Pre-Calculus Notes 1.5-1.9
1.5 Analyzing Graphs
I. Graphs
Y = 2x – 3
means the same as
f ( x ) = 2x - 3
Is y 2  x a function? Can we write it in function notation?
y
Let’s graph it…
x
II. Zeros of a Function
x – intercepts
(x,0)
Example:
1. Find the zero(s) for each of the following functions.
a) f  x   2 x2  x  1
b)
g  x   3 x 1
III. Increasing and Decreasing
c)
h  x 
x 5
x2
y
f  x   x 3  3x 2
Increasing:
Decreasing:
x
2. Graph f  x   3x2  2 x  1 on the graphing calculator and find the maximum,
minimum, zeros and the intervals for which the function increases and decreases.
y
IV. Rate of Change & Average Rate of Change
x
3. Find the average rate of change for f  x   x2  2 x from
a)
x1  1 to x2  3
b) x1  0 to x2  1
y
x
1.6 Parent Graphs
I. Linear and Squaring Functions
A) Linear
f ( x ) = mx + b
D :  ,  
R :  ,  
Increa sin g : m  0
Decrea sin g : m  0
Cons tan t : m  0
** f ( x ) = c is a constant function m = 0
** f ( x ) = x is the identity function m = 1
Examples:
1. Write the linear function given that f ( 2 ) = 4 and f ( 5 ) = 1. Now graph.
y
D:
R:
x
y
B) Squaring Function
f ( x)  x 2
D:
x
R:
Increa sin g :  0,  
Decea sin g :  , 0 
Relative Minimum: ( 0 , 0 )
II. Cubic, Square Root and Reciprocal
A)
y
f ( x)  x 3
D:
R:
x
y
B) f ( x)  x
D:
R:
x
y
1
C) f ( x ) 
x
D:
R:
x
III. Step Functions and Piecewise Functions
y
A) Step Function
f ( x)   x 
x
**show in grapher
Examples:
1. For f ( x)   x  4 evaluate:
a) f ( -2 )
b) f ( 4 )
B) Piecewise Functions
c) f ( 2.5 )
d) f ( ½ )
y
 x  1ifx  2 
2. Graph f ( x)  

1  xifx  2 
x
IV. All Parent Graphs
1.
2.
f ( x)  c
f ( x)  x
y
y
x
3.
x
4.
f ( x)  x
y
f ( x)  x
y
x
5.
x
6.
f ( x)  x 2
y
f ( x)  x 3
y
x
7.
f ( x) 
1
x
x
8.
y
x
f ( x)   x 
y
x
1.7 Transformations of Functions
I. Shifting Graphs
f ( x)  x 2
f ( x)  x 2  2
f ( x)   x  2 
f ( x)  x 2  2
y
y
y
x
f ( x)   x  2 
2
y
x
x
x
Examples
1. For the function f ( x)  x3 , sketch the following transformations:
b) f ( x)   x  3  1
a) f ( x)  x3  3
3
y
y
x
2
x
II. Reflections
a) Reflection in the x-axis means h( x)   f ( x)
b) Reflections in the y-axis means h( x)  f ( x)
Example
2. Compare the following graphs with f ( x)  x  2 .
a) g ( x)   x  2
c) p( x)    x  2 
b) h( x)   x  4
III. Non-Rigid Transformations ( Changes the Shape )
Example
3. Compare the following functions with
f ( x)  x 2
1
b) h( x)  x 2
3
a) g ( x)  3x 2
y
y
x
x
4. Compare the following functions with f ( x)  x 2  3
1 
b) h( x)  f  x 
3 
a) g ( x)  f (3x)
5. For the function g ( x)  2  x  1  3 ,
2
a) Name the Parent Graph f ( x )
b) Describe transformations from f ( x ) to g ( x )
c) Sketch the graph of g ( x )
y
x
1.8 Combinations of Functions
I. Arithmetic Combinations
Examples:
1. Let
f ( x) 
5
and
x 1
g ( x) 
1
find:
x 1
2
a)
 f  g  x  
b)
 f  g  x  
c)
 f  g  x  
f 
d)    x  
g
II. Composition of Functions
f
g  x   f  g  x  
Examples:
2. Let f ( x)  3x  4 and g ( x)  2 x 2  3 , find the following:
a)
f
g  x  
c) f  g  1  
b) g  f  x   
d)
g
f  2 
3. Find the Domain of f(g(x)) given:
a) f ( x)  x and g ( x)  x  5
b) f ( x)  x  4 and g ( x)  3  x
III. Decomposing a Composite Function
a) Find f ( x ) and g ( x ) given that f ( g ( x)) 
4  x2
.
9
1.9 Inverse Functions
I. Definition
Let f and g be 2 functions such that f ( g ( x))  x for every x in the domain
of g and g ( f ( x))  x for every x in the domain of f.
f 1  f  x    x and
f  f 1  x    x
Examples:
1. Find the inverse.
a) f ( x)  8 x
b) f ( x)  3x  5
2. Verify that f ( x)  5 x  8 and g ( x ) 
1
8
x  are inverses of one another.
5
5
3. Find the inverse function of f ( x)  2 x  3 and sketch both.
y
x
II. Horizontal Line Test and One-to-One Functions
4. Graph
Find f
1
 x
f ( x)  x  2 . Does the graph of f(x) pass the horizontal line test?
and graph both. Should we restrict the domain of f 1  x  so that we
have a function?
y
x