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Section 2.5 Transformations of Functions
* Graphs of Common Functions
Table 2.3 (page 242 on the textbook)
* Vertical Shifts
Let f be a function and c a positive real number.
The graph of y  f (x)  c is the graph of y  f (x) shifted c units vertically upward.
The graph of y  f (x)  c is the graph of y  f (x) shifted c units vertically downward.
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Example 1) Use the graph of f (x)  x to obtain the graph of g(x)  x  3.
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* Horizontal Shifts
Let f be a function and c a positive real number.
The graph of y  f (x  c) is the graph of y  f (x) shifted to the left c units.
The graph of y  f (x  c) is the graph of y  f (x) shifted to the right c units.
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Example 2) Use the graph of f (x)  x to obtain the graph
of g(x)  x  4 .
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Example 3) Use the graph of f (x)  x 2 to obtain the graph of g(x)  (x  2) 2  3 .
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*Reflections of Graphs
The graph of y   f (x) is the graph of y  f (x) reflected about the x-axis.
Example 4) Use the graph of f (x)  x to obtain the graph of g(x)   x .
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The graph of y  f (x) is the graph of y  f (x) reflected about the y-axis.
Example 5) Use the graph of f (x)  3 x to obtain the graph of g(x)  3 x .
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* Vertical Stretching and Shrinking
Let f be a function and c a positive real number.
If c  1, the graph of y  cf (x) is the graph of y  f (x) vertically stretched by
multiplying each of its y-coordinates by c.
If 0  c  1, the graph of y  cf (x) is the graph of y  f (x) vertically shrunk by
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multiplying each of its y-coordinates by c.
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Example 6) Use
the graph of f (x)  x to
 obtain the graph of g(x)  2 x .
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* Horizontal Stretching and Shrinking
Let f be a function and c a positive real number.
If c  1, the graph of y  f (cx) is the graph of y  f (x) horizontally shrunk by dividing
each of its x-coordinates by c.
If 0  c  1, the graph of y  f (cx) is the graph of y  f (x) horizontally stretched by
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dividing each of its y-coordinates by c.
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Example 7) Use the graph of f (x)  x to obtain the graphs of g(x)  2x and
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h(x)  x .
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* Sequences of Transformations (HSRV)
Let f (x) be a function and c be a positive real number.
To Graph:
Draw the Graph of f
and:
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Horizontal shifts
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Stretching Vertical
or
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Shrinking
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Horizontal
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Reflection
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About
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x-axis
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About the
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y-axis
Vertical shifts
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Changes of
Coordinates
(each point)
y  f (x  c)
Shift c units Left.
Subtract c from each xcoordinate.
y  f (x  c)
Shift c units Right.
Add c from each xcoordinate.
Multiply each yy  cf (x),  Stretch vertically by
multiplying y by c .
coordinate by c .
c 1
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Shrink
vertically
by
Multiply each yy  cf (x),
multiplying y by c .
coordinate by c.
0  c 1
Shrink 
horizontally by Divide
y  f (cx) ,
 each x-coordinate
dividing
x
by
c.
by
c.
c 1
Stretch
horizontally by Divide each x-coordinate
y  f (cx) ,
dividing x by c.
by c.
0  c 1
y   f (x)
Reflect about x-axis.
Replace each ycoordinate with its
opposite.
y  f (x)
Reflect about y-axis.
Replace each xcoordinate with its
opposite.
y  f (x)  c
Raise by c units.
Add c from each ycoordinate.
y  f (x)  c
Lower by c units.
Subtract c from each ycoordinate.
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A function involving more than one transformation can be graphed by performing
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transformations inthe following order.
1. Horizontal shifting
2. Stretching or shrinking
3. Reflecting
4. Vertical shifting
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Example 8) Use the graph of f (x)  x 2 to graph g(x)  2(x  3) 2  3 .
Step 1 Find the basic function unless it is given.
 of the changesfrom the basic function to the given function in
Step 2 Find the sequence
the HSRV order.
x2
2(x  3) 2  3
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Step 3 Describe the changes of graph of each transformation.
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2.
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Step 4 Pick several points of the basic function of graph and find the transformations of
each point
Step 5 Draw graphs with the points.
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Example 9) Use the graph of y  f (x) to graph g(x)  
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f (2x)  3 .
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Step 1 Find the basic function unless it is given.
Step 2 Find the sequence of the changes from the basic function to the given function in
the HSRV order.
f (x)
1
f (2x)  3
2
Step 3 Describe the changes of graph of each transformation.
1.
2.
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4.
Step 4 Pick several points of the basic function of graph and find the transformations of
each point
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Step 5 Draw graphs with the points.
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