Transformation of Graphs

1
Transformation of Graphs

• Lesson 3.5

2
Tools for Exploration

• Consider the function f(x) 0.1(x3 9x2)
• Enter this function into your calculator on the
  y screen
• Set the window to be  -10 lt x lt 10   and   -20 lt
  y lt 20
• Graph the function

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Shifting the Graph

• Enter the following function calls of our
  original function on the y screen
• y1  0.1 (x3 - 9x2) 
• y2   y1(x 2) 
• y3   y1(x) 2
• Before you graph the other two lines, predict
  what you think will be the result.

Use different styles for each of the functions
4
Shifting the Graph

• How close wereyour predictions?
• Try these functions again, predict results
• y1  0.1 (x3 - 9x2) 
• y2   y1(x - 2) 
• y3   y1(x) - 2

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Which Way Will You Shift? 
Matching -- match the letter of the list on the
right with the function on the left.
f(x) a f(x - a) f(x)a f(x a) f(x) - a A) shift down a units  B) shift right a units  C) shift left a units  D) shift up a units  E) turn upside down  F) none of these
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Which Way Will It Shift? 

• It is possible to combine more than one of the
  transformations in one function
• What is the result of graphing this
  transformation of our function, f(x)?
• f(x - 3) 5

7
Numerical Results

• Given the functiondefined by a table
• Determine the value of the following
  transformations

x -3 -2 -1 0 1 2 3
f(x) 7 4 9 3 12 5 6
(x) 3              
f(x 1)               
f(x - 2)                      
8
Sound Waves

• Consider a sound wave
• Represented by the function y sin x) 
• Place the function in your Y screen
• Make sure the mode is set to radians
• Use the ZoomTrig option

The rise and fall of the graph model the
vibration of the object creating or transmitting
the sound. What should be altered on the graph to
show increased intensity or loudness?
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Sound Waves

• To model making the sound LOUDER we increase the
  maximum and minimum values (above and below the
  x-axis)
• We increase the amplitude of the function
• We seek to "stretch" the function vertically
• Try graphing the following functions.  Place them
  in your Y screen

Function Style
y1sin x y2(1/2)sin(x) y33sin(x) dotted thick normal
Predict what you think will happen before you
actually graph the functions
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Sound Waves

• Note the results of graphing the three functions.
• The coefficient 3  in  3 sin(x)  stretches the
  function vertically
• The coefficient 1/2  in  (1/2) sin (x) compresses
  the function vertically

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Compression

• The graph of f(x) (x - 2)(x 3)(x - 7) with a
  standard zoom graphs as shown to the right.
• Enter the function in for y1(x - 2)(x 3)(x -
  7) in your Y screen.
• Graph it to verify you have the right function.  

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Compression

• What can we do (without changing the zoom) to
  force the graph to be within the standard zoom?
• We wish to compress the graph by a factor of 0.1
• Enter the altered form of your y1(x) function
  into y2  your Y screen which will  do this.

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Compression

• When we multiply the function by a positive
  fraction less than 1,
• We compress the function
• The local max and min are within the bounds of
  the standard zoom window.

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Flipping the Graph of a Function

• Given the function below
• We wish to manipulate it by reflecting it across
  one of the axes

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Flipping the Graph of a Function

• Consider the function
• f(x) 0.1(x3 - 9x2 5) place it in y1(x)
• graphed on the window   -10 lt x lt 10  and  -20 lt
  y lt 20

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Flipping the Graph of a Function

• specify the following functions on the Y screen
  
• y2(x) y1(-x)                dotted style
• y3(x) -y1(x)                thick style
• Predict which of these will rotate the function
• about the x-axis
• about the y-axis

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Flipping the Graph of a Function

• Results
• To reflect f(x) in the x-axis       or
  rotate about  
• To reflect f(x) in the y-axis         or
  rotate about

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Assignment

• Lesson 3.5
• Page 235
• Exercises 1 73 EOO 95, 99