Polynomial Graph Features: 1. Must be a smooth continuous curve (no holes, or sharp corners) Graphing Polynomial Functions

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1 3.4 - Graphing Polynomial Functions 1. Notice that the graph is a smooth continuous curve. 2. The graph also has several "turning points", which are local maximums and minimums. P(x)=(1/30)(x+3)(x-2) 2 (x-5) 3. And it crosses the x-axis at x=-3 and x=5, while it bounces off the x-axis at x=2. 4. The end behavior of the graph is heading upwards at both tails. Polynomial Graph Features: 1. Must be a smooth continuous curve (no holes, or sharp corners)

2 2. If a polynomial with degree n, then the graph has at most (n-1) turning points Why at most? Why not exactly n-1 turning points? 3. If a polynomial graph has n turning points, then the degree of the polynomial is at least (n+1). Why is it at least (n+1) and not exactly (n+1)?

3 End Behavior of a Polynomial Graph The end behavior of a graph occurs at the tails of the graph Example 1: Give the end behavior of the graph in words and limit notation a) Words: to the left; to the right Limit notation: as x, y ; as x -, y b) c) Words: to the left; to the right Limit notation: as x, y ; as x -, y Words: to the left; to the right Limit notation: as x, y ; as x -, y

4 d) e) Words: to the left; to the right Limit notation: as x, y ; as x -, y Words: to the left; to the right Limit notation: as x, y ; as x -, y Have you noticed the pattern from what you've seen in the past and the recent examples?

5 End Behavior Rules: Given a polynomial with leading term ax n : 1) If n is even, then the ends will point in the same direction. a > 0 a < 0 2) If n is odd, then the ends will point in the opposite directions. a > 0 a < 0 Example 2: Without graphing, tell the end behavior of each graph. a) -5x 5 +3x 3-6x b) 12x 10 +7x 5-9x 3 as x, y ; as x -, y as x, y ; as x -, y

6 Attributes of Zeroes of Multiplicity: Odd multiplicities: The graph will pass through the x-axis at the zero. y = x y = x 3 y = x 5 Each graph has an odd multiplicity for a zero located at x = 0. What happens as you increase the multiplicity?

7 Even multiplicities: The graph will bounce off the x-axis at the zero. (touching the x-axis at a single point) y = x 2 y = x 4 y = x 6 Each graph has an even multiplicity for a zero located at x = 0. What happens as you increase the multiplicity?

8 Example 3: Use the given polynomial. a) Is the degree even or odd? b) State the zeroes and their least possible multiplicity. c) What is the minimum possible degree of the graph? d) What is a possible equation for the polynomial? Example 4: Use the given polynomial. a) Is the degree even or odd? b) State the zeroes and their least possible multiplicity. c) What is the minimum possible degree of the graph?

9 Example 5: Write a possible equation for the polynomial in factored form. a) b)

10 c) Graphing Polynomials Tips: 1. Determine the End Behavior 2. Find the y-intercept (0, y) 3. Find zeroes (use factoring, remainder theorem, quadratic formula, etc) 4. Use the above to sketch the graph. Use mid-interval points as needed to sketch a smooth, continuous curve.

11 Example 6: Sketch the graph of g(x)=(x+1) 2 (x-2) Degree/End Behavior y-int: Zeroes: Example 7: Sketch the graph of g(x)=(1/4)x 5 - (3/4)x 4 + (1/4)x 3 - (3/4)x 2 Degree/End Behavior y-int: Zeroes:

12 Example 8: Sketch the graph of g(x)=-x 5 +4x 4-5x 3 +4x 2-4x Degree/End Behavior y-int: Zeroes: Example 9: Sketch the graph of g(x)= x 5 -x 4-8x 3 +8x 2 +7x-7 Degree/End Behavior y-int: Zeroes:

13 Example 10: Sketch the graph of g(x)= x 4-4x 3-7x 2 +8x+10 Degree/End Behavior y-int: Zeroes: Describe the multiplicity of zeroes describe the behavior of the graph degree: end behaviour: describe multiplicity of zeroes: degree: end behaviour: describe multiplicity of zeroes: other points- y-intercept: other points- y-intercept:

14 (0,12) (0,4) (0,12)

15 degree: lc: zeroes: end behav: x=1 bounces off (x2) x axis x=-1, 2 crosses x axis- odd times find y int: x=0 find another point ex. x=1.5

16 Factor since it is not in factored form. degree: lc: end behav: find y int: x= zeroes:p/q test x=±1 synthetic div: find 1or 2 other points ex. x= Factor since it is not in factored form.

17 For the polynomial graph, (a) state whether the degree of the function is even or odd; (b) use the graph to name the zeroes of f, then state whether their multiplicity is even or odd; (c) state the minimum possible degree of f and write it in factored form; (d) estimate the domain and range. Assume all zeroes are real. e. Write the equation in factored form.

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