to each point

More information

Graphing f ( x) = ax 2 + bx + c

Graphing f ( x) = ax 2 + bx + c 8.3 Graphing f ( ) = a + b + c Essential Question How can ou find the verte of the graph of f () = a + b + c? Comparing -Intercepts with the Verte Work with a partner. a. Sketch the graphs of = 8 and =

More information

13.2. General Angles and Radian Measure. What you should learn

13.2. General Angles and Radian Measure. What you should learn Page 1 of 1. General Angles and Radian Measure What ou should learn GOAL 1 Measure angles in standard position using degree measure and radian measure. GOAL Calculate arc lengths and areas of sectors,

More information

Graphing Absolute Value Functions. Objectives To graph an absolute value function To translate the graph of an absolute value function

Graphing Absolute Value Functions. Objectives To graph an absolute value function To translate the graph of an absolute value function 5-8 CC-0 CC-6 Graphing Absolute Value Functions Content Standards F.BF.3 Identif the effect on the graph of replacing f () b f () k, kf (), f (k), and f ( k) for specific values of k (both positive and

More information

Rational Functions with Removable Discontinuities

Rational 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 information

Unit 2: Function Transformation Chapter 1

Unit 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 information

Unit 4 Part 1: Graphing Quadratic Functions. Day 1: Vertex Form Day 2: Intercept Form Day 3: Standard Form Day 4: Review Day 5: Quiz

Unit 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 information

2-3. Attributes of Absolute Value Functions. Key Concept Absolute Value Parent Function f (x)= x VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

2-3. Attributes of Absolute Value Functions. Key Concept Absolute Value Parent Function f (x)= x VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING - Attributes of Absolute Value Functions TEKS FOCUS TEKS ()(A) Graph the functions f() =, f() =, f() =, f() =,f() = b, f() =, and f() = log b () where b is,, and e, and, when applicable, analze the ke

More information

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED FOM 11 T9 GRAPHING LINEAR EQUATIONS REVIEW - 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) -INTERCEPT = the point where the graph touches or crosses the -ais. It occurs when = 0. ) -INTERCEPT = the

More information

SECONDARY MATH TRANSFORMATIONS

SECONDARY MATH TRANSFORMATIONS SECONDARY MATH 3 3-3 TRANSFORMATIONS WARM UP WHAT YOU WILL LEARN How to transform functions from the parent function How to describe a transformation How to write an equation of a transformed function

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;

More information

Answers Investigation 4

Answers Investigation 4 Answers Investigation Applications. a. At seconds, the flare will have traveled to a maimum height of 00 ft. b. The flare will hit the water when the height is 0 ft, which will occur at 0 seconds. c. In

More information

Find the Relationship: An Exercise in Graphical Analysis

Find the Relationship: An Exercise in Graphical Analysis Find the Relationship: An Eercise in Graphical Analsis In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables. For eample,

More information

Section 4.3 Features of a Line

Section 4.3 Features of a Line Section.3 Features of a Line Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif the - and -intercepts of a line. Plotting points in the --plane

More information

Transformation of curve. a. reflect the portion of the curve that is below the x-axis about the x-axis

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

More information

Four Ways to Represent a Function: We can describe a specific function in the following four ways: * verbally (by a description in words);

Four 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 information

0 COORDINATE GEOMETRY

0 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 information

A function: A mathematical relationship between two variables (x and y), where every input value (usually x) has one output value (usually y)

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

More information

1.1. 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? 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 information

The Quadratic function f(x) = x 2 2x 3. y y = x 2 2x 3. We will now begin to study the graphs of the trig functions, y = sinx, y = cosx and y = tanx.

The Quadratic function f(x) = x 2 2x 3. y y = x 2 2x 3. We will now begin to study the graphs of the trig functions, y = sinx, y = cosx and y = tanx. Chapter 7 Trigonometric Graphs Introduction We have alread looked at the graphs of various functions : The Linear function f() = The Quadratic function f() = The Hperbolic function f() = = = = We will

More information

Advanced Functions Unit 4

Advanced Functions Unit 4 Advanced Functions Unit 4 Absolute Value Functions Absolute Value is defined by:, 0, if if 0 0 - (), if 0 The graph of this piecewise function consists of rays, is V-shaped and opens up. To the left of

More information