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Hat04 0203

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  • 1. Copyright © 2007 Pearson Education, Inc. Slide 2-1 2.3 Stretching, Shrinking, and Reflecting Graphs Vertical Stretching of the Graph of a Function If c > 1, the graph of is obtained by vertically stretching the graph of by a factor of c. In general, the larger the value of c, the greater the stretch. )(xfcy ⋅= )(xfy = .1units,stretched )(ofgraphGeneral > = cc xfy .2.3and,4.2 ,3.4,ofgraphThe 43 21 xyxy xyxy == ==
  • 2. Copyright © 2007 Pearson Education, Inc. Slide 2-2 2.3 Vertical Shrinking Vertical Shrinking of the Graph of a Function If the graph of is obtained by vertically shrinking the graph of by a factor of c. In general, the smaller the value of c, the greater the shrink. ,10 << c )(xfcy ⋅= )(xfy = .10units,shrunk )(ofgraphGeneral << = cc xfy . 3 4 3 3 3 2 3 1 3.and,5. ,1.,ofgraphThe xyxy xyxy == ==
  • 3. Copyright © 2007 Pearson Education, Inc. Slide 2-3 2.3 Reflecting Across an Axis Reflecting the Graph of a Function Across an Axis For a function (a) the graph of is a reflection of the graph of f across the x-axis. (b) the graph of is a reflection of the graph of f across the y-axis. )(xfy −= ),(xfy = )( xfy −=
  • 4. Copyright © 2007 Pearson Education, Inc. Slide 2-4 2.3 Example of Reflection Given the graph of sketch the graph of (a) (b) Solution (a) (b) ),(xfy = )(xfy −= )( xfy −= ).,(isso ,graphon theis),(pointIf ba ba − If point ( , ) is on the graph, so is ( , ). a b a b−
  • 5. Copyright © 2007 Pearson Education, Inc. Slide 2-5 2.3 Reflection with the Graphing Calculator ).( and, ,126Set 13 12 2 1 xyy yy xxy −= −= ++= .andofgraphthehaveWe 21 yy .andofgraphthehaveWe 31 yy
  • 6. Copyright © 2007 Pearson Education, Inc. Slide 2-6 2.3 Combining Transformations of Graphs Example Describe how the graph of can be obtained by transforming the graph of Sketch its graph. Solution Since the basic graph is the vertex of the parabola is shifted right 4 units. Since the coefficient of is –3, the graph is stretched vertically by a factor of 3 and then reflected across the x-axis. The constant +5 indicates the vertex shifts up 5 units. 5)4(3 2 +−−= xy .2 xy = ,2 xy = 2 )4( −x 2 )4(3 −− x 2 ) 53( 4xy −−= + shift 4 units right shift 5 units up vertical stretch by a factor of 3 reflect across the x-axis
  • 7. Copyright © 2007 Pearson Education, Inc. Slide 2-7 Graphs: 5)4(3 2 +−−= xy 2 ( 4)y x= − 2 3( 4)y x= − 2 3( 4)y x= − −
  • 8. Copyright © 2007 Pearson Education, Inc. Slide 2-8 2.3 Caution in Translations of Graphs • The order in which transformations are made is important. If they are made in a different order, a different equation can result. – For example, the graph of is obtained by first stretching the graph of by a factor of 2, and then translating 3 units upward. – The graph of is obtained by first translating horizontally 3 units to the left, and then stretching by a factor of 2. 32 += xy xy = 32 += xy
  • 9. Copyright © 2007 Pearson Education, Inc. Slide 2-9 2.3 Transformations on a Calculator- Generated Graph Example The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph. Solution The first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is Thus, the equation of the transformed graph is xy = First View Second View .3 1 3 43 12 = − − = − −− =m .233 −−= xy
  • 10. Copyright © 2007 Pearson Education, Inc. Slide 2-9 2.3 Transformations on a Calculator- Generated Graph Example The figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph. Solution The first view indicates the lowest point is (3,–2), a shift 3 units to the right and 2 units down. The second view shows the point (4,1) on the graph of the transformation. Thus, the slope of the ray is Thus, the equation of the transformed graph is xy = First View Second View .3 1 3 43 12 = − − = − −− =m .233 −−= xy