1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.
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Transcript of 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.
![Page 1: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/1.jpg)
1 10/02/11
Section 1.6Section 1.6
Dr. Mark Faucette
Department of Mathematics
University of West Georgia
![Page 2: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/2.jpg)
2 10/02/11
Transformations of Functions Transformations of Functions
How does the graph of a function change if you add a constant to the independent variable? To the dependent variable?
How does the graph of a function change if you multiply a constant times the independent variable? To the dependent variable?
How does the graph of a function change if you change the sign of the independent variable? Of the dependent variable?
![Page 3: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/3.jpg)
3 10/02/11
TopicsTopics
Vertical shiftsHorizontal shiftsReflection about the x-axisReflection about the y-axisVertical stretching and shrinkingHorizontal stretching and shrinking
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4 10/02/11
Vertical ShiftsVertical ShiftsGiven the graph
you get the graph
by moving the first graph up k units (for k>0)
Adding a constant outside the function moves the graph of the function up.
y = f ( x)
y = f ( x) + k
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5 10/02/11
Vertical ShiftingVertical Shifting
y = x 2
y = x 2 + 1
y = x 2 + 2
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6 10/02/11
Vertical Shifting Vertical Shifting (Continued)(Continued)Given the graph
you get the graph
by moving the first graph down k units (for k>0)
Subtracting a constant outside the function moves the graph of the function down.
y = f ( x)
y = f ( x) - k
![Page 7: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/7.jpg)
7 10/02/11
Vertical Shifting Vertical Shifting (Continued)(Continued)
y = x 2
y = x 2 - 1
y = x 2 - 2
![Page 8: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/8.jpg)
8 10/02/11
Horizontal ShiftingHorizontal ShiftingGiven the graph
you get the graph
by moving the first graph left k units (for k>0)
Adding a constant inside the function moves the graph of the function left.
y = f ( x)
y = f ( x + k )
![Page 9: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/9.jpg)
9 10/02/11
Horizontal ShiftingHorizontal Shifting
y = x 2
y = (x + 1)2
y = (x + 2)2
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10 10/02/11
Horizontal Shifting Horizontal Shifting (Continued)(Continued)Given the graph
you get the graph
by moving the first graph right k units (for k>0)
Subtracting a constant inside the function moves the graph of the function right.
y = f ( x)
y = f ( x - k )
![Page 11: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/11.jpg)
11 10/02/11
Horizontal Shifting Horizontal Shifting (Continued)(Continued)
y = x 2
y = (x - 1)2
y = (x - 2)2
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12 10/02/11
Reflections about the Reflections about the xx-axis-axisGiven the graph
you get the graph
by reflecting the first graph across the x-axis.
Changing y to y reflects the graph across the x-axis.
y = f ( x)
y = - f (x )
![Page 13: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/13.jpg)
13 10/02/11
Reflections about the Reflections about the xx-axis-axis
y = x 2
y = - x 2
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14 10/02/11
Reflections about the Reflections about the yy-axis-axisGiven the graph
you get the graph
by reflecting the first graph across the y-axis.
Changing x to x reflects the graph across the y-axis.
y = f ( x)
y = f (- x )
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15 10/02/11
Reflections about the Reflections about the yy-axis-axis
y = x 2 - 2x
y = (- x )2 - 2(- x ) = x 2 + 2x
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16 10/02/11
Vertical StretchingVertical StretchingGiven the graph
you get the graph
by stretching the first graph vertically by a factor of k.
Multiplying y by k>1 stretches the graph vertically by a factor of k.
y = f ( x)
y = k f ( x ) fo r k > 1
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17 10/02/11
Vertical StretchingVertical Stretching
y = x 3 - x
y = 2(x 3 - x)
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18 10/02/11
Vertical ShrinkingVertical ShrinkingGiven the graph
you get the graph
by shrinking the first graph vertically by a factor of k.
Multiplying y by 0<k<1 shrinks the graph vertically by a factor of k.
y = f ( x)
y = k f ( x ) fo r 0 < k < 1
![Page 19: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/19.jpg)
19 10/02/11
Vertical ShrinkingVertical Shrinking
y = x 3 - x
y =12
(x 3 - x)
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20 10/02/11
Horizontal StretchingHorizontal StretchingGiven the graph
you get the graph
by stretching the first graph horizontally by a factor of k.
Multiplying x by 0<k<1 stretches the graph horizontally by a factor of k.
y = f ( x)
y = f ( k x ) fo r 0 < k < 1
![Page 21: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/21.jpg)
21 10/02/11
Horizontal StretchingHorizontal Stretching
y = x 3 - x
y = ((12
x)3 - (12
x))
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22 10/02/11
Horizontal ShrinkingHorizontal ShrinkingGiven the graph
you get the graph
by shrinking the first graph horizontally by a factor of k.
Multiplying x by k>1 shrinks the graph horizontally by a factor of k.
y = f ( x)
y = f ( k x) fo r k > 1
![Page 23: 1 10/02/11 Section 1.6 Dr. Mark Faucette Department of Mathematics University of West Georgia.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f2b5503460f94c451d6/html5/thumbnails/23.jpg)
23 10/02/11
Horizontal ShrinkingHorizontal Shrinking
y = x 3 - x
y = ((2x )3 - (2x ))
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24 10/02/11
Why Are We Learning This?Why Are We Learning This?
Question: Why do we have to know this when our very expensive graphing calculator will graph for us?
Answer: You need to have some idea what the graph should look like in case you enter the function incorrectly into your graphing calculator. In other words, garbage in, garbage out
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25 10/02/11
SummarySummaryWe have learned about vertical and
horizontal shiftingWe have learned about vertical and
horizontal stretching and shrinkingWe have learned about reflections about
both axes
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26 10/02/11
Things To DoThings To Do
Reread Section 1.6Do the homework for Section 1.6Read Section 1.7
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27 10/02/11
One More Thing . . .One More Thing . . .
Have a nice day