STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Axis Combining Transformations of Graphs"

Transcription

1 6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous section, we saw how adding or subtracting a constant can cause a vertical or horizontal shift. Now we will see how multipling b a constant alters the graph of a function. Vertical Stretching TECHNOLOGY NOTE B defining Y as directed in parts A, B, and C, and defining Y, Y, and Y 4 as shown here, ou can minimize our kestrokes. (These graphs will not appear unless Y is defined.) = f() = c f(), c > FIGURE 8 FOR DISCUSSION In each group, we give four related functions. Graph the four functions in the first group (Group A), and then answer the questions regarding those functions. Repeat the process for Group B and Group C. Use the window specified for each group. A B C 5, 5 b 5, 5, 5 b 5,, b, How does the graph of compare to the graph of?. How does the graph of compare to the graph of?. How does the graph of 4 compare to the graph of? 4. If we choose c 4, how do ou think the graph of 5 c would compare to the graph of 4? Provide support b choosing such a value of c. In each group of functions in the preceding activit, we started with a basic function and observed how the graphs of functions of the form c compared with the graph of for positive values of c that began at and became progressivel larger. In each case, we obtained a vertical stretch of the graph of the basic function with which we started. These observations can be generalized to an function. Vertical Stretching of the Graph of a Function If c, the graph of c f is obtained b verticall stretching the graph of f b a factor of c. In general, the larger the value of c, the greater the stretch. In Figure 8, we graphicall interpret the statement above. EXAMPLE Recognizing Vertical Stretches Figure 9 shows the graphs of four functions. The graph labeled is that of the function defined b f. The other three functions,,, and 4, are defined as follows, but not necessaril in the given order:.4,., and 4..

2 . Stretching, Shrinking, and Reflecting Graphs 7 Determine the correct equation for each graph. 4 5 = f() = FIGURE 9 Solution The values of c here are.4,., and 4.. The vertical heights of the points with the same -coordinates of the three graphs will correspond to the magnitudes of these c values. Thus, the graph just above will be that of.4, the highest graph will be that of 4., and the graph of. will lie between the others. Therefore, If we were to trace to an point on the graph of and then move to the other graphs one b one, we would see that the -values of the points would be multiplied b the appropriate values of c. You ma wish to eperiment with our calculator in this wa. Vertical Shrinking 4.,.4, and 4.. TECHNOLOGY NOTE You can use a screen such as this to minimize our kestrokes in parts A, B, and C. Again, Y must be defined in order to obtain the other graphs. FOR DISCUSSION This discussion parallels the one given earlier in this section. Follow the same general directions. (Note: The fractions 4,, and 4 ma be entered as their decimal equivalents when plotting the graphs.) A B C 5, 5 b 5, 5, 5 b, 5, b, How does the graph of compare to the graph of?. How does the graph of compare to the graph of?. How does the graph of 4 compare to the graph of? 4. If we choose c 4, how do ou think the graph of 5 c would compare to the graph of 4? Provide support b choosing such a value of c. In this For Discussion activit, we began with a basic function and observed the graphs of c, as we chose progressivel smaller values of c, with c. In each case, the graph of was verticall shrunk (or compressed). These observations can also be generalized to an function.

3 8 CHAPTER Analsis of Graphs of Functions = f() Vertical Shrinking of the Graph of a Function If c, the graph of c f is obtained b verticall shrinking the graph of f b a factor of c. In general, the smaller the value of c, the greater the shrink. = c f(), < c < FIGURE TECHNOLOGY NOTE Figure shows a graphical interpretation of vertical shrinking. EXAMPLE Recognizing Vertical Shrinks Figure shows the graphs of four functions. The graph labeled is that of the function defined b f. The other three functions,,, and 4, are defined as follows, but not necessaril in the given order:.5,., and.. Determine the correct equation for each graph. = f() = 4 This method of defining Y and Y using a list of coefficients in Y will allow ou to duplicate Figure. TECHNOLOGY NOTE B defining Y as directed in parts A, B, C, and D, and defining Y as shown here, ou can minimize our kestrokes FIGURE Solution The smaller the positive value of c, where c, the more compressed toward the -ais the graph will be. Since we have c.5,., and., the function rules must be as follows: Reflecting Across an Ais.,.5, and 4.. We have seen how graphs can be transformed b shifting, stretching, and shrinking. We now eamine how graphs can be reflected across an ais. FOR DISCUSSION In each pair, we give two related functions. Graph f and f in the standard viewing window, and then answer the questions below for each pair. A B C D With respect to the -ais,. how does the graph of compare to the graph of?. how would the graph of compare with the graph of, based on our answer to Item? Confirm our answer b graphing.

