Transformations of Functions and their Graphs Ms. P.
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Transcript of Transformations of Functions and their Graphs Ms. P.
![Page 1: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/1.jpg)
Transformations of Functions and their Graphs
Ms. P
![Page 2: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/2.jpg)
Linear Transformations
Translations (shifts) Reflections Dilations (stretches or shrinks)
We examine the mathematics: Graphically Numerically Symbolically Verbally
These are the common linear transformations used in high school algebra courses.
![Page 3: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/3.jpg)
Translations
![Page 4: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/4.jpg)
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
![Page 5: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/5.jpg)
TranslationsThis picture might help.
![Page 6: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/6.jpg)
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
Here are the alternate numerical representations of the line graphs above.
1 1
1 2
1 3
2 3
2 2
1 2
4 2
4 3
4 4
5 4
5 3
4 3
![Page 7: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/7.jpg)
Translations
How do we get the flag figure in the left graph to move to the position in the right graph?
This does it!
1 1
1 2
1 3
2 3
2 2
1 2
4 2
4 3
4 4
5 4
5 3
4 3
3 1
3 1
3 1
3 1
3 1
3 1
+
=
![Page 8: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/8.jpg)
Translations
Alternately, we could first add 1 to the y-coordinates and then 3 to the x-coordinates to arrive at the final image.
![Page 9: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/9.jpg)
TranslationsWhat translation could be applied to the left graph to obtain the right graph?
2y x y = ???
![Page 10: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/10.jpg)
Translations
Following the vertex, it appears that the vertex, and hence all the points, have been shifted up 1 unit and right 3 units.
Graphic Representations:
![Page 11: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/11.jpg)
Translations
Numerically, 3 has been added to each x-coordinate and 1 has been added to each y coordinate of the function on the left to produce the function on the right. Thus the graph is shifted up 1 unit and right 3 units.
Numeric Representations:
![Page 12: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/12.jpg)
Translations
To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.
![Page 13: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/13.jpg)
Translations
To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.
The graph on the left above has the equation y = x2.
To translate 1 unit up, we must add 1 to every y-coordinate. We can alternately add 1 to x2 as y and x2 are equal. Thus we have
y = x2 + 1
![Page 14: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/14.jpg)
Translations
We verify our results below:
The above demonstrates a vertical shift up of 1.
y = f(x) + 1 is a shift up of 1 unit that was applied to the graph y = f(x).
How can we shift the graph of y = x2 down 2 units?
![Page 15: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/15.jpg)
Translations
We verify our results below:
The above demonstrates a vertical shift down of 2.
y = f(x) - 2 is a shift down 2 unit to the graph y = f(x)
Vertical Shifts
If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.
Did you guess to subtract 2 units?
![Page 16: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/16.jpg)
Vertical Translation Example
Graph y = |x|
22
11
00
11
22
|x|x
![Page 17: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/17.jpg)
Aside: y = |x| on the TI83/84
![Page 18: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/18.jpg)
Vertical Translation Example
Graph y = |x| + 2
422
311
200
311
422
|x|+2|x|x
![Page 19: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/19.jpg)
Vertical Translation Example
Graph y = |x| - 1
122
011
-100
011
122
|x| -1|x|x
![Page 20: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/20.jpg)
Example Vertical Translations
y = 3x2
![Page 21: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/21.jpg)
Example Vertical Translations
y = 3x2
y = 3x2 – 3
y = 3x2 + 2
![Page 22: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/22.jpg)
Example Vertical Translations
y = x3
![Page 23: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/23.jpg)
Example Vertical Translations
y = x3
y = x3 – 3
y = x3 + 2
![Page 24: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/24.jpg)
Translations
Vertical Shift Animation:http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalshift.html
![Page 25: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/25.jpg)
Translations
The vertex has been shifted up 1 unit and right 3 units.
Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units?
Getting back to our unfinished task:
![Page 26: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/26.jpg)
Translations
Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units?
We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
![Page 27: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/27.jpg)
Translations
We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x y=x2
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
x y=x2+1
-3 10
-2 5
-1 2
0 1
1 2
2 5
3 10
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
![Page 28: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/28.jpg)
TranslationsWe need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
So, let’s try y = (x + 3)2 + 1 ???
Oops!!!
![Page 29: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/29.jpg)
TranslationsWe need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?
x+3 y=x2+1
0 10
1 5
2 2
3 1
4 2
5 5
6 10
So, let’s try y = (x - 3)2 + 1 ???
