Translations and Combinations
Algebra 5/Trigonometry
Shifting
What does it mean to shift?
(interactive)
Vertical Shift c Units Up: h(x) = f(x) + c
Vertical Shift c Units Down: h(x) = f(x) – c
Vertical Shift c Units Right: h(x) = f(x - c)
Vertical Shift c Units Left: h(x) = f(x + c)
Try These
f ( x)  x
g ( x)  x  2
m( x )  x  2
h( x) | x |
j ( x) | x  1 |
n( x) | x | 1
Reflecting
What does it mean to reflect?
(interactive)
Reflection in the x-axis: h(x) = -f(x)
Reflection in the y-axis: h(x) = f(-x)
Try These
f ( x)  x
g ( x)   x
h( x) | x |
j ( x) |  x |
Stretching
Try These
f ( x)  x
g ( x)  3 x
m( x ) 
1
x
2
h( x ) | x |
1
|x|
2
n( x )  2 | x |
j ( x) 
Arithmetic Combinations of
Functions
Sum:
(f + g)(x) = f(x) + g(x)
Difference: (f - g)(x) = f(x) - g(x)
Product:
(fg)(x) = f(x) ∙ g(x)
f ( x)
f
Quotient: ( )(x) = g ( x) , g ( x)  0
g
Compute
f(x) = 4x + 2 and g(x) = 2x – 1
f(x) + g(x)
f(x) - g(x)
fg(x)
( f )(x)
g
Now Try These
f(x) = x2 - 9 and g(x) = 3x – 1
f(x) + g(x)
f(x) - g(x)
fg(x)
( f )(x)
g
Composition of Two Functions
The composition of the function f with the
function g is given by (f o g)(x) = f(g(x))
The domain of (f o g) is the set of all x in the
domain of g such that g(x) is in the domain
of f.
Compositions
Given: f(x) = x2 - 9 and g(x) = 3m,
find f(g(x)).
find g(f(x)).
Now Try These
Given f(x) = 6x + 2 and g(x) = 2x – 1
find f(g(x)).
find g(f(x)).