2.2 Power Function w/ modeling2.3 Polynomials of higher degree with modeling

Power functionA 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 functionsTo do you will find the following:

domainrangecontinuous/discontinuousincreasing/decreasing intervalseven (symmetric over y-axis)/odd function (symmetric over origin)boundednesslocal 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 behaviorWhen 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’sFind the 0’s of f(x) = 3x3 – x2 -2xAlgebraically: factor

3x3 – x2 -2x = 0x(3x2 – x -2) = 0x(3x + 2)(x - 1) = 0x = 0, 3x + 2 = 0, x – 1= 0x = 0, x = -2/3, x = 1

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

Zeros of Polynomial functionMultiplicity 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=cif 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

MultiplicityState 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 – 8degree: 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’s0, -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, 27-30p. 193-194 #2-6 even, 25-28 all,

34-38 even, 39-42 all, 49-52 all, 53, 54