Power Functions
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Transcript of Power Functions
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:
1. n mn
m
aa 3
2
8 4643 3 28
Write in radical form.
Write in radical form.
Write each expression in radical form.
a.
b.
Answer:
Answer:
Write using rational exponents.
Answer:
Write using rational exponents.
Answer:
Write each radical using rational exponents.
a.
b.
Answer:
Answer:
Examples
.
7
4
5
5 98 5
9
8
11
3
4x 11 3)4( x
7 45
Power Function
• Definition Where k and p
are non zero constants
• Power functions are seen when dealing with areas and volumes
• Power functions also show up in gravitation (falling bodies)
py k x
34
3v r
216velocity t
Direct Proportions
• The variable y is directly proportional to x when: y = k * x• (k is some constant value)
• Alternatively
• As x gets larger, y must also get larger• keeps the resulting k the same
yk
x
This is a power function
This is a power function
Direct Proportions
• Example: The harder you hit the baseball The farther it travels
• Distance hit is directlyproportional to theforce 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
to x when
• Alternatively y = k * x -1
• As x gets larger, y must get smaller to keep the resulting k the same
ky
x
Again, this is a power function
Again, this is a power function
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 4
yx
Power Function
• Looking at the definition
• Recall from the chapter on shifting and stretching, what effect the k will have? Vertical stretch or compression
py k x
for k < 1
Power Functions
• Parabola y = x2
• Cubic function y = x3
• Hyperbola y = x-1
Power Functions
• y = x-2
•
•
1
2y x
133y x 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
x5x4
x3
x2
x
Variations for Different Powers of p
• For 0 < x < 1, small powers of x dominate
x5x4
x3x2
x
Variations for Different Powers of p
• Note asymptotic behavior of y = x -3 is more extreme
y = x -3 approaches x-axis more rapidly
0.5
0.510
20
y = x -3 climbs faster near the y-axis
1
x
2x
2x
1
x
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
• Find the values of m, t, and k
4
3( )g x kx
1
3( )f x mx(8,t)
Homework
• Pg. 189 1-49 odd