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### 2.2 Power Function w/ modeling

2.3 Polynomials of higher degree with modeling

Power function

- A power function is any function that can be written in the form:

f(x) = kxa, where k and a ≠ 0

- Monomial functions: f(x) = k or f(x) = kxn
- Cubing function: f(x) = x3
- Square root function: f(x) = x
- Higher degree: (standard form)

p(x) = anxn + an-1xn-1 + an-2xn-2 +… + a1x + a0

Analyzing Power functions

- To do you will find the following:

domain

range

continuous/discontinuous

increasing/decreasing intervals

even (symmetric over y-axis)/odd function (symmetric over origin)

boundedness

local extrema (max/min)

asymptotes (both VA and HA)

end behavior (as it approaches both -∞ & ∞

Example: analyze f(x) = 2x4

- Power: 4, constant: 2
- Domain: (-∞,∞)
- Range: [0, ∞)
- Continuous
- Increasing: [0,∞), decreasing: (-∞, 0]
- Even: symmetric w/ respect to y-axis
- Bounded below
- Local minimum: (0, 0)
- Asymptotes: none
- End behavior: lim 2x4 = ∞, lim 2x4 = ∞

x -∞ x ∞

(in other words, on the left the graph goes up, on the right the graph goes up)

Try one: f(x) = 5x3

Investigating end behavior

- When determining end behavior you need to determine if the graph is going to -∞ (down) or ∞ (up) as x -∞ & as x ∞
- Example: graph each function in the window [-5, 5] by [-15, 15] . Describe the end behavior using lim f(x) & lim f(x)

x -∞ x ∞

- f(x)=x3+2x2-5x-6: -∞,∞
- Do exploration

Finding 0’s

- Find the 0’s of f(x) = 3x3 – x2 -2x
- Algebraically: factor

3x3 – x2 -2x = 0

x(3x2 – x -2) = 0

x(3x + 2)(x - 1) = 0

x = 0, 3x + 2 = 0, x – 1= 0

x = 0, x = -2/3, x = 1

- Graphically: graph & find the 0’s (2nd trace #2)

Zeros of Polynomial function

- Multiplicity of a Zero

If f is a polynomial function and (x-c)m is a factor of f but (x – c)m+1 is not, then c is a zero of multiplicity m of f.

- Zero of Odd and Even Multiplicity:

If a poly. funct. f has a real 0 c of odd multiplicity, then the graph of f crosses the x-axis at (c, 0) & the value of f changes sign at x=c

if a poly. funct. f has a real 0 c of even multiplicity, then the graph doesn’t cross the x-axis at (c, 0) & the value of f doesn’t change sign at x=0

Multiplicity

- State the degree, list the 0’s, state the multiplicity of each 0 and what the graph does at each corresponding 0. then sketch the graph by hand (look at exponents: if 1 then crosses, if odd then wiggles, if even then tangent)
- Examples: 1)f(x) = x2 + 2x – 8

degree: 2, 0’s: 2, -4, graph: crosses at x=-4 & x= 2

2) f(x) = (x + 2)3(x-1)(x-3)2

Now finish paper

Writing the functions given 0’s

- 0, -2, 5

1) rewrite w/ a variable & opposite sign:

x(x + 2)(x – 5)

2) simplify

x(x2 -3x + -10)

x3 – 3x2 – 10x

Try one: -6, -4, 5

Homework:

- p. 182 #1-10, 27-30
- p. 193-194 #2-6 even, 25-28 all, 34-38 even, 39-42 all, 49-52 all, 53, 54

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