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