4 . Stretching, Shrinking, and Reflecting Graphs 9 TECHNOLOGY NOTE B defining Y as directed in parts E, F, and G, and defining Y as shown here (using function notation), ou can minimize our kestrokes. Again, in each pair, we give two related functions. Graph f and f in the standard viewing window, and then answer the questions below for each pair. E F G 4 4 With respect to the -ais,. how does the graph of compare to the graph of? 4. how would the graph of compare with the graph of, based on our answer to Item? Confirm our answer b graphing. The results of the preceding discussion can be formall summarized. Reflecting the Graph of a Function Across an Ais For a function defined b f, (a) the graph of f is a reflection of the graph of f across the -ais. (b) the graph of f is a reflection of the graph of f across the -ais. Figure shows how the reflections just described affect the graph of a function in general. (, 6) (7, 6) = f() ( 4, ) (, ) FIGURE (, ) ( 4, ) = f() (, 6) (7, 6) FIGURE 4 = f () = f () = f ( ) b (a, b) b ( a, b) (a, b) a a a (a, b) b = f () Reflection across the -ais Reflection across the -ais (a) (b) FIGURE EXAMPLE Appling Reflections across Aes Figure shows the graph of a function f. (a) Sketch the graph of f. (b) Sketch the graph of f. Solution (a) We must reflect the graph across the -ais. This means that if a point a, b lies on the graph of f, then the point a, b must lie on the graph of f. Using the labeled points, we find the graph of f in Figure 4.

5 CHAPTER Analsis of Graphs of Functions ( 7, 6) (, 6) = f( ) (4, ) (, ) FIGURE 5 (b) Here we must reflect the graph across the -ais, meaning that if a point a, b lies on the graph of f, then the point a, b must lie on the graph of f. Thus, we obtain the graph of f as shown in Figure 5. To illustrate reflections on calculator-generated graphs, observe Figure 6. Figure 6(a) shows that Y has been defined b 6 and Y Y, which means that the graph of Y is a reflection across the -ais. Figure 6(b) shows the graphs of Y and Y, confirming this fact. Notice that Y Y, indicating that the graph of is a reflection across the -ais. This is confirmed b Figure 6(c). Y Y = Y = Y = Y ( ) Y = Y (a) (b) FIGURE 6 (c) What Went WRONG? To see how negative values of a affect the graph of a, a student entered three functions Y, Y, and Y as in the accompaning screen. The calculator graphed the first two as shown, but gave a snta error when it attempted to graph the third. What Went Wrong? graph for? What must the student do in order to obtain the desired Answers to What Went Wrong? The student used a subtraction sign to define Y rather than a negative sign. Notice the difference between the signs in Y and as compared to Y. The student must re-enter using a negative sign. Y Y