Hurray!!!!!!
![Page 30: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/30.jpg)
Translations
Vertical Shifts
If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.
Horizontal Shifts
If h is a real number and y = f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h follows a minus sign, then the shift is right and if h follows a + sign, then the shift is left.
![Page 31: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/31.jpg)
Example Horizontal Translation
Graph g(x) = |x|
22
11
00
11
22
|x|x
![Page 32: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/32.jpg)
Example Horizontal Translation
Graph g(x) = |x + 1|
322
211
100
011
122
|x + 1||x|x
![Page 33: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/33.jpg)
Example Horizontal Translation
Graph g(x) = |x - 2|
022
111
200
311
422
|x - 2||x|x
![Page 34: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/34.jpg)
Horizontal Translation
y = 3x2
![Page 35: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/35.jpg)
Horizontal Translation
y = 3x2
y = 3(x+2)2
y = 3(x-2)2
![Page 36: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/36.jpg)
Horizontal Shift Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalshift.html
![Page 37: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/37.jpg)
Summary of Shift Transformations
To Graph: Shift the Graph ofy = f(x) by c units
y = f(x) + c UP
y = f(x) - c DOWN
y = f(x + c) LEFT
y = f(x - c) RIGHT
![Page 38: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/38.jpg)
Translations – Combining ShiftsInvestigate Vertex form of a Quadratic Function: y = x2 + bx + c
y = x2
vertex: (0, 0)
y = (x – 3)2 + 1
vertex: (3, 1)
Vertex Form of a Quadratic Function (when a = 1):
The quadratic function: y = (x – h)2 + k
has vertex (h, k).
![Page 39: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/39.jpg)
TranslationsCompare the following 2 graphs by explaining what to do to the graph of the first function to obtain the graph of the second function.
f(x) = x4
g(x) = (x – 3)4 - 2
![Page 40: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/40.jpg)
Warm-up
If 0 < x < 1, rank the following in order from smallest to largest:
![Page 41: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/41.jpg)
Warm-up
![Page 42: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/42.jpg)
Reflections
![Page 43: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/43.jpg)
Reflections
How do we get the flag figure in the left graph to move to the position in the right graph?
![Page 44: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/44.jpg)
Reflections
How do we get the flag figure in the left graph to move to the position in the right graph? The numeric representations of the line graphs are:
1 1
1 2
1 3
2 3
2 2
1 2
1 -1
1 -2
1 -3
2 -3
2 -2
1 -2
![Page 45: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/45.jpg)
ReflectionsSo how should we change the equation of the function, y = x2 so that the result will be its reflection (across the x-axis)?
Try y = - (x2) or simply y = - x2 (Note: - 22 = - 4 while (-2)2 = 4)
![Page 46: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/46.jpg)
Reflection:Reflection: (across the x-axis)
The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).
![Page 47: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/47.jpg)
Example Reflection over x-axis
f(x) = x2
![Page 48: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/48.jpg)
Example Reflection over x-axis
f(x) = x2
f(x) = -x2
![Page 49: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/49.jpg)
Example Reflection over x-axis
f(x) = x3
![Page 50: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/50.jpg)
Example Reflection over x-axis
f(x) = x3
f(x) = -x3
![Page 51: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/51.jpg)
Example Reflection over x-axis
f(x) = x + 1
![Page 52: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/52.jpg)
Example Reflection over x-axis
f(x) = x + 1
f(x) = -(x + 1) = -x - 1
![Page 53: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/53.jpg)
More Reflections
Reflection in x-axis: 2nd coordinate is negated
Reflection in y-axis: 1st coordinate is negated
![Page 54: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/54.jpg)
Reflection:Reflection: (across the x-axis)
The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).
Reflection: (across the y-axis)
The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).
![Page 55: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/55.jpg)
Example Reflection over y-axis
f(x) = x2
![Page 56: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/56.jpg)
Example Reflection over y-axis
f(x) = x2
f(-x) = (-x)2 = x2
![Page 57: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/57.jpg)
Example Reflection over y-axis
f(x) = x3
![Page 58: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/58.jpg)
Example Reflection over y-axis
f(x) = x3
f(-x) = (-x)3 = -x3
![Page 59: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/59.jpg)
Example Reflection over y-axis
f(x) = x + 1
![Page 60: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/60.jpg)
Example Reflection over y-axis
f(x) = x + 1
f(-x) = -x + 1
http://www.mathgv.com/
![Page 61: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/61.jpg)
Dilations
![Page 62: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/62.jpg)
How do we get the flag figure in the left graph to move to the position in the right graph?
