2 5 bzca5e
TRANSCRIPT
![Page 1: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/1.jpg)
Section 2.5Transformation of Functions
![Page 2: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/2.jpg)
Graphs of Common Functions
![Page 3: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/3.jpg)
![Page 4: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/4.jpg)
![Page 5: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/5.jpg)
![Page 6: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/6.jpg)
x
y
Reciprocal Function
Domain: - ,0 0,
Range: - ,0 0,
Decreasing on - ,0 0,
Odd function
and
1( )f x
x
![Page 7: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/7.jpg)
Vertical Shifts
![Page 8: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/8.jpg)
Vertical Shifts
Let be a function and be a positive real number.
The graph of is the graph of shifted units
vertically upward.
The graph of is the graph of shifted
f c
y f x c y f x c
y f x c y f x c
units
vertically downward.
![Page 9: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/9.jpg)
Vertical Shifts
![Page 10: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/10.jpg)
![Page 11: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/11.jpg)
Example
Use the graph of f(x)=|x| to obtain g(x)=|x|-2
x
y
![Page 12: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/12.jpg)
Horizontal Shifts
![Page 13: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/13.jpg)
Horizontal Shifts
Let be a function and a positive real number.
The graph of is the graph of shifted
to the left units.
The graph of is the graph of shifted
to the
f c
y f x c y f x
c
y f x c y f x
right units.c
![Page 14: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/14.jpg)
Horizontal Shifts
![Page 15: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/15.jpg)
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2
x
y
![Page 16: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/16.jpg)
Combining Horizontal and Vertical Shifts
![Page 17: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/17.jpg)
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2+2
x
y
![Page 18: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/18.jpg)
Reflections of Graphs
![Page 19: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/19.jpg)
Refection about the -Axis
The graph of is the graph of reflected
about the -axis.
x
y f x y f x
x
![Page 20: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/20.jpg)
Reflections about the x-axis
![Page 21: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/21.jpg)
Reflection about the y-Axis
The graph of is the graph of reflected
about - axis.
y f x y f x
y
![Page 22: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/22.jpg)
Example
Use the graph of f(x)=x3 to obtain the graph of g(x)= (-x)3.
x
y
![Page 23: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/23.jpg)
Example
x
y
Use the graph of f(x)= x to graph g(x)=- x
![Page 24: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/24.jpg)
Vertical Stretching and Shrinking
![Page 25: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/25.jpg)
![Page 26: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/26.jpg)
Vertically Shrinking
![Page 27: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/27.jpg)
Vertically Stretching
x
y
x
yGraph of f(x)=x3
Graph of g(x)=3x3
This is vertical stretching – each y coordinate is multiplied by 3 to stretch the graph.
![Page 28: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/28.jpg)
Example
Use the graph of f(x)=|x| to graph g(x)= 2|x|
x
y
![Page 29: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/29.jpg)
Horizontal Stretching and Shrinking
![Page 30: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/30.jpg)
![Page 31: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/31.jpg)
Horizontal Shrinking
![Page 32: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/32.jpg)
Horizontal Stretching
![Page 33: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/33.jpg)
Example
x
y
Use the graph of f(x)= to obtain the
1graph of g(x)=
3
x
x
![Page 34: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/34.jpg)
Sequences of Transformations
![Page 35: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/35.jpg)
A function involving more than one transformation can be graphed by performing transformations in the following order:
1. Horizontal shifting
2. Stretching or shrinking
3. Reflecting
4. Vertical shifting
![Page 36: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/36.jpg)
Summary of Transformations
![Page 37: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/37.jpg)
A Sequence of Transformations
Move the graph to the left 3 units
Starting graph.
Stretch the graph vertically by 2.
Shift down 1 unit.
![Page 38: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/38.jpg)
Example
x
y
1Given the graph of f(x) below, graph ( 1).
2f x
![Page 39: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/39.jpg)
Example
x
y
Given the graph of f(x) below, graph - ( 2) 1.f x
![Page 40: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/40.jpg)
Example
Given the graph of f(x) below, graph 2 ( ) 1.f x
![Page 41: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/41.jpg)
(a)
(b)
(c)
(d)
x
y
Use the graph of f(x)= x to graph g(x)= -x.
The graph of g(x) will appear in which quadrant?
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
( )f x x
![Page 42: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/42.jpg)
(a)
(b)
(c)
(d)
x
y
Write the equation of the given graph g(x). The original function was f(x) =x2
g(x)
2
2
2
2
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
g x x
g x x
g x x
g x x
![Page 43: 2 5 Bzca5e](https://reader035.vdocuments.mx/reader035/viewer/2022062419/558b4ef7d8b42a7e688b4616/html5/thumbnails/43.jpg)
(a)
(b)
(c)
(d)
Write the equation of the given graph g(x). The original function was f(x) =|x|
g(x)
( ) 4
( ) 4
( ) 4
( ) 4
g x x
g x x
g x x
g x x
x
y