6 . Stretching, Shrinking, and Reflecting Graphs = = FIGURE 7 Combining Transformations of Graphs The graphs of and are shown in the same viewing window in Figure 7. In terms of the tpes of transformations we have studied, the graph of is obtained b verticall stretching the graph of b a factor of and then reflecting across the -ais. Thus, we have a combination of transformations. As ou might epect, we can create an infinite number of functions b verticall stretching or shrinking, shifting upward, downward, left, or right, and reflecting across an ais. The net eample investigates eamples of this tpe of function. In determining the order in which the transformations are made, use the order of operations. EXAMPLE 4 Describing a Combination of Transformations of a Graph (a) Describe how the graph of 4 5 can be obtained b transforming the graph of. Illustrate with a graphing calculator. (b) Give the equation of the function that would be obtained b starting with the graph of, shifting units to the left, verticall shrinking the graph b a factor of, reflecting across the -ais, and shifting the graph 4 units downward, in this order. Illustrate with a graphing calculator. Analtic Solution (a) The presence of 4 in the definition of the function indicates that the graph of must be shifted 4 units to the right. Since the coefficient of 4 is (a negative number with absolute value greater than ), the graph is stretched verticall b a factor of and then reflected across the -ais. Finall, the constant 5 indicates that the graph is shifted upward 5 units. These ideas are summarized below. ➂ Reflect across the -ais. ➁ Stretch b a factor of. ➀ Shift 4 units to the right. 4 5 ➃ Shift 5 units upward. (b) Shifting units to the left means that is transformed to. Verticall shrinking b a factor of means multipling b, and reflecting across the -ais changes to. Finall, shifting 4 units downward means subtracting 4. Putting this all together leads to the equation 4. Graphing Calculator Solution (a) Figure 8 supports the discussion in the analtic solution. FIGURE 8 (b) Figure 9 supports the discussion in the analtic solution. = ( 4) + 5 = + 4 FIGURE 9 = =

7 CHAPTER Analsis of Graphs of Functions CAUTION The order in which the transformations are made is important. If the are made in a different order, a different equation can result. See the diagram that follows. ➀ Stretch b a factor of. ➁ Shift units upward. ➀ Shift units to the left. ➁ Stretch b a factor of. EXAMPLE 5 Recognizing a Combination of Transformations Figure 4 shows two views of the graph of and another graph illustrating a combination of transformations. Find the equation of the transformed graph.. =. = (a). (b) FIGURE 4 Solution Figure 4(a) shows that the lowest point on the transformed graph has coordinates,, indicating that the graph has been shifted units to the right and units downward. Figure 4(b) shows that a point on the right side of the transformed graph has coordinates 4,, and thus the slope of this ra is m 4 Thus, the stretch factor is. This information leads to as the equation of the transformed graph... EXERCISES Write the equation that results in the desired transformation.. The squaring function, stretched b a factor of. The square root function, reflected across the -ais. The cubing function, shrunk b a factor of 4. The cube root function, reflected across the -ais

8 . Stretching, Shrinking, and Reflecting Graphs 5. The absolute value function, stretched b a factor of and reflected across the -ais 7. The cubing function, shrunk b a factor of.5 and reflected across the -ais 6. The absolute value function, shrunk b a factor of and reflected across the -ais 8. The square root function, shrunk b a factor of. and reflected across the -ais Use the concepts of this chapter to draw a rough sketch of the graphs of,, and. Do not plot points. In each case, and can be graphed b one or more of these: a vertical and/or horizontal shift of the graph of, a vertical stretch or shrink of the graph of, or a reflection of the graph of across an ais. After ou have made our sketches, check b graphing them in an appropriate viewing window of our calculator. 9.., 4,,, 4.,,.., 6, ,, 6.,,,,.5,, 7.,, 4 8. Concept Check Suppose that the graph of f is smmetric with respect to the -ais and it is reflected across the -ais. How will the new graph compare with the original one? Fill in each blank with the appropriate response. (Remember that the vertical stretch or shrink factor is positive.) 9. The graph of 4 can be obtained from the graph of b verticall stretching b a factor of and reflecting across the -ais.. The graph of 6 can be obtained from the graph of b verticall stretching b a factor of and reflecting across the -ais.. The graph of 4 can be obtained from the graph of b shifting horizontall units to the, verticall shrinking b a factor of, reflecting across the -ais, and shifting verticall units in the direction.. The graph of 5 6 can be obtained from the graph of b reflecting across the -ais, verticall shrinking b a factor of, reflecting across the -ais, and shifting verticall units in the direction.. The graph of 6 can be obtained from the graph of b shifting horizontall units to the and stretching verticall b a factor of. 4. The graph of.5 can be obtained from the graph of b shifting horizontall units to the and shrinking verticall b a factor of. Give the equation of each function whose graph is described. 5. The graph of is verticall shrunk b a factor of and the resulting graph is shifted 7 units downward. 6. The graph of is verticall stretched b a factor of. This graph is then reflected across the -ais. Finall, the graph is shifted 8 units upward. 7. The graph of is shifted units to the right. This graph is then verticall stretched b a factor of 4.5. Finall, the graph is shifted 6 units downward. 8. The graph of is shifted units to the left. This graph is then verticall stretched b a factor of.5. Finall, the graph is shifted 8 units upward.,

