Power Functions and
Radical Equations
Lesson 4.8
Properties of Exponents

Given


m and m positive integers
r, b and p real numbers
r
b b  b
r
p
b 
r
b
m/ n
 b
p

b
m 1/ n
b
r p

b
bp
r p
1
b  r
b
r
r p
 b

1/ n m
b
m/ n
 b 
n
m
 b
n
m
Power Function

Definition


Where k and p
are constants
y  kx
p
Power functions are seen when dealing with
areas and volumes
4
3
v   r
3

Power functions also show up in gravitation
(falling bodies)
velocity  16t 2
Special Power Functions

Parabola
y = x2

Cubic function
y = x3

Hyperbola
y = x-1
Special Power Functions


y = x-2
yx
1
2
Text calls them
"root" functions

1
3
yx  x
3
Special Power Functions





Most power functions are similar to one of
these six
xp with even powers of p are similar to x2
xp with negative odd powers of p are similar
to x -1
xp with negative even powers of p are similar
to x -2
Which of the functions have symmetry?

What kind of symmetry?
Variations for Different Powers of p

For large x, large powers of x dominate
x5
x4
x3
x2
x
Variations for Different Powers of p

For 0 < x < 1, small powers of x dominate
x
x2
x3
x4
x5
Variations for Different Powers of p

Note asymptotic behavior of y = x -3 is more
extreme
0.5
20
10
y = x -3 approaches x-axis
more rapidly
0.5
y = x -3 climbs faster
near the y-axis
Think About It…


Given y = x –p for p a positive integer
What is the domain/range of the function?




Does it make a difference if p is odd or even?
What symmetries are exhibited?
What happens when x approaches 0
What happens for large positive/negative values
of x?
Equations with Radicals

Consider


Strategy is to square both sides to solve
May require this step twice


2 x  1  13
x  x  5 1
Make sure to check for extraneous roots

The squaring process can produce results that do
not satisfy the original equation
Assignment



Lesson 4.7
Page 350
Exercises 1 – 65 EOO
89, 91, 93