Transcript Transformations - Physicsservello

Transformation of
Graphs
2010
Tools for Exploration
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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 < x < 10 and -20 < y < 20
Graph the function
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Shifting the Graph
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Enter the following function calls of our original
function on the y= screen:
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y1= 0.1 (x3 - 9x2)
y2= y1(x + 2)
y3= y1(x) + 2
Use different styles for
each of the functions
Before you graph the other two lines, predict
what you think will be the result.
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Shifting the Graph
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How close were
your predictions?
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Try these functions – again, predict results
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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.
1.
2.
3.
4.
5.
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?
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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
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Numerical Results
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Given the function
defined by a table
x
-3
-2
-1
0
1
2
3
f(x)
7
4
9
3
12
5
6
Determine the value of the following
transformations
(x) + 3
f(x + 1)
f(x - 2)
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Sound Waves
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Consider a sound wave
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Represented by the function y = sin x)
Place the function in your Y= screen
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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
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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
y1=sin x
 y2=(1/2)*sin(x)
 y3=3*sin(x)
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Predict what you think will
happen before you actually
graph the functions
Style
dotted
 thick
 normal
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Sound Waves
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Note the results of graphing the three functions.
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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
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The graph of f(x) = (x - 2)(x + 3)(x - 7) with a
standard zoom graphs as shown to the right.
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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
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What can we do (without changing the zoom) to
force the graph to be within the standard zoom?
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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
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When we multiply the function by a positive
fraction less than 1,
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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
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Given the function below
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We wish to manipulate it by reflecting it across one
of the axes
Across the x-axis
Across the y-axis
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Flipping the Graph of a
Function
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Consider the function
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f(x) = 0.1*(x3 - 9x2 + 5) : place it in y1(x)
graphed on the window -10 < x < 10 and -20 < y <
20
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Flipping the Graph of a
Function
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specify the following functions on the Y= screen:
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y2(x) = y1(-x)
y3(x) = -y1(x)
dotted style
thick style
Predict which of these will rotate the function
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about the x-axis
about the y-axis
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Flipping the Graph of a
Function
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Results
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To reflect f(x) in the x-axis
or rotate about
use -f(x)
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To reflect f(x) in the y-axis
or rotate about
use f(-x)
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Assignment
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Lesson 3.4A
Page 209
Exercises 1 – 35 odd
Lesson 3.4B
Page 210
Exercises 37 – 51 odd
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