9 4 CHAPTER Analsis of Graphs of Functions Shown on the left is the graph of Y in the standard viewing window of a graphing calculator. Si other functions, Y through Y 7, are graphed according to the rules shown in the screen on the right. Y = ( ) + Match each function with its calculator-generated graph from choices A F first without using a calculator, b appling the techniques of this chapter. Then, confirm our answer b graphing the function on our calculator Y Y Y 4 Y 5 Y 6 Y 7 A. B. C. D. E. F. In Eercises 5 and 6, the graph of f has been transformed to the graph of g. No shrinking or stretching is involved. Give the equation of g = f() = = g() (5, ) (4, ) = g() = f() =

10 . Stretching, Shrinking, and Reflecting Graphs 5 In Eercises 7 4, each figure shows the graph of a function f. Sketch b hand the graphs of the functions in parts (a), (b), and (c), and answer the question of part (d). 7. (a) f (b) f (c) f 8. (a) f (b) f (c) f (d) What is f? (d) What is f 4? 9. (a) f (b) f (c) f 4. (a) f (b) f (c) f (d) What are the -intercepts of f? (d) On what interval of the domain is f? (, ) (, ) = f() (6, ) (4, ) = f() (.5,.5) (, ) (, ) (, ) (,.5) (, ) ( 4, ) (, ) = f() (, ) (6, ) (, ) = f() 4. (a) f (b) f (c).5f 4. (a) f (b) f (c) f (d) What smmetr does the graph of f ehibit? (d) What smmetr does the graph of f ehibit? = f() (, ) (, ) (, ) (, ) (, ) (, ) 4. Concept Check If r is an -intercept of the graph of f, what statement can be made about the -intercept of the graph of each of the following? (Hint: Make a sketch.) (a) f (b) f (c) f (, ) = f () (, ) 44. Concept Check If b is the -intercept of the graph of f, what statement can be made about the -intercept of the graph of each of the following? (Hint: Make a sketch.) (a) f (b) f (c) 5f (d) f Concept Check The sketch shows an eample of a function defined b f that increases on the interval a, b. Use this graph as a visual aid, and appl the concepts of reflection introduced in this section to answer each question. (Make our own sketch if ou wish.) = f() 45. Does the graph of f increase or decrease on the interval a, b? 46. Does the graph of f increase or decrease on the interval b, a? 47. Does the graph of f increase or decrease on the interval b, a? a b 48. If c, does the graph of c f increase or decrease on the interval a, b?

11 6 CHAPTER Analsis of Graphs of Functions State the intervals over which each function is (a) increasing, (b) decreasing, and (c) constant. 49. The function graphed in Figure 5. The function graphed in Figure 4 5. The function graphed in Figure (See Figure 9.) In Eercises 5 55, each function has a graph with an endpoint (a translation of the point, ). Enter each into our calculator in an appropriate viewing window, and using our knowledge of the graph of, determine the domain and range of the function. (Hint: Locate the endpoint.) Concept Check Based on our observations in Eercise 5, what are the domain and range of f a h k, if a, h, and k? Concept Check Shown here are the graphs of and 5. The point whose coordinates are given at the bottom of the screen lies on the graph of. Use this graph, not our calculator, to find the value of for the same value of shown = 5 = = = Reviewing Basic Concepts (Sections..). Suppose that f is defined for all real numbers, and f 6. For the given assumptions, find another function value. (a) The graph of f is smmetric with respect to the origin. (b) The graph of f is smmetric with respect to the -ais. (c) For all, f f. (d) For all, f f.. Match each equation in Column I with a description of its graph from Column II as it relates to the graph. I II (a) 7 A. a shift of 7 units to the left (b) 7 B. a shift of 7 units to the right (c) 7 C. a shift of 7 units upward (d) 7 D. a shift of 7 units downward (e) 7 E. a vertical stretch b a factor of 7

