Lesson 3.4

Transformation of Graphs

Lesson 3.4

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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 < x < 10 and -20 < y < 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

Use different styles for each of the functions

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Shifting the Graph How close were

your 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?

1. f(x) + a

2. f(x - a)

3. f(x)*a

4. f(x + a)

5. 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

Matching -- match the letter of the list on the right with the function on the left.

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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

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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)

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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?

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

y1=sin x y2=(1/2)*sin(x) y3=3*sin(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

Across the x-axis Across the y-axis

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Consider the function 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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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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Results

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

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

use -f(x)

use f(-x)

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Assignment

Lesson 3.4A Page 209 Exercises 1 – 35 odd

Lesson 3.4B Page 210 Exercises 37 – 51 odd

Lesson 3.4

Education

Transcript of Lesson 3.4