Barnett/Ziegler/Byleen
Precalculus: Functions & Graphs, 5th Edition
Chapter Two
Graphs & Functions
Copyright © 2001 by the McGraw-Hill Companies, Inc.
Symmetry
y
(–a, b )
y
(a, b )
(a, b )
x
x
(a, –b )
(a)
Symmetry with respect to y axis
(b) Symmetry with respect to x axis
2-1-13-1
Symmetry
y
y
(–a, b )
(a, b )
(a, b )
x
x
(–a, –b )
(–a, –b )
(c)
Symmetry with respect to origin
(a, –b )
(d) Symmetry with respect to y axis,
x axis, and origin
2-1-13-2
Distance Between Two Points
y
y2
P2 (x 2 , y 2 )
d(P1 , P2 )
0
x1
y1
P1 (x 1 , y 1 )
|x 2
d(P1, P2)
=
x
|y 2
– y 1 |
x2
(x 2 , y 1 )
– x 1 |
(x2 – x1)2 + (y2 – y1)2
2-1-14
Circle
A circle is the set of all points in a plane equidistant from a fixed
point. The fixed distance is called the radius, and the fixed point is
called the center.
Standard Equation of a Circle
y
1. Circle with radius r and center at (h, k):
P(x, y )
(x – h)2 + ( y – k)2 = r 2
r>0
r
2. Circle with radius r and center at (0, 0):
x2 + y2 = r 2
C(h, k )
r>0
x
2-1-15
Geometric Interpretation of Slope
Line
Rising as x moves
from left to right
Slope
Example
Positive
Falling as x moves
from left to right
Negative
Horizontal
0
Vertical
Not defined
2-2-16
Equations of a Line
Standard form
Ax + By = C
A and B not both 0
Slope-intercept form
y = mx + b
Slope: m; y intercept: b
Point-slope form
y – y1 = m(x – x1)
Slope: m; Point: (x1, y1)
Horizontal line
y=b
Slope: 0
Vertical line
x=a
Slope: Undefined
2-2-17
Vertical Line Test for a Function
An equation defines a function if each vertical line in the rectangular coordinate
system passes through at most one point on the graph of the equation.
If any vertical line passes through two or more points on the graph of an equation,
then the equation does not define a function.
y
y
3
6
(a) y3 – x = 1
–3
0
3
x
–6
0
6
x
–6
–3
(b) y2 – x2 = 9
2-3-18
Increasing, Decreasing, and Constant Functions
f (x)
g(x)
5
10
f (x) = – x 3
–5
g(x) = 2 x + 2
x
0
5
–10
(a) Decreasing on (–  )
–5
0
x
5
–5
(b) Increasing on (– ,  )
2-4-19-1
Increasing, Decreasing, and Constant Functions
h(x)
p(x)
5
5
h(x) = 2
–5
0
5
–5
(c) Constant on (– )
x
–5
5
2
p(x) = x – 1
x
–5
(d) Decreasing on (–, 0]
Increasing on [0, )
2-4-19-2
Properties of a Quadratic Function and Its Graph
f(x) = ax2 + bx + c = a(x - h)2 + k a  0
1.
The graph of f is a parabola:
2. Vertex: (h, k) (Parabola increases on one side of the vertex and decreases on the other.)
3. Axis (of symmetry): x = h (Parallel to y axis)
4. f(h) = k is the minimum if a > 0 and the maximum if a < 0.
5. Domain: All real numbers
Range: (–, k] if a < 0 or [k, ) if a > 0
2-4-20
Six Basic Functions
Absolute Value Function
g(x)
Identity Function
f(x)
5
5
–5
5
x
–5
–5
1.
5
x
g(x) = |x|
f(x) = x
2.
2-5-21-1
Six Basic Functions
Square Function
Cube Function
m(x)
h(x)
5
5
x
–5
h(x) = x
3.
2
–5
5
x
5
–5
m(x) = x
3
4.
2-5-21-2
Six Basic Functions
Cube-Root Function
Square-Root Function
p(x)
n(x)
5
5
x
5
n(x) =
5.
x
–5
5
–5
p(x) =
3
x
x
6.