12 . Stretching, Shrinking, and Reflecting Graphs 7. Match each equation in parts (a) (h) with the sketch of its graph. The basic graph,, is shown here. (a) (b) (c) (d) = 4 (e) (f) (g) (h) A. B. C. D. 4 4 E. F. G. H (, ) (, ) 4. Match each equation with its calculator-generated graph. (a) 6 (b) 6 (c) (d) 4 (e) 4 6 A. B. C. D. E.

13 8 CHAPTER Analsis of Graphs of Functions 5. Each graph is obtained from the graph of f or g b appling the transformations discussed in Sections. and.. Describe the transformations, and then give the equation for the graph. (a) (b) Suppose F is changed to F h. How are the graphs of these equations related? Is the graph of F h the same as the graph of F h? If not, how do the differ? 8. Suppose the equation F is changed to c F, for some constant c. What is the effect on the graph of F? Discuss the effect depending on whether c or c, and c or c. 9. Complete the table if (a) f is an even function and (b) f is an odd function. (c) 5 (d) 6. Consider the two functions in the figure. (a) Find a value of c for which g f c. (b) Find a value of c for which g f c. 7 = g() f() (Modeling) Carbon Monoide Levels The 8-hour maimum carbon monoide levels (in parts per million) for the United States from 98 to 99 can be modeled b the function defined b f , where corresponds to 98. (Source: U.S. Environmental Protection Agenc, 99.) Find a function represented b g that models the same carbon monoide levels ecept that is the actual ear between 98 and 99. For eample, g 985 f and g 99 f 8. (Hint: Use a horizontal translation.) = f()

2 Analysis of Graphs of

2 Analysis of Graphs of ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,

More information

Investigating Horizontal Stretches, Compressions, and Reflections

Investigating Horizontal Stretches, Compressions, and Reflections .7 YOU WILL NEED graph paper (optional) graphing calculator Investigating Horizontal Stretches, Compressions, and Reflections GOAL Investigate and appl horizontal stretches, compressions, and reflections

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

3.1 Quadratic Functions

3.1 Quadratic Functions Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

Transformations of Function Graphs

Transformations of Function Graphs - - - 0 - - - - - - - Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),

More information

1.5 Shifting, Reflecting, and Stretching Graphs

1.5 Shifting, Reflecting, and Stretching Graphs 7_00.qd /7/0 0: AM Page 7. Shifting, Reflecting, and Stretching Graphs Section. Shifting, Reflecting, and Stretching Graphs 7 Summar of Graphs of Parent Functions One of the goals of this tet is to enable

More information

Review Exercises. Review Exercises 83

Review Exercises. Review Exercises 83 Review Eercises 83 Review Eercises 1.1 In Eercises 1 and, sketch the lines with the indicated slopes through the point on the same set of the coordinate aes. Slope 1. 1, 1 (a) (b) 0 (c) 1 (d) Undefined.,

More information

Modifying Functions - Families of Graphs

Modifying Functions - Families of Graphs Worksheet 47 Modifing Functions - Families of Graphs Section Domain, range and functions We first met functions in Sections and We will now look at functions in more depth and discuss their domain and

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

5.3 Graphing Cubic Functions

5.3 Graphing Cubic Functions Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

More information

SECTION 2-5 Combining Functions

SECTION 2-5 Combining Functions 2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

Graphing and transforming functions

Graphing and transforming functions Chapter 5 Graphing and transforming functions Contents: A B C D Families of functions Transformations of graphs Simple rational functions Further graphical transformations Review set 5A Review set 5B 6

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Analyzing the Graph of a Function

Analyzing the Graph of a Function SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the

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

A Summary of Curve Sketching. Analyzing the Graph of a Function

A Summary of Curve Sketching. Analyzing the Graph of a Function 0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph

