1.3 - New Functions From
Old Functions
1
Translation: f (x) + k
Graph y1 = x2 on your graphing calculator and then graph
y2 given below to determine the movement of the graph
of y2 as compared to y1. Generalize the effect of k.
y2
Direction of
Translation
Units
Translated
Value of k
x2 – 4
x2 – 2
x2 + 2
x2 + 4
2
Translation: f (x – h)
Graph y1 = x2 on your graphing calculator and then
graph y2 given below to determine the movement of the
graph of y2 as compared to y1. Generalize the effect of h.
y2
Direction of
Translation
Units
Translated
Value of h
(x – 4)2
(x – 2)2
(x + 2)2
(x + 4)2
3
Horizontal and Vertical Translations
f (x – h) + k
• If h < 0, the graph shifts h units _______. If
h > 0, the graph shifts h units _______. The
value of h causes a _________________
translation.
• If k > 0, the graph shifts k units _______. If
k < 0 then the graph shifts k units ______.
The value of k causes a ______________
translation.
4
Examples
Use the library of functions to sketch a graph of each
of the following without using your graphing
calculator.
(a) f(x) = (x + 3)2 – 2
(b) g(x) = | x – 2 | + 3
(c) h( x)  x  1  2
3
(d) h( x)  ( x  3)  1
5
x- and y-Axis Reflections
• The graph of y = - f(x) is the
same as graph of f(x) but
reflected about the __-axis.
• The graph of y = f(-x) is the
same as graph of f(x) but
reflected about the __-axis.
Graph with your
graphing calculator:
y1  x
y2   x
y2   x
6
Compression and Stretches
Sketch the following using your graphing calculator
y1 = |x|
y2 = 3|x|
y2 = (⅓) |x|
• If |a| > 1, the graph of y = af(x) is ______________
vertically or _____________ horizontally…
• 0 < |a| < 1 the graph of y = af(x) is _____________
vertically or _____________ horizontally…
• … as compared to y = f(x).
7
Examples
Use the library of functions to sketch a graph of each
of the following without using your graphing
calculator.
(a) f(x) = - (x + 3)2 – 2
(b) g(x) = 2| x – 1 | + 3
(c) h( x)  12 x  1  2
3
(d)h( x)  2( x  3)  1
8
The Algebra of Functions
9
The sum f + g is the function defined by
(f + g)(x) = f(x) + g(x)
The domain of f + g consists of numbers x that are in the
domain of both f and g (the intersection of the domains).
The difference f - g is the function defined by
(f - g)(x) = f(x) - g(x)
The domain of f - g consists of numbers x that are in the
domain of both f and g (the intersection of the domains).
10
The product f ∙ g is the function defined by
(f ∙ g)(x) = f(x) ∙ g(x)
The domain of f ∙ g consists of numbers x that are in the
domain of both f and g (the intersection of the domains).
The quotient f / g is the function
f 
f ( x)
  ( x) 
g ( x)
g
The domain of f / g consists of all x such that x is in the
domain of f and g and g(x) ≠ 0 (the intersection of the
domains).
11
Composition of Functions
Given two functions f and g, the composite
function is defined by
(f ◦ g)(x) = f(g(x))
Read “f composite g of x”
The domain of f ◦ g is the set of all numbers x in
the domain of g such that g(x) is in the domain of
f. Note: In general (f ◦ g)(x) ≠ (g ◦ f)(x)
12
Range of g
(equal to or a subset of)
Domain of g
x
Domain of f
g
g(x)
Range of f
f
f(g(x))
Domain of f(g)
(f ◦ g)(x) = f(g(x))
Or f(g)
Range of f(g)
13
Composition of Functions
Let f(x) = 2x – 3 and g(x) = x 2 – 5x. Determine
(f ◦ g)(x).
(f ◦ g)(x) = f(g(x)) = f(x 2 – 5x)
= 2(xx 2 – 5x) – 3
= 2x 2 – 10x – 3
14
Examples
(a) Let f ( x) 
1 x
x 1
and g(x) = x2 – 2. Determine (i)
(f ◦ g)(x) and (ii) (g ◦ f)(x). Determine the domains of each.
(b)If y = cos (x2 – 2) and y = (f ◦ g)(x), determine f and g.
(c) If y = esin(x+5) and y = (f ◦ g ◦ h)(x), determine f, g, and h.
15