2-5-21-3
Composite Functions
Given functions f and g, then f ° g is called their composite and is defined
by the equation
(f ° g)(x) = f [g (x)]
The domain of f ° g is the set of all real numbers x in the domain of g where
g(x) is in the domain of f.
Domain f o g
(f o
g )( x) = f [g(x)]
f og
Range f og
g
f
x
Domain g
g( x )
Range g
Domain f
Range f
2-5-22
y
Vertical Shifts
5
(1, 1)
–5
x
0
5
(a) f: y = x 2
–5
y
y
5
5
(1, 3)
–5
x
0
5
–5
x
0
5
(1, –2)
–5
(b) F: y = x 2 + 2
The graph of y = x 2 + 2 is the same
as the graph of y = x 2 shifted up
two units.
–5
(c) G: y = x 2 – 3
The graph of y = x 2 – 3 is the same
as the graph of y = x 2 shifted down
three units.
2-5-23
y
5
Horizontal Shifts
(1, 1)
–5
0
x
5
(a) f: y = x 2
y
–5
y
5
5
(4, 1)
(–1, 1)
–5
0
–5
(b) P: y = (x + 2) 2
The graph of y = (x + 2) 2 is the same
as the graph of y = x 2 shifted to the
left two units.
x
5
–5
x
0
5
–5
(c) Q: y = (x – 3) 2
The graph of y = (x – 3) 2 is the same
as the graph of y = x 2 shifted to the
right three units.
2-5-24
Reflection, Expansion, and Contraction
y
y
10
10
(b) R: y = –x2
Reflection
(a) f: y = x2
(2, 4)
–10 x
0
10
–10 –10 x
0
(2, –4)
10
–10 y
y
10
10
(d) T: y = x2
Contraction
(2, 8)
(c) S: y = 2x2
Expansion
–10 x
0
–10 10
(2, 2)
–10 0
x
10
–10 2-5-25
One-to-One Functions
A function is one-to-one if no two ordered pairs in the function have the same
second component and different first components.
Horizontal Line Test
A function if one-to-one if and only if each horizontal line intersects the graph
of the function in at most one point.
y
y
y = f (x)
y = f (x)
(a, f (a))
a
(a)
(b, f (b))
b
f(a) = f(b) for a  b;
f is not one-to-one
(a, f (a))
x
a
(b)
x
Only one point has ordinate f(a);
f is one-to-one
2-6-26
Increasing and Decreasing
Functions
y
If a function f is increasing throughout its
domain or decreasing throughout its
domain, then f is a one-to-one function.
x
(a) An increasing function
is always one-to-one
y
y
x
(b) A decreasing function
is always one-to-one
x
(c) A one-to-one function is not
always increasing or decreasing
2-6-27
Finding the Inverse of a Function f
Step 1.
Find the domain of f and verify that f is one-to-one. If f is not oneto-one, then stop, since f–1 does not exist.
Step 2.
Solve the equation y = f(x) for x. The result is an equation of the
form x = f–1(y).
Step 3.
Interchange x and y in the equation found in step 2. This
expresses f–1 as a function of x.
Step 4.
Find the domain of f–1. Remember, the domain of f–1 must be the
same as the range of f.
Check your work by verifying that:
f–1[f(x)] = x
for all x in the domain of f
f[f–1(x)] = x
for all x in the domain of f–1
and
2-6-28
Symmetry Property for the
Graphs of f and f –1
y
5
x
–5
5
The graphs of y = f(x) and y = f–
1(x) are symmetric with respect
to the line y = x.
–5
(a) (a, b) and (b, a) are symmetric with
respect to the line y = x
y
y
y = f (x) y = x
y = f -1 (x)
y=x
10
5
y = f -1 (x)
x
–5
5
y = f (x)
–5
(b) f(x) = 2x – 1
f–1(x) = x +
x
0
(c) f(x) = x – 1
f –1(x) = x2 + 1, x  0
2-6-29