More information

3 Functions and Graphs

3 Functions and Graphs 54617_CH03_155-224.QXP 9/14/10 1:04 PM Page 155 3 Functions and Graphs In This Chapter 3.1 Functions and Graphs 3.2 Smmetr and Transformations 3.3 Linear and Quadratic Functions 3.4 Piecewise-Defined Functions

More information

2.3 TRANSFORMATIONS OF GRAPHS

2.3 TRANSFORMATIONS OF GRAPHS 78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 Inequalities with Absolute Value and Quadratic Functions 08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we

More information

3.1 Quadratic Functions

3.1 Quadratic Functions 33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial

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

4.4 Concavity and Curve Sketching

4.4 Concavity and Curve Sketching Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at

More information

Graphs of Functions of Two Variables and Contour Diagrams

Graphs of Functions of Two Variables and Contour Diagrams Graphs of Functions of Two Variables and Contour Diagrams In the previous section we introduced functions of two variables. We presented those functions primaril as tables. We eamined the differences between

More information

13 Graphs, Equations and Inequalities

13 Graphs, Equations and Inequalities 13 Graphs, Equations and Inequalities 13.1 Linear Inequalities In this section we look at how to solve linear inequalities and illustrate their solutions using a number line. When using a number line,

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Translating Points. Subtract 2 from the y-coordinates

Translating Points. Subtract 2 from the y-coordinates CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

More information

HI-RES STILL TO BE SUPPLIED

HI-RES STILL TO BE SUPPLIED 1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

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

Families of Quadratics

Families of Quadratics Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish

More information

In this section you will learn how to draw the graph of the quadratic function defined by the equation. f(x) = a(x h) 2 + k. (1)

In this section you will learn how to draw the graph of the quadratic function defined by the equation. f(x) = a(x h) 2 + k. (1) Section.1 The Parabola 419.1 The Parabola In this section ou will learn how to draw the graph of the quadratic function defined b the equation f() = a( h) 2 + k. (1) You will quickl learn that the graph

More information

2.3 Quadratic Functions

2.3 Quadratic Functions . Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the

More information

4.1 Piecewise-Defined Functions

4.1 Piecewise-Defined Functions Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

More information

The Rectangular Coordinate System

The Rectangular Coordinate System The Mathematics Competenc Test The Rectangular Coordinate Sstem When we write down a formula for some quantit,, in terms of another quantit,, we are epressing a relationship between the two quantities.

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Inverse Relations and Functions

Inverse Relations and Functions 11.1 Inverse Relations and Functions 11.1 OBJECTIVES 1. Find the inverse of a relation 2. Graph a relation and its inverse 3. Find the inverse of a function 4. Graph a function and its inverse 5. Identif

More information

Algebra II Quadratic Functions and Equations- Transformations Unit 05a

Algebra II Quadratic Functions and Equations- Transformations Unit 05a Previous Knowledge: (What skills do the need to have to succeed?) Squares and Square Roots Simplif Radical Expressions Multipl Binomials Solve Multi-Step Equations Identif and graph linear functions Transform

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane.

INTRODUCTION TO GRAPHS AND FUNCTIONS Introductory Topics in Graphing Distance Between Two Points On The Coordinate Plane. P ( 1, 1 ) INTRODUCTION TO GRAPHS AND FUNCTIONS Introductor Topics in Graphing Distance Between Two Points On The Coordinate Plane Q (, ) R (, 1 ) The straight line distance between two points P ( 1, 1

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest

More information

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it 0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form

More information

Write seven terms of the Fourier series given the following coefficients. 1. a 0 4, a 1 3, a 2 2, a 3 1; b 1 4, b 2 3, b 3 2

Write seven terms of the Fourier series given the following coefficients. 1. a 0 4, a 1 3, a 2 2, a 3 1; b 1 4, b 2 3, b 3 2 36 Chapter 37 Infinite Series Eercise 5 Fourier Series Write seven terms of the Fourier series given the following coefficients.. a 4, a 3, a, a 3 ; b 4, b 3, b 3. a.6, a 5., a 3., a 3.4; b 7.5, b 5.3,

