REVIEW, pages

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1 REVIEW, pages 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 the -ais, the part of the graph that is below the -ais. From the graph, the -intercept is., the -intercept is, the domain of is ç, and the range is». 8 Draw the graph of ( )( ). It has -intercepts and. The ais of smmetr is, so the verte is at (, 9). Reflect, in the -ais, the part of the graph that is below the -ais. From the graph, the -intercepts are and, the -intercept is 8, the domain of ( )( ) is ç, and the range is».. Write each absolute value function in piecewise notation. a) = ƒ ƒ b) = ƒ ( + ) ƒ 9 when 9»» 9 9 ( 9), or 9 when 9< <9 > 9 So, using piecewise notation: 9, e 9, if 9 if > 9 The -intercepts of the graph of ( ) are and. The graph opens up, so between the -intercepts, the graph is below the -ais. For the graph of ( ) : For or», the value of ( )» For <<, the value of ( )< ; that is, ( ). So, using piecewise notation: ( ), if or» e ( ), if << Chapter 8: Absolute Value and Reciprocal Functions Review Solutions DO NOT COPY. P

2 8. 3. Solve b graphing. a) 3 = ƒ - + ƒ b) ƒ - ƒ = To graph, graph, then reflect, in the -ais, the part of the graph that is below the -ais. The line 3 intersects at (., 3) and (3., 3). So, the solutions are. and 3.. Enter and 6 in the graphing calculator. The line 6 appears to intersect at points: (, 6), (, 6), (3, 6), and (6, 6). So, the equation has solutions:,, 3, and 6. Use algebra to solve each equation. a) = ƒ ( - ) - ƒ When ( )» : When ( ) <: ( ) (( ) ) ( ) ( ) 3 or ( ) So,,, and 3 are the solutions. b) = ƒ 3 - ƒ 3 3 if 3» that is, if» 3 (3 ) if 3 < that is, if < 3 When» : 3 When < : 3 3 (3 ) is not greater than or equal to, so is not a solution. < 3 so is a root The solution is. 7 P DO NOT COPY. Chapter 8: Absolute Value and Reciprocal Functions Review Solutions

3 8.3. Identif the equation of the vertical asmptote of the graph of =, then graph the function. - + The graph of has slope, -intercept, and -intercept. The graph of has a horizontal asmptote and a vertical asmptote. Points (3, ) and (, ) are common to both graphs. Some points on are (, 3), (, ), (, 3), and (, ). So, points on are a,, (,.), a,, and (,.). 3 b 3 b 6 6. Use the graph of = f () to sketch a graph of =. f () Identif the equations of the asmptotes of the graph of each reciprocal function. a) b) (, ) f( ) f() 6 8 f( ) f() Horizontal asmptote: -intercept is, so vertical asmptote is. Points (, ) and (, ) are common to both graphs. Some points on f() are: (, 3) and (, 3). So, points on f() Horizontal asmptote: -intercept is 6, so vertical asmptote is 6. Points (7, ) and (, ) are common to both graphs. Some points on f() are: (8, ) and (, ). So, points on are (8,.) and (,.). f() are a, and a,. 3 b 3 b 6 Chapter 8: Absolute Value and Reciprocal Functions Review Solutions DO NOT COPY. P

4 7. Use the graph of = to graph the linear function = f (). f () Describe our strateg. f() f() 6 Vertical asmptote is., so graph of f() has -intercept.. Mark points at and on graph of, then draw a line f() through these points for the graph of f() Use a graphing calculator or graphing software. For which values of q does the graph of = have: -( - 3) + q a) no vertical asmptotes? Look at the quadratic function ( 3) q. The graph opens down. For the graph of its reciprocal function to have no vertical asmptotes, the quadratic function must have no -intercepts. So, the verte of the quadratic function must be below the -ais; that is, q<. b) one vertical asmptote? Look at the quadratic function ( 3) q. The graph opens down. For the graph of its reciprocal function to have one vertical asmptote, the quadratic function must have one -intercept. So, the verte of the quadratic function must be on the -ais; that is, q. c) two vertical asmptotes? Look at the quadratic function ( 3) q. The graph opens down. For the graph of its reciprocal function to have two vertical asmptotes, the quadratic function must have two -intercepts. So, the verte of the quadratic function must be above the -ais; that is, q>. P DO NOT COPY. Chapter 8: Absolute Value and Reciprocal Functions Review Solutions 7

5 9. Determine the equations of the vertical asmptotes of the graph of each reciprocal function. Graph to check the equations. a) = b) = ( - ) - 9 -( - ) - 9 ( ) 9 when ( ) 9 when ( ) 9; that is, when or. So, the ( ) 9. Since the square of a number is never negative, graph of has the graph of ( ) 9 ( ) 9 has no -intercepts, and the vertical asmptotes, graph of has and. I used m graphing ( ) 9 calculator to show that m equations are correct. no vertical asmptotes. I used m graphing calculator to show that m equations are correct. 8.. On the graph of each quadratic function = f (), sketch a graph of the reciprocal function =. Identif the vertical asmptotes, f () if the eist. a) b) f() 6 f() 6 f() 6 8 f() The graph of f() has no The graph of f() has -intercepts, so the graph of -intercept, so the graph of has no vertical has vertical asmptote, f() f() asmptotes and has Shape., and has Shape. Horizontal asmptote: Horizontal asmptote: Points (, ), (3, ), and Plot points where the line (, ) lie on f(), so intersects the graph of f(). points (,.), (3,.), These points are common to both and (,.) lie on graphs.. f() 8 Chapter 8: Absolute Value and Reciprocal Functions Review Solutions DO NOT COPY. P

6 . On the graph of each reciprocal function =, sketch a graph of f () the quadratic function = f (). a) b) 6 f() f() f() f() The graph has one vertical asmptote,, so the graph of f() has verte (, ). The line intersects the graph at points that are common to both graphs. The graph has vertical asmptotes, so the graph of f() has -intercepts. Plot points where the asmptotes intersect the -ais. The point (, ) is on the line of smmetr, so (, ) is the verte of f(). P DO NOT COPY. Chapter 8: Absolute Value and Reciprocal Functions Review Solutions 9