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.
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Transcript 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.
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
r p
b
b
1/ n m
b
m/ n
r
1
r
b
b
n
m
b
n
m
Power Function
Definition
Where k and p
are constants
y kx
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
yx
1
2
Text calls them
"root" functions
1
3
yx 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