More information

+ k, and follows all the same rules for determining. y x 4. = + c. ( ) 2

+ k, and follows all the same rules for determining. y x 4. = + c. ( ) 2 The Quadratic Function The quadratic function is another parent function. The equation for the quadratic function is and its graph is a bowl-shaped curve called a parabola. The point ( 0,0) is called the

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

2.5 Library of Functions; Piecewise-defined Functions

2.5 Library of Functions; Piecewise-defined Functions SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,

More information

Solving Absolute Value Equations and Inequalities Graphically

Solving Absolute Value Equations and Inequalities Graphically 4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM . Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

More information

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form

Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. Slope-Intercept Form 8 (-) Chapter Linear Equations in Two Variables and Their Graphs In this section Slope-Intercept Form Standard Form Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications

More information

Section 1.3: Transformations of Graphs

Section 1.3: Transformations of Graphs CHAPTER 1 A Review of Functions Section 1.3: Transformations of Graphs Vertical and Horizontal Shifts of Graphs Reflecting, Stretching, and Shrinking of Graphs Combining Transformations Vertical and Horizontal

More information

5-1. Lesson Objective. Lesson Presentation Lesson Review

5-1. Lesson Objective. Lesson Presentation Lesson Review 5-1 Using Transformations to Graph Quadratic Functions Lesson Objective Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x h) 2 + k. Lesson Presentation Lesson

More information

GRAPHS OF RATIONAL FUNCTIONS

GRAPHS OF RATIONAL FUNCTIONS 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique

More information

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study

Functions and Graphs CHAPTER INTRODUCTION. The function concept is one of the most important ideas in mathematics. The study Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Why? Describe how the graph of each function is related to the graph of f(x) = x 2. a. h(x) = x 2 + 3

Why? Describe how the graph of each function is related to the graph of f(x) = x 2. a. h(x) = x 2 + 3 Transformations of Quadratic Functions Then You graphed quadratic functions b using the verte and ais of smmetr. (Lesson 9-1) Now 1Appl translations to quadratic functions. 2Appl dilations and reflections

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Chapter 2 Analysis of Graphs of Functions

Chapter 2 Analysis of Graphs of Functions Chapter Analysis o Graphs o Functions Chapter Analysis o Graphs o Functions Covered in this Chapter:.1 Graphs o Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations

More information

Attributes and Transformations of Reciprocal Functions VOCABULARY

Attributes and Transformations of Reciprocal Functions VOCABULARY TEKS FOCUS - Attributes and Transformations of Reciprocal Functions VOCABULARY TEKS (6)(G) Analze the effect on the graphs of f () = when f () is replaced b af (), f (b), f ( - c), and f () + d for specific

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

Rationale/Lesson Abstract: Students will graph exponential functions, identify key features and learn how the variables a, h and k in f x a b

Rationale/Lesson Abstract: Students will graph exponential functions, identify key features and learn how the variables a, h and k in f x a b Grade Level/Course: Algebra Lesson/Unit Plan Name: Graphing Eponential Functions Rationale/Lesson Abstract: Students will graph eponential functions, identif ke features h and learn how the variables a,

More information

Higher. Functions and Graphs. Functions and Graphs 18

Higher. Functions and Graphs. Functions and Graphs 18 hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

More information

More on Functions CHAPTER CONNECTIONS CHAPTER OUTLINE

More on Functions CHAPTER CONNECTIONS CHAPTER OUTLINE cob97_ch0_0-9.qd // 9: AM Page 0 CHAPTER CONNECTIONS More on Functions CHAPTER OUTLINE. Analzing the Graph of a Function 06. The Toolbo Functions and Transformations 0. Absolute Value Functions, Equations,

More information

Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Exercise in Graphing Analysis Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

More information

Section C Non Linear Graphs

Section C Non Linear Graphs 1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,

More information

Systems of linear equations (simultaneous equations)

Systems of linear equations (simultaneous equations) Before starting this topic ou should review how to graph equations of lines. The link below will take ou to the appropriate location on the Academic Skills site. http://www.scu.edu.au/academicskills/numerac/inde.php/1