Dilations (Vertical Stretches and Shrink)
1 1
1 2
1 3
2 3
2 2
1 2
1 2
1 4
1 6
2 6
2 4
1 4
![Page 63: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/63.jpg)
Dilations (Stretches and Shrinks)Definitions: Vertical Stretching and Shrinking
The graph of y = af(x) is obtained from the graph of y = f(x) by
a). shrinking the graph of y = f ( x) by a when a > 1, or
b). stretching the graph of y = f ( x) by a when 0 < a < 1.
Vertical Stretch Vertical Shrink
![Page 64: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/64.jpg)
Example Vertical Stretching/Shrinking
y = |x|
![Page 65: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/65.jpg)
Example: Vertical Stretching/Shrinking
y = |x|
y = 0.5|x|
y = 3|x|
![Page 66: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/66.jpg)
Vertical Stretching / Shrinking Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalstretch.html
![Page 67: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/67.jpg)
What is this?
Base Function
y = |x|
y = ????
y = -2|x -1| + 4
![Page 68: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/68.jpg)
Warm-up
Explain how the graph of
can be obtained from the graph of .
![Page 69: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/69.jpg)
How do we get the flag figure in the left graph to move to the position in the right graph?
Dilations (Horizontal Stretches and Shrink)
1 1
1 2
1 3
2 3
2 2
1 2
2 1
2 2
2 3
4 3
4 2
2 2
![Page 70: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/70.jpg)
Horizontal Stretching / Shrinking Animation
http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalstretch.html
![Page 71: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/71.jpg)
Procedure: Multiple TransformationsGraph a function involving more than one transformation in
the following order:1. Horizontal translation2. Stretching or shrinking3. Reflecting4. Vertical translation
Multiple Transformations
![Page 72: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/72.jpg)
Graphing with More than One Transformation
Graph -|x – 2| + 1First graph f(x) = |x|
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Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
![Page 74: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/74.jpg)
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|
![Page 75: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/75.jpg)
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|4. Perform vertical translation:
f(x) = -|x-2| + 1The graph shifts up 1 unit.
![Page 76: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/76.jpg)
Graphing with More than One Transformation
Graph f(x) = -|x – 2| + 1First graph f(x) = |x|
1. Perform horizontal translation: f(x) = |x-2|The graph shifts 2 to the right.
2. There is no stretch3. Reflect in x-axis:
f(x) = -|x-2|4. Perform vertical translation:
f(x) = -|x-2| + 1The graph shifts up 1 unit.
![Page 77: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/77.jpg)
Questions?
![Page 78: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/78.jpg)
Time for worksheet
![Page 79: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/79.jpg)
Other Transformation: Shears
(x, y) (x+y, y)
![Page 80: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/80.jpg)
Can we Apply this Shear to y = x2?
Look at a line graph first!
Apply the shear:
(x, y) (x+y, y)
![Page 81: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/81.jpg)
Can we Apply this Shear to y = x2?
Apply the shear:
(x, y) (x+y, y)
![Page 82: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/82.jpg)
Can we Apply this Shear to y = x2?Apply the shear:
(x, y) (x+y, y)
![Page 83: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/83.jpg)
Yes we CAN Apply this Shear to y = x2.Apply the shear:
(x, y) (x+y, y)
BUT…Can we write the symbolic equation in terms of x and y?
![Page 84: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/84.jpg)
Shear Example
2y t t y
Apply the shear:
(x, y) (x+y, y) to y = x2
Parametrically we have:
x = t + t2 Our job is to eliminate t.
y = t2 We will use the substitution method.
Now substitute t back into the x equation and we have.
2
2
2
2
2
2
22
2 0
x y y
x y y
x y y
x y y
x xy y y
x xy y y
![Page 85: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/85.jpg)
Shears
Horizontal Shear for k a constant
(x, y ) (x+ky, y)
Vertical Shear for k a constant
(x, y ) (x, kx+y)
![Page 86: Transformations of Functions and their Graphs Ms. P.](https://reader034.fdocuments.in/reader034/viewer/2022051516/56649e165503460f94b00a88/html5/thumbnails/86.jpg)
Other Linear Transformations?
Rotations