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Power Functions - Morgan Park High School

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Power Functions
Objectives
• Students will:
 Have a review on converting radicals to
exponential form
 Learn to identify, graph, and model power
functions
Converting Between Radical and
Rational Exponent Notation
•
An exponential expression with exponent of the form “m/n”
can be converted to radical notation with index of “n”, and
vice versa, by either of the following formulas:
m
1.
a
n
2
пЂЅ
n
a
m
8 пЂЅ
3
3
8
2
пЂЅ
3
64 пЂЅ 4
Write
in radical form.
Write
in radical form.
Write each expression in radical form.
a.
Answer:
b.
Answer:
Write
using rational exponents.
Answer:
Write
using rational exponents.
Answer:
Write each radical using rational exponents.
a.
Answer:
b.
Answer:
Examples
4
7
5 пЂЅ
7
5
4
9
5
8 пЂЅ
9
85
3
4x
11
пЂЅ
11
(4 x)
3
.
Power Function
• Definition
 Where k and p
y пЂЅ kпѓ—x
p
are non zero constants
• Power functions are seen when dealing with
areas and volumes
4
пЃ° пѓ—r
vпЂЅ
3
3
• Power functions also show up in gravitation
(falling bodies) velocity пЂЅ пЂ­ 16 t 2
Direct Proportions
• The variable y is directly proportional to x
when:
This is a power
function
y=k*x
• (k is some constant value)
• Alternatively
k пЂЅ
y
x
• As x gets larger, y must also get larger
• keeps the resulting k the same
Direct Proportions
• Example:
 The harder you hit the baseball
 The farther it travels
• Distance hit is directly
proportional to the
force of the hit
Direct Proportion
• Suppose the constant of proportionality is 4
 Then y = 4 * x
 What does the graph of this function look like?
Inverse Proportion
• The variable y is inversely proportional
k
yпЂЅ
to x when
x
Again, this is a power
• Alternatively
function
y = k * x -1
• As x gets larger, y must get smaller to keep
the resulting k the same
Inverse Proportion
• Example:
If you bake cookies
at a higher
temperature,
they take less time
• Time is inversely proportional to temperature
Inverse Proportion
• Consider what the graph looks like
 Let the constant or proportionality k = 4
 Then
yпЂЅ
4
x
Power Function
p
• Looking at the
y пЂЅ kпѓ—x
definition
• Recall from the chapter on shifting and
stretching, what effect the k will have?
 Vertical stretch or compression
for k < 1
Power Functions
• Parabola
y = x2
• Cubic function
y = x3
• Hyperbola
y = x-1
Power Functions
• y = x-2
1
2
y
пЂЅ
x
•
1
• y  x3 
3
x
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
1
1
x
x
пЂ­2
x
пЂ­2
x
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?
Finding Values
4
g ( x ) пЂЅ kx 3
1
п‚џ
(8,t)
• Find the values of m, t, and k
f (x) пЂЅ mx 3
Homework
• Pg. 189 1-49 odd
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