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Analyzing Graphs of Polynomials

Analyzing Graphs of Polynomials. Section 3.2. First a little review…. Given the polynomial function of the form: If k is a zero, Zero: __________ Solution: _________ Factor: _________ If k is a real number, then k is also a(n) __________________.

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Analyzing Graphs of Polynomials

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  1. Analyzing Graphs of Polynomials Section 3.2

  2. First a little review… Given the polynomial function of the form: If k is a zero, Zero: __________ Solution: _________ Factor: _________ If k is a real number, then k is also a(n) __________________. x = k x = k (x – k) x - intercept

  3. Sharp corner – This graph must not be a polynomial function. Hole Break What kind of curve? All polynomials have graphs that are smooth continuous curves. A smooth curve is a curve that does not have sharp corners. A continuous curve is a curve that does not have a break or hole. This is not a continuous curve!

  4. y y y y x x x x , f(x) , f(x) , f(x) , f(x) , f(x) , f(x) , f(x) , f(x) As x  + As x  - As x  - As x  - As x  + As x  + As x  - As x  + End Behavior An < 0 , Even Degree An > 0 , Odd Degree An > 0 , Odd Degree An < 0 , Odd Degree An > 0 , Even Degree

  5. Examples of End Behaviors 1. 2. 4. 3.

  6. Relative maximum The graph “turns” Relative minimums The graph “turns” What happens in the middle? ** This graph is said to have 3turning points. ** The turning points happen when the graph changes direction. This happens at the vertices. ** Vertices are minimums and maximums. ** The lowest degree of a polynomial is (# turning points + 1). So, the lowest degree of this polynomial is 4 !

  7. , f(x) As x  - , f(x) As x  + What’s happening? Relative Maximums Relative Minimums 4 The number of turning points is _____ . 5 The lowest degree of this polynomial is _____ . positive The leading coefficient is __________ .

  8. Negative-odd polynomial of degree 3 ( -x * x * x) , f(x) As x  - , f(x) As x  + Graphing by hand Step 1: Plot the x-intercepts Step 2: End Behavior? Number of Turning Points? Step 3: Check in Calculator!!! Example #1: Graph the function: f(x) = -(x + 4)(x + 2)(x - 3) and identify the following. End Behavior: _________________________ Lowest Degree of polynomial: ______________ # Turning Points: _______________________ 3 2 You can check on your calculator!! x-intercepts

  9. Positive-even polynomial of degree 4 Relative max Relative minimum Absolute minimum , f(x) As x  - , f(x) As x  + Graphing with a calculator Example #2: Graph the function: f(x) = x4 – 4x3 – x2 + 12x – 2 and identify the following. End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ y-intercept: _______ 4 3 (0, -2) • Plug equation into y= • Find minimums and maximums • using your calculator Real Zeros

  10. Positive-odd polynomial of degree 3 , f(x) As x  - , f(x) As x  + Graphing without a calculator Example #3: Graph the function: f(x) = x3 + 3x2 – 4x and identify the following. End Behavior: _________________________ Degree of polynomial: ______________ # Turning Points: _______________________ 3 2 1. Factor and solve equation to find x-intercepts f(x)=x(x2 + 3x – 4) = x(x - 4)(x + 1) 2. Plot the zeros. Sketch the end behaviors.

  11. P(b) is positive. (The y-value is positive.) a b Therefore, there must be at least one real zero in between x = a & x = b! P(a) is negative. (The y-value is negative.) 0 4 -1 -6 1 0 0 0 4 -1 -6 1 1 4 -1 -6 1 4 3 -3 4 3 -3 -2 Zero Location Theorem Given a function, P(x) and a & b are real numbers. If P(a) and P(b) have opposite signs, then there is at least one real number c between a and b such that P(c) = 0. Example #4: Use the Zero Location Theorem to verify that P(x) = 4x3 - x2 – 6x + 1 has a zero between a = 0 and b = 1. The graph of P(x) is continuous because P(x) is a polynomial function. P(0)= 1 and P(1) = -2  Furthermore, -2 < 0 < 1 The Zero Location Theorem indicates there is a real zero between 0 and 1!

  12. Polynomial Functions:Real Zeros, Graphs, and Factors (x – c) • If P is a polynomial function and c is a real root, then each of the following is equivalent. • (x – c) __________________________________ . • x = c __________________________________ . • x = c __________________________________ . • (c, 0) __________________________________ . is a factor of P is a real solution of P(x) = 0 is a real zero of P is an x-intercept of the graph of y = P(x)

  13. Even & Odd Powers of (x – c) The exponent of the factor tells if that zero crosses over the x-axis or is a vertex. If the exponent of the factor is ODD, then the graph CROSSES the x-axis. If the exponent of the factor is EVEN, then the zero is a VERTEX. Try it. Graph y = (x + 3)(x – 4)2 Try it. Graph y = (x + 6)4 (x + 3)3

  14. Assignment: Write the questions and show all work for each. pp. 301-302 #1-13 ODD, 17 & 19 (TI-83), 21-29 ODD, 33, 35, 41, & 43

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