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# Power Functions - Morgan Park High School

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```Power Functions
Objectives
вЂў Students will:
п‚§ Have a review on converting radicals to
exponential form
п‚§ Learn to identify, graph, and model power
functions
Converting Between Radical and
Rational Exponent Notation
вЂў
An exponential expression with exponent of the form вЂњm/nвЂќ
can be converted to radical notation with index of вЂњnвЂќ, 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
вЂў Definition
п‚§ Where k and p
y пЂЅ kпѓ—x
p
are non zero constants
вЂў Power functions are seen when dealing with
areas and volumes
4
пЃ° пѓ—r
vпЂЅ
3
3
вЂў Power functions also show up in gravitation
(falling bodies) velocity пЂЅ пЂ­ 16 t 2
Direct Proportions
вЂў The variable y is directly proportional to x
when:
This is a power
function
y=k*x
вЂў (k is some constant value)
вЂў Alternatively
k пЂЅ
y
x
вЂў As x gets larger, y must also get larger
вЂў keeps the resulting k the same
Direct Proportions
вЂў Example:
п‚§ The harder you hit the baseball
п‚§ The farther it travels
вЂў Distance hit is directly
proportional to the
force of the hit
Direct Proportion
вЂў Suppose the constant of proportionality is 4
п‚§ Then y = 4 * x
п‚§ What does the graph of this function look like?
Inverse Proportion
вЂў The variable y is inversely proportional
k
yпЂЅ
to x when
x
Again, this is a power
вЂў Alternatively
function
y = k * x -1
вЂў As x gets larger, y must get smaller to keep
the resulting k the same
Inverse Proportion
вЂў Example:
If you bake cookies
at a higher
temperature,
they take less time
вЂў Time is inversely proportional to temperature
Inverse Proportion
вЂў Consider what the graph looks like
п‚§ Let the constant or proportionality k = 4
п‚§ Then
yпЂЅ
4
x
Power Function
p
вЂў Looking at the
y пЂЅ kпѓ—x
definition
вЂў Recall from the chapter on shifting and
stretching, what effect the k will have?
п‚§ Vertical stretch or compression
for k < 1
Power Functions
вЂў Parabola
y = x2
вЂў Cubic function
y = x3
вЂў Hyperbola
y = x-1
Power Functions
вЂў y = x-2
1
2
y
пЂЅ
x
вЂў
1
вЂў y пЂЅ x3 пЂЅ
3
x
Power Functions
вЂў Most power functions are similar to one of
these six
вЂў xp with even powers of p are similar to x2
вЂў xp with negative odd powers of p are
similar to x -1
вЂў xp with negative even powers of p are
similar to x -2
вЂў Which of the functions have symmetry?
п‚§ What kind of symmetry?
Variations for Different Powers of p
вЂў For large x, large powers of x dominate
x5
x4
x3
x2
x
Variations for Different Powers of p
вЂў For 0 < x < 1, small powers of x dominate
x
x2
x3
x4
x5
Variations for Different Powers of p
вЂў 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 ItвЂ¦
вЂў Given y = x вЂ“p for p a positive integer
вЂў What is the domain/range of the function?
п‚§ Does it make a difference if p is odd or even?
вЂў What symmetries are exhibited?
вЂў What happens when x approaches 0
вЂў What happens for large positive/negative
values of x?
Finding Values
4
g ( x ) пЂЅ kx 3
1
п‚џ
(8,t)
вЂў Find the values of m, t, and k
f (x) пЂЅ mx 3
Homework
вЂў Pg. 189 1-49 odd
```
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