Absolute Value Functions - ClassZone

1 Page 1 of 2122 Chapter 2 Linear equations and FunctionsAbsolute Value FunctionsREPRESENTINGABSOLUTEVALUEFUNCTI ONSIn Lesson you learned that the Absolute Value of xis defined by:|x| = The graph of this piecewise function consists of two rays, is V-shaped, and opens corner point of the graph, called the occurs at the that the graph of y= |x| is symmetric in the y-axis because for every point(x, y) on the graph, the point ( x, y) is also on the in the activity you investigated the effects of a, h, and kon the graph of y= a|x h| + kseparately, these effects can be combined. For example, the graph ofy=2|x 4|+3is shown in red along with the graph ofy= |x|in blue. Notice that thevertex of the red graph is (4, 3) and that thered graph is narrower than the blue the left of x 0,the graph is given bythe line y the right of x 0,the graph is given bythe line y ( 2, 2)2(2, 2)1xvertex,x, if x> 00,if x = 0 x, if x< 0 GOAL1 Representabsolute Value Absolute valuefunctions to model real-lifesituations, such as playingpool in Example 4.

2 To solve real-lifeproblems, such as when anorchestra should reach adesired sound level in Exs. 44 and 45. Why you should learn itGOAL2 GOAL1 What you should of Absolute Value FunctionsIn the same coordinate plane, graph y= a|x| for a= 2, 12 , 12 , and effect does ahave on the graph of y= a|x|? What is the vertex ofthe graph of y= a|x|?In the same coordinate plane, graph y= |x h| for h= 2, 0, and effect does hhave on the graph of y= |x h|? What is the vertexof the graph of y= |x h|?In the same coordinate plane, graph y= |x| + kfor k= 2, 0, and effect does khave on the graph of y= |x| + k? What is the vertexof the graph of y= |x| + k?321 DevelopingConceptsACTIVITYPage 1 of Value Functions123To graph an Absolute Value function you may find it helpful to plot the vertex and oneother point. Use symmetry to plot a third point and then complete the an Absolute Value FunctionGraph y= |x+ 2| + graph y= |x+ 2| + 3, plot the vertex at ( 2, 3). Then plot another point on the graph, such as ( 3, 2).

3 Use symmetry to plot a third point, ( 1, 2). Connect these three points with a V-shapedgraph. Note that a= 1 < 0 and |a| = 1, so the graph opens down and is the same width as the graph of y=|x|.Writing an Absolute Value FunctionWrite an equation of the graph vertex of the graph is (0, 3), so the equation hasthe form:y=a|x 0| + ( 3) ory= a|x| 3To find the Value of a, substitute the coordinates of thepoint (2, 1) into the equation and a|x| 3 Write a|2| 3 Substitute 1 for y and 2 for = 2a = 2aAdd 3 to each = aDivide each side by 2. An equation of the graph is y= 2|x| 3. CHECK Notice the graph opens up and is narrower than the graph of y= |x|,so 2 is a reasonable Value for (2, 1)(0, 3)EXAMPLE 2 EXAMPLE 1 The graph of y= a|x h| + khas the following characteristics. The graph has vertex (h, k) and is symmetric in the line x= h. The graph is V-shaped. It opens up if a> 0 and down if a< 0. The graph is wider than the graph of y= |x| if |a| < 1. The graph is narrower than the graph of y= |x| if |a| > Absolute Value FUNCTIONSSTUDENTHELPS kills Review For help withsymmetry, see p.

4 919.( 3, 2)( 2, 3)42yx( 1, 2)Page 1 of 2124 Chapter 2 Linear equations and FunctionsUSINGABSOLUTEVALUEFUNCTIONS INREALLIFEI nterpreting an Absolute Value FunctionThe front of a camping tent can be modeled by the functiony= |x | + xand yare measured in feet and the x-axis represents the the the domain and range of the function in the given graph of the function is shown. The vertex is ( , ) and the graph opens down. It isnarrower than the graph of y=|x|. domain is 0 x 5, so the tent is 5 feet range is 0 y , so the tent is feet an Absolute Value GraphWhile playing pool, you try to shoot the eight ballinto the corner pocket as shown. Imagine that acoordinate plane is placed over the pool table. The eight ball is at 5, 54 and the pocket you are aiming for is at (10, 5). You are going to bank theball off the side at (6, 0). an equation for the path of the you make your shot? vertex of the path of the ball is (6, 0), so the equation has the form y=a|x 6|.

5 Substitute the coordinates of the point 5, 54 into the equationand solve for a. 54 = a|5 6|Substitute }54}for yand 5 for x. 54 = aSolve for a. An equation for the path of the ball is y= 54 |x 6|. will make your shot if the point (10, 5) lies on the path of the 54 |10 6|Substitute 5 for yand 10 for = 5 Simplify. The point (10, 5) satisfies the equation, so you do make your 4 EXAMPLE 3 GOAL2 REALLIFEREALLIFEC ampingREALLIFEREALLIFEB illiards5 ft10 ftyx31yx( , )(0, 0)(5, 0)Page 1 of Value do the coordinates (h, k) represent on the graph of y= a|x h| + k? do you know if the graph of y= a|x h| + kopens up or down? How do you know if it is wider,narrower, or the same width as the graph of y=|x|? why the graph shown is not the graph of y= |x+ 3| + the function. Then identify the vertex, tell whether the graph opens upor down, and tell whether the graph is wider, narrower, or the same width asthe graph of y= |x|. 12 |x| |x+ 5| |x| |x| + 2|x+ 6| |x 12 | an equation for the function whose graph is that the tent in Example 3is 7 feet wide and 5 feet tall.

