Transformations of Equations and Functions Section 1.3
Shifting
2
Consider the graph of the relation x  y
2
 1 , the unit circle.
2
2
If we shift the center to the point (a,b), we have the relation ( x  a )  ( y  b )  1 .
2
2
Example 1: The circle ( x  2 )  ( y  3 )  1 has center ( 2 ,  3 ) . The unit circle has been shifted
2 units to the right and 3 units down.
Rule: To shift a graph to the right a units for a > 0, replace x with ( x  a ) in the relation.
To shift the graph to the left a units for a > 0, replace x with ( x  a ) in the relation.
Similarly, To shift a graph b units up, for b > 0, replace y with ( y  b ) .
To shift down b units, for b > 0, replace y with ( y  b ) .
The same rules work in functions.
Example 2: y 
y3
1 x
1  ( x  2)
2
2
shifted to the right 2 units and down 3 units becomes
which we write as y 
2
1  ( x  2)  3
Distortions/ Stretching & Shrinking
Let us transform the unit circle into an ellipse centered at (0,0). If the horizontal axis is the interval
[-2,2] and the vertical axis is [-3,3] we are stretching horizontally by a factor of 2 and vertically by a
2
2
x
 y
factor of 3. The equation of the ellipse is       1 .
2
3
Rule: To stretch a graph horizontally by a factor of c, for c > 1, replace x by
x
in the relation or function.
c
To stretch vertically by a factor of c, for c > 1, replace y by
y
c
Shrinking is the same except c < 1.
in the relation or function.
Example 3: Stretch the function y 
x vertically by a factor of 3 and horizontally by a factor of 4.
Write the new function as y = f(x).
Now shift this function to the right 5 units and down 6 units. Write the new function as y = g(x).
Reflections To reflect the graph across the x-axis, replace y with (-y) in the equation.
To reflect the graph across the y-axis replace x with (-x) in the equation.
Example 4: Sketch the graphs to see how reflections work.
2
a) y  x ,
Compositions
y  x
2
b) y 
x  2,
y 
2 x
f  g ( x )  f ( g ( x )) means to substitute g(x) for the variable in the function f.
We perform the operation of g on x and then perform the operation f on the result.
Example 5: f ( u ) 
u
g ( x)  1  x
2
f ( g ( x )) 
1 x
2
Example 6: Find g  f ( x )  g ( f ( x )) for the functions of example 5.
3
Example 7: Find f and g so that h ( x )  ( 2  sin x )  4  f ( g ( x )) There are different possibilities.