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Understanding Power Functions & Higher Degree Polynomials: Analysis and Modeling Techniques

This guide covers the fundamentals of power functions and higher-degree polynomials, exploring their forms, properties, and analyses. A power function is defined as ( f(x) = kx^a ) where both ( k ) and ( a ) are non-zero constants. The document discusses essential concepts like domain, range, continuity, symmetry (even/odd functions), boundedness, local extrema, asymptotes, and end behavior. It includes examples to illustrate how to analyze graphs and find zeros, multiplicity of roots, and sketch corresponding graphs.

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Understanding Power Functions & Higher Degree Polynomials: Analysis and Modeling Techniques

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  1. 2.2 Power Function w/ modeling 2.3 Polynomials of higher degree with modeling

  2. 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

  3. 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 -∞ & ∞

  4. 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

  5. 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

  6. 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)

  7. 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

  8. 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

  9. 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

  10. Homework: • p. 182 #, 28, 29 • p. 193-194 #2-6 even, 25-28 all, 34-38 even, 39-42 all, 49-52 all, 53, 54

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