More information

1.6 Graphs of Functions

1.6 Graphs of Functions .6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time

More information

Quadratic Functions ESSENTIAL QUESTIONS EMBEDDED ASSESSMENTS

Quadratic Functions ESSENTIAL QUESTIONS EMBEDDED ASSESSMENTS Quadratic Functions 5 01 College Board. All rights reserved. Unit Overview In this unit ou will stud a variet of was to solve quadratic functions and sstems of equations and appl our learning to analzing

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Chapter 3 Sullivan 8th Edition Practice for the Eam Kincade MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the relation represents

More information

8 Graphs of Quadratic Expressions: The Parabola

8 Graphs of Quadratic Expressions: The Parabola 8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be

More information

4-1. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

4-1. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 4-1 Quadratic Functions and Transformations Vocabular Review 1. Circle the verte of each absolute value graph. Vocabular Builder parabola (noun) puh RAB uh luh Related Words: verte, ais of smmetr, quadratic

More information

lim f ( x) may be given by a table, formula, or a graph. For this handout, we ll assume the function is given by a graph.

lim f ( x) may be given by a table, formula, or a graph. For this handout, we ll assume the function is given by a graph. Finding imits Graphicall To find a limit graphicall, we must understand each component of the limit to insure the graph is used properl to evaluate the limit et s look at the smbols used in a limit: lim

More information

Quadratic Functions. MathsStart. Topic 3

Quadratic Functions. MathsStart. Topic 3 MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE

More information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

17.1 Connecting Intercepts and Zeros

17.1 Connecting Intercepts and Zeros Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

7.3 Graphing Rational Functions

7.3 Graphing Rational Functions Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

Algebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Algebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete

More information

Solving x < a. Section 4.4 Absolute Value Inequalities 391

Solving x < a. Section 4.4 Absolute Value Inequalities 391 Section 4.4 Absolute Value Inequalities 391 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute

More information

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran College Algera - MAT 161 Page: 1 Copright 009 Killoran Quadratic Functions The graph of f./ D a C C c (where a,,c are real and a 6D 0) is called a paraola. Paraola s are Smmetric over the line that passes

More information

Exponential Functions

Exponential Functions CHAPTER Eponential Functions 010 Carnegie Learning, Inc. Georgia has two nuclear power plants: the Hatch plant in Appling Count, and the Vogtle plant in Burke Count. Together, these plants suppl about

More information

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

More information

Lesson 8.3 Exercises, pages

Lesson 8.3 Exercises, pages Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

More information

Transformations of y = 1/x 2

Transformations of y = 1/x 2 Transformations of y = / In Eample on page 6 we saw that some simple rational functions can be graphed by shifting, stretching, and/or reflecting the graph of y = /. Similarly, some rational functions

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest y-value of a function. Minimum value the least

Attributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest y-value of a function. Minimum value the least - Attributes and Transformations of Quadratic Functions TEKS FCUS TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.

More information

A Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0.

A Library of Parent Functions. Linear and Squaring Functions. Writing a Linear Function. Write the linear function f for which f 1 3 and f 4 0. 0_006.qd 66 /7/0 Chapter.6 8:0 AM Page 66 Functions and Their Graphs A Librar of Parent Functions What ou should learn Identif and graph linear and squaring functions. Identif and graph cubic, square root,

More information

Filling in Coordinate Grid Planes

Filling in Coordinate Grid Planes Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the

More information

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza

Rational Functions. 7.1 A Rational Existence. 7.2 A Rational Shift in Behavior. 7.3 A Rational Approach. 7.4 There s a Hole In My Function, Dear Liza Rational Functions 7 The ozone laer protects Earth from harmful ultraviolet radiation. Each ear, this laer thins dramaticall over the poles, creating ozone holes which have stretched as far as Australia

More information

Lesson 17: Graphing the Logarithm Function

Lesson 17: Graphing the Logarithm Function Lesson 17 Name Date Lesson 17: Graphing the Logarithm Function Exit Ticket Graph the function () = log () without using a calculator, and identify its key features. Lesson 17: Graphing the Logarithm Function

More information