1-2 Transformations of Graphs
(learn the graphs of six “elementary functions”, p. 23)
Horizontal shifts
reference:
f(x) = x2



shift left 2:
f(x + 2) = (x + 2)2
y



shift right 2:
f(x - 2) = (x - 2)2
y
x



y
x
   

       

       










x
    
Vertical shifts
reference:
f(x) = x2



shift up 2:
f(x) + 2 = x2 + 2
y



x
shift down 2:
f(x) - 2 = x2 - 2
y



x
   

       

       










1-2
y
x
    
p. 1
Expansions and Contractions (Omit)
Reflections across the X-axis
reference:
f(x) = x2
reflected in the x-axis:
-f(x) = - x2



y



x
   

y
x
   

    
    






REFLECTIONS ACROSS THE Y-AXIS
reference:
f(x) = x3
reflected in the y-axis:
f(-x) = (-x)3



   




1-2
y



x
    
   

y
x
    



p. 2
Combinations of transformations
g(x) = -(x - 3)2 + 2
to graph:
(1) note that basic function is f(x) = x2
(2) find transformations that get from x2  -(x - 3)2 + 2
x2

2
(x - 3) 
-(x - 3)2 
(x - 3)2
-(x - 3)2
-(x - 3)2 + 2
x2



   

(x - 3)2
y



x
    



   




y
x
   

    



-(x - 3)2



Transformation
shift right by 3
reflect across x-axis
shift up by 2
-(x - 3)2 + 2
y



x
    
   

y
x
    



When you do problems of this type:
 you must state the transformations as shown above
 but you do not need to draw the intermediate graphs
1-2
p. 3
Piecewise-Defined Functions
Example:
f(x) = x
= 1+x2
x0
x>0






f is defined using functional notation and a formula
different formula for different parts of the domain
we say that f is piecewise-defined
it "jumps" at x = 0 from 0 to 1
x = 0 is a point of discontinuity
the "solid dot" at (0,0) is used to indicate clearly that the
straight line goes all the way to (0,0) inclusive
 that is, that f(0) = 0
 the "open dot" at (0,1) indicates that the curve extends all the
way to (0,1), but doesn't include that point
 discontinuities are always drawn in this way to indicate the
value of the function at the point of discontinuity
1-2
p. 4