unit 2: polynomial functions graphs of polynomial functions 2.2 jmerrill 2005 revised 2008

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  • Slide 1
  • Slide 2
  • Unit 2: Polynomial Functions Graphs of Polynomial Functions 2.2 JMerrill 2005 Revised 2008
  • Slide 3
  • Learning Goal To find zeros and use transformations to sketch graphs of polynomial functions To use the Leading Coefficient Test to determine end behavior
  • Slide 4
  • Significant features The graphs of polynomial functions are continuous (no breaksyou draw the entire graph without lifting your pencil). This is opposed to discontinuous functions (remember piecewise functions?). This data is continuous as opposed to discrete.
  • Slide 5
  • Examples of Polynomials DegreeNameExample 0Constant5 1Linear3x+2 2QuadraticX 2 4 3CubicX 3 + 3x + 1 4Quartic-3x 4 + 4 5QuinticX 5 + 5x 4 - 7
  • Slide 6
  • Significant features The graph of a polynomial function has only smooth turns. A function of degree n has at most n 1 turns. AA 2 nd degree polynomial has 1 turn AA 3 rd degree polynomial has 2 turns AA 5 th degree polynomial has
  • Slide 7
  • Cubic Parent Function Draw the parent functions on the graphs. f(x) = x 3 XY -3-27 -2-8 00 11 28 327
  • Slide 8
  • XY -381 -216 1 00 11 216 381 Quartic Parent Function Draw the parent functions on the graphs. f(x) = x 4
  • Slide 9
  • Graph and Translate Start with the graph of y = x 3. Stretch it by a factor of 2 in the y direction. Translate it 3 units to the right. XY -3-27 -2-8 00 11 28 327 XY 0-54 1-16 2-2 30 42 516 654
  • Slide 10
  • XY -381 -216 1 00 11 216 381 Graph and Translate Start with the graph of y = x 4. Reflect it across the x-axis. Translate it 2 units down. XY -3-83 -2-18 -3 0-2 1-3 2-18 3-83 XY -3-81 -2-16 00 1 2-16 3-81
  • Slide 11
  • Slide 12
  • Max/Min A parabola has a maximum or a minimum Any other polynomial function has a local max or a local min. (extrema) Local max Local min max min
  • Slide 13
  • Polynomial Quick Graphs From yesterdays activity: f(x) = x 2 + 2x f(x) =
  • Slide 14
  • Look at the root where the graph of f(x) crossed the x-axis. What was the power of the factor? A.3 B.2 C.1
  • Slide 15
  • Look at each root where the graph of a functionwiggled at the x-axis. Were the powers even or odd? A.Even B.Odd
  • Slide 16
  • Look at each root where the graph of a function was tangent to the x-axis. What was the power of the factor? A.4 B.3 C.2 D.1
  • Slide 17
  • Describe the end behavior of a function if a > 0 and n is even. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right
  • Slide 18
  • Describe the end behavior of a function if a < 0 and n is even. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right
  • Slide 19
  • Describe the end behavior of a function if a > 0 and n is odd. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right
  • Slide 20
  • Describe the end behavior of a function if a < 0 and n is odd. A.Rise left, rise right B.Fall left, fall right C.Rise left, fall right D.Fall left, rise right
  • Slide 21
  • Leading Coefficient Test As x moves without bound to the left or right, the graph of a polynomial function eventually rises or falls like this: In an odd degree polynomial: IIf the leading coefficient is positive, the graph falls to the left and rises on the right IIf the leading coefficient is negative, the graph rises to the left and falls on the right In an even degree polynomial: IIf the leading coefficient is positive, the graph rises on the left and right IIf the leading coefficient is negative, the graph falls to the left and right
  • Slide 22
  • End Behavior If the leading coefficient of a polynomial function is positive, the graph rises to the right. y = x 2 y = x 3 + y = x 5 +
  • Slide 23
  • Finding Zeros of a Function If f is a polynomial function and a is a real number, the following statements are equivalent: x = a is a zero of the function x = a is a solution of the polynomial equation f(x)=0 (x - a) is a factor of f(x) (a, 0) is an x-intercept of f
  • Slide 24
  • Example Find all zeros of f(x) = x 3 x 2 2x Set function = 00 = x 3 x 2 2x Factor0 = x(x 2 x 2) Factor completely0 = x(x 2)(x + 1) Set each factor = 0, solve0 = x 0 = x 2; so x = 2 0 = x +1; so x = -1
  • Slide 25
  • You Do Find all zeros of f(x) = - 2x 4 + 2x 2 X = 0, 1, -1
  • Slide 26
  • Multiplicity (repeated zeros) How many roots?How many roots? 3 roots; x = 1, 3, 3. 4 roots; x = 1, 3, 3, 4. 3 is a double root. It is tangent to the x- axis
  • Slide 27
  • Roots of Polynomials How many roots? How many roots? 5 roots: x = 0, 0, 1, 3, 3. 0 and 3 are double roots 3 roots; x = 2, 2, 2 Double roots Double roots (tangent ) Triple root lies flat then crosses axis (wiggles)
  • Slide 28
  • Given Roots, Find a Polynomial Function There are many correct solutions. Our solutions will be based only on the factors of the given roots: Ex: Find a polynomial function with roots 2, 3, 3 Turn roots into factors: f(x) = (x 2)(x 3)(x 3) Multiply factors: f(x) = (x 2)(x 2 6x + 9) Finish multiplying: f(x) = x 3 8x 2 + 21x -18
  • Slide 29
  • You Do Find a polynomial with roots , 3, 3 One answer might be: f(x) = 2x 3 11x 2 + 12x +9
  • Slide 30
  • Sketch graph f(x) = (x - 4)(x - 1)(x + 2) Step 1: Find zeros. Step 2: Mark the zeros on a number line. Step 3: Determine end behavior Step 4: Sketch the graph Fall left, rise right
  • Slide 31
  • Sketch graph f(x)= -(x-4)(x-1)(x+2)
  • Slide 32
  • You Do f(x) = (x+1) 2 (x-2)
  • Slide 33
  • You Do f(x) = - (x-4) 3
  • Slide 34
  • Sketch graph. f(x) = (x-2) 2 (x+3)(x+2) roots: -3, -2 and 2 Rise left, rise right
  • Slide 35
  • Roots: -3, 2 and 6 Factors: (x+3), (x-2) and (x-6) Factored Form: f(x) = (x+3)(x-2)(x-6) Write an equation. Polynomial Form: f(x) = (x+3)(x 2 8x + 12) = x3 x3 5x 2 12x + 36
  • Slide 36
  • Write equation. Zeros: -2, -1, 3 and 5 Factors: (x+2), (x+1), (x-3) and (x-5) Factored Form: f(x) = (x + 2)(x + 1)(x 3)(x 5) Polynomial Form:
  • Slide 37
  • Gateway Problem Sketch the graph of f(x) = x 2 (x 4)(x + 3) 3 Double root at x = 0 Root at x = 4 Triple root at x = -3 Roots? Degree of polynomial? 6 End Behavior? Rise left Rise right

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