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Copyright © 2011 Pearson Education, Inc. Slide Vertical Stretching Vertical Stretching of the Graph of a Function If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical stretching of the graph of by applying a factor of c.TRANSCRIPT
Copyright © 2011 Pearson Education, Inc. Slide 2.3-1
Copyright © 2011 Pearson Education, Inc. Slide 2.3-2
Chapter 2: Analysis of Graphs of Functions
2.3 Stretching, Shrinking, and Reflecting Graphs
Goals: Recognize difference between x-axis & y-axis reflections Apply vertical stretches (compressions) Analyze functions and determine domains & ranges
graphically and analytically. Apply series of transformations to a parent function to
produce a graphical representation of a function efficiently. Synthesize a functions equation from graphical
representations
Copyright © 2011 Pearson Education, Inc. Slide 2.3-3
2.3 Vertical Stretching
.1 units, stretched
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Vertical Stretching of the Graph of a Function
If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical stretching of the graph of by applying a factor of c.
( ).y cf x( ),y f x ( , )x cy( , )x y
1,c ( )y cf x( )y f x
Copyright © 2011 Pearson Education, Inc. Slide 2.3-4
2.3 Vertical Shrinking
.10 units, shrunk
)( ofgraph General
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Vertical Shrinking of the Graph of a Function
If a point lies on the graph of then the point lies on the graph of If then the graph of is a vertical shrinking of the graph of by applying a factor of c.
( ).y cf x( ),y f x ( , )x cy( , )x y
0 1,c ( )y cf x( )y f x
Copyright © 2011 Pearson Education, Inc. Slide 2.3-5
2.3 Horizontal Stretching and Shrinking
( )y f cx
Horizontal Stretching and Shrinking of the Graph of a Function
If a point lies on the graph of then the point lies on the graph of
(a) If then the graph of is a horizontal stretching of the graph of
(b) If then the graph of is a horizontal shrinking of the graph of
( ).y f cx( ),y f x( , )x y
0 1,c ( )y f cx( ).y f x
( / , )x c y
( ).y f x1,c
This topic is very difficult to recognize when looking at a graph and trying to determine the equation of the function because a horizontal stretch looks very similar to a vertical
compression. And a horizontal compression looks like a vertical stretch (horizontal stretching and shrinking will be dealt with in greater detail when we discuss
trigonometric (periodic) functions.
Copyright © 2011 Pearson Education, Inc. Slide 2.3-6
2.3 Reflecting Across an Axis
Reflecting the Graph of a Function Across an Axis
For a function defined by the following are true.(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
If (3,4) is a point on f(x), where does that point go in –f(x)?
If (3,4) is a point on f(x), where does that point go in f(-x)?
Copyright © 2011 Pearson Education, Inc. Slide 2.3-7
2.3 Example of Reflection
Given the graph of sketch the graph of(a) (b)
Solution(a) (b)
),(xfy
)(xfy )( xfy
Copyright © 2011 Pearson Education, Inc. Slide 2.3-8
2.3 Combining Transformations of Graphs
ExampleDescribe how the graph of can be obtained by transforming the parent function. Sketch its graph.
What is the parent function?
Solution:What is the vertex of the original parent function?What is the new vertex after transformations?What was the domain and range of parent function?What is the new domain and range?
2( ) 3( 4) 5f x x
2) 53( 4xy
Copyright © 2011 Pearson Education, Inc. Slide 2.3-9
• Why is this set of steps important? If we take y=-x2+2 and graph it and apply the vertical translation first and then reflect it, the process will provide us a graph that is not consistent with what our technology provides. If we do some algebra we can rewrite it as y=-(x2-2) thus we can graph the parent function, slide it down two units and then reflect over x axis. However some people do not like doing the preliminary algebra. SO what many people do is a vertical shift up 2 units and then reflect over the x-axis which provides a completely different (incorrect) graph. HOWEVER, if you follow the steps below and basically save all vertical shifts for last you will never make this mistake.
(L/R SHIFT)
(U/D SHIFT)
Copyright © 2011 Pearson Education, Inc. Slide 2.3-10
5)4(3 2 xy2( 4)y x 23( 4)y x
Determine domains and ranges after quickly sketching the graphs, can these be found without graphing?
Parent functionCritical points?
Domain:Range:
Domain:Range:
Domain:Range:
Domain:Range:
Copyright © 2011 Pearson Education, Inc. Slide 2.3-11
2.3 Caution in Translations of Graphs
• The order in which transformations are made can be important. Changing the order of a stretch and shift can result in a different equation and graph.
– 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 xyxy
32 xy
Copyright © 2011 Pearson Education, Inc. Slide 2.3-12
2.3 Transformations on a Calculator-Generated Graph
ExampleThe figures show two views of the graph and another graph illustrating a combination of transformations. Find the equation of the transformed graph.
Solution: In order to find the scalar that does the stretching we must find the slope of one of the rays of the curve.
xy
First View Second View
Copyright © 2011 Pearson Education, Inc. Slide 2.3-13
Now lets complete transformations on an abstract function f(x)
Copyright © 2011 Pearson Education, Inc. Slide 2.3-14
Extend your thinkingReinforce learned concepts
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