6 Write a function thatmodels the front of the tent. Let the x-axis representthe ground. Then graph the function and identify thedomain and range of the THEEFFECT OFaMatch the function with its (x) = 3|x|13. (x) = 3|x|14. (x) = 13 |x| THEEFFECTS OFhANDkMatch the function with its |x 2| |x| |x+ 2| Check Concept Check Skill Check STUDENTHELPE xtra Practice to help you masterskills is on p. HELPE xample 1:Exs. 12 25 Example 2:Exs. 34 39 Example 3:Exs. 40 45 Example 4:Exs. 46 48Ex. 3yx11Ex. 10 Page 1 of 2126 Chapter 2 Linear equations and FunctionsGRAPHINGABSOLUTEVALUEFUNCTIONSG raph the function. Then identifythe vertex, tell whether the graph opens up or down, and tell whether thegraph is wider, narrower, or the same width as the graph of y= |x|. 6|x 7| |x| + |x 8| + |x+ 2| + 13 |x 3| + 2|x+ 9| + |x| 52 12 |x+ 6|ABSOLUTEVALUEOn many graphing calculators |x| is denoted by ABS(x).Use a graphing calculator to graph the Absolute Value function.

7 Then usethe Tracefeature to find the corresponding x- Value (s) for the given |x| + 4; y= |x+ 14|; y= 15|x|; y= 32 |x+ 74 | 5; y= |x 2| + 5; y= |x| + 7; y= |x+ | 5; y= |x 3| + 6; y= an equation of the graph Exercises 40 and 41, use the following musical group s new single is released. Weekly sales s(in thousands) increasesteadily for a while and then decrease as given by the Functions = 2|t 20| + 40where tis the time (in weeks). the was the maximum number of singles sold in one week?RAINSTORMSIn Exercises 42 and 43, use the following rainstorm begins as a drizzle, builds up to a heavy rain, and then drops back to a drizzle. The rate r(in inches per hour) at which it rains is given by the function r= |t 1| + where t is the time (in hours). the how long does it rain and when does it rain the hardest?yx105yx22yx11yx12yx31yx12 MUSIC SINGLESA musical group ssingle will change position in the charts from week toweek. The Beatles were atNo.

8 1 most often with a totalof 22 hit ONAPPLICATIONSKEYSTROKE HELPV isit our Web see keystrokes forseveral models 1 of Value Functions127 SOUNDLEVELSIn Exercises 44 and 45, use the following a musical piece calls for an orchestra to start at fortissimo(about 90decibels), decrease in loudness to pianissimo(about 50 decibels) in four measures,and then increase back to fortissimoin another four measures. The sound level s(indecibels) of the musical piece can be modeled by the function s= 10|m 4| + 50where mis the number of the function for 0 m how many measures should the orchestra be at the loudness of mezzo forte(about 70 decibels)? are trying to make a hole-in-one on the miniature golf green shown. Imagine thata coordinate plane is placed over the golf green. Thegolf ball is at ( , 2) and the hole is at ( , 2). You aregoing to bank the ball off the side wall of the green at(6, 8). Write an equation for the path of the ball anddetermine if you make your are sitting in a boat on a lake.

9 You can get asunburn from sunlight that hits you directly and from sunlight that reflects off thewater. Sunlight reflects off the water at the point (2, 0) and hits you at the point( , 3). Write and graph the function that shows the path of the Transamerica Pyramid, shown at the right, is an office building in San Francisco. It stands 853 feet tall and is 145 feet wide at its base. Imagine that a coordinate plane is placed over a side of the building. In the coordinate plane, each unit represents one foot, and the origin is at the center of the building s base. Write an Absolute Value function whose graph is the V-shaped outline of the sides of the building, ignoring the shoulders of the statement is true about the graph of the function y= |x+ 2| + 3? AIts vertex is at (2, 3). BIts vertex is at ( 2, 3). CIt opens down. DIt is wider than the graph of y= |x|. function isrepresented by the graph shown? Ay= |x 10| + 2 By= |x+ 10| 2 Cy= |x 2| 10 Dy= |x+ 2| + 10 GRAPHINGG raph the |2x| and y= 2|x| | 5x| and y= 5|x| |x+ 6| and y= |x| + |x+ ( 3)| and y= |x| + on your answers to Exercises 51 54, do you think |ab| = |a| |b| and|a+ b| = |a| + |b| are true statements?

10 Ft12 ftyx HELPV isit our Web help with problemsolving in Ex. 1 of 2 REWRITINGEQUATIONSS olve the equation for y. (Review ) 5y= + 2y= 958. 15 x 23 y= 1 GRAPHINGEQUATIONSG raph the equation. (Review for ) x 6x+ 12 x+ + y= + y= y= 1 FITTING ALINE TODATADraw a scatter plot of the data. Then approximatethe best-fitting line for the data. (Review ) the inequality in a coordinate plane. (Lesson ) > 3x + 5y< 15 Evaluate the function for the given Value of x. (Lesson )5. (5) where (x) = 6. (0) where (x) = Graph the function. (Lesson ) 3|x 2| |x| + 5|x+ 3| 8 Write an equation of the graph shown. (Lesson ) and four friends have $15 to spend on snacks at thebeach. A medium box of popcorn costs $ and a medium soft drink costs$ Write and graph an inequality that represents the numbers of mediumboxes of popcorn and medium soft drinks you can buy. (Lesson ) ACARA local car rental company charges a weekly rate of $200with 1000 free miles.