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# Polynomial Functions

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1. Polynomial Functions

2. Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so positive integers would be 0, 1, 2 … A polynomial function is a function of the form: n must be a positive integer All of these coefficients are real numbers The degree of the polynomial is the largest power on any x term in the polynomial.

3. Polynomial Functions The largest exponent within the polynomial determines the degree of the polynomial.

4. Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. We can write in an x0 since this = 1. x 0 A polynomial of degree 0. Not a polynomial because of the square root since the power is NOT an integer Not a polynomial because of the x in the denominator since the power is negative

5. Graphs of polynomials are smooth and continuous. No gaps or holes, can be drawn without lifting pencil from paper No sharp corners or cusps This IS the graph of a polynomial This IS NOT the graph of a polynomial

6. Two important features of the graphs of polynomial functions are that they are smooth and continuous. By smooth, we mean that the graph contains only rounded curves with no y y sharp corners. By continuous, we mean that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. These ideas are illustrated in the figure. Smooth rounded corner Smooth rounded corner x x Smooth rounded corner Smooth rounded corner Smooth, Continuous Graphs

8. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Even functions are symmetrical over the y-axis 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

9. A function is even if f( -x) = f(x) for every number x in the domain. All exponents must be even!! So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

10. Let’s look at the graph of where n is an even integer. and grows steeper on either side Notice each graph looks similar to x2 but is wider and flatter near the origin between –1 and 1 The higher the power, the flatter and steeper

11. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Odd functions are symmetrical over the origin 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

12. A function is odd if f( -x) = - f(x) for every number x in the domain. All exponents must be odd!! So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

13. Let’s look at the graph of where n is an odd integer. Notice each graph looks similar to x3 but is wider and flatter near the origin between –1 and 1 and grows steeper on either side The higher the power, the flatter and steeper

14. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

15. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7

16. Let’s graph Translates up 2 Reflects about the x-axis Looks like x2 but wider near origin and steeper after 1 and -1 So as long as the function is a transformation of xn, we can graph it, but what if it’s not? We’ll learn some techniques to help us determine what the graph looks like in the next slides.

17. and RIGHT LEFT HAND BEHAVIOR OF A GRAPH The degree of the polynomial along with the sign of the coefficient of the term with the highest power will tell us about the left and right hand behavior of a graph.

18. Even Degree: Positive coefficient Negative Coefficient If the leading coefficient is positive, the graph rises to the left and to the right. If the leading coefficient is negative, the graph falls to the left and to the right. Rises right Rises left Falls left Falls right The Leading Coefficient Test

19. Even degree polynomials rise on both the left and right hand sides of the graph (like x2) if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term. left hand behavior: rises right hand behavior: rises

20. Even degree polynomials fall on both the left and right hand sides of the graph (like - x2) if the coefficient is negative. turning points in the middle left hand behavior: falls right hand behavior: falls

21. For an odd degree: Positive coefficient negative coefficient If the leading coefficient is positive, the graph falls to the left and rises to the right. If the leading coefficient is negative, the graph rises to the left and falls to the right. Rises right Rises left Falls right Falls left The Leading Coefficient Test

22. Odd degree polynomials fall on the left and rise on the right hand sides of the graph (like x or x3) if the coefficient is positive. turning Points in the middle right hand behavior: rises left hand behavior: falls

23. Odd degree polynomials rise on the left and fall on the right hand sides of the graph (like -x or -x3) if the coefficient is negative. turning points in the middle left hand behavior: rises right hand behavior: falls

24. Determining End Behavior Match each function with its graph. B. A. C. D.

25. Graph A Graph B Quartic Polynomials Look at the two graphs and discuss the questions given below. 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 3. What is the end behavior for each graph? 4. Which graph do you think has a positive leading coeffient? Why? 5. Which graph do you think has a negative leading coefficient? Why?

26. doesn’t mean it has that many turning points but that’s the most it can have A polynomial of degree n can have at mostn-1 turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn). Let’s determine left and right hand behaviour for the graph of the function: degree is 4 which is even and the coefficient is positive so the graph will look like x2 looks off to the left and off to the right. How do we determine what it looks like near the middle? The graph can have at most 3 turning points

27. x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x. To find the x intercept we put 0 in for y. Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay) (0,30)

28. We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = 0. These are also called the roots of the polynomial. Can you find the zeros of the polynomial? There are repeated factors. (x-1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times). What are the other zeros and their multiplicities? -2 is a zero of multiplicity 2 3 is a zero of multiplicity 1

29. Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

30. So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph: What would the left and right hand behavior be? You don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x3. Notice the negative out in front. What would the y intercept be? 1 of mult. 1(so crosses axis at 1) -2 of mult. 2(so touches at 2) (0, 4) Find the zeros and their multiplicity

31. Rises right y x Falls left Text Example Use the Leading Coefficient Test to determine the end behavior of the graph of Graph the quadratic functionf(x)= x3+3x2-x- 3. SolutionBecause the degree is odd (n=3) and the leading coefficient, 1, is positive, the graph falls to the left and rises to the right, as shown in the figure.

32. Solution Step 1 Determine end behavior. Because the degree is even (n = 4) and the leading coefficient, 1, is positive, the graph rises to the left and the right: y Rises left Rises right x Text Example Graph: f(x)=x4-2x2+ 1.

33. Text Example cont. Graph: f(x)=x4-2x2+ 1. Solution Step 2 Find the x-intercepts (zeros of the function) by settingf(x)=0. x4-2x2+ 1 = 0 (x2- 1)(x2- 1) = 0 Factor. (x+ 1)(x -1)(x+ 1)(x -1) = 0 Factor completely. (x+ 1)2(x- 1)2= 0 Express the factoring in more compact notation. (x+ 1)2= 0 or (x- 1)2= 0 Set each factor equal to zero. x=-1 x= 1Solve for x.

34. y Rises right Rises left x 1 1 Text Example cont. Graph: f(x)=x4-2x2+ 1. Solution Step 2We see that -1 and 1 are both repeated zeros with multiplicity 2. Because of the even multiplicity, the graph touches the x-axis at -1 and 1 and turns around. Furthermore, the graph tends to flatten out at these zeros with multiplicity greater than one:

35. y Rises right Rises left 1 x 1 1 Text Example cont. Graph: f(x)=x4-2x2+ 1. Solution Step 3 Find the y-intercept. Replace xwith 0 in f(x)=-x+ 4x - 1. f(0) =04- 2 • 02+ 1=1 There is a y-intercept at 1, so the graph passes through (0, 1).

36. y x Text Example cont. Graph: f(x)=x4-2x2+ 1. Solution

37. Graphing a Polynomial Function f (x)=anxn+ an-1xn-1+ an-2xn-2 + ¼ +a1x + a0 (an¹ 0) • Use the Leading Coefficient Test to determine the graph's end behavior. • Find x-intercepts by setting f (x) = 0 and solving the resulting polynomial equation. If there is an x-intercept at r as a result of (x-r)k in the complete factorization of f (x), then: a. If k is even, the graph touches the x-axis at r and turns around. b. If k is odd, the graph crosses the x-axis at r. c. If k > 1, the graph flattens out at (r, 0). 3. Find the y-intercept by setting x equal to 0 and computing f (0).

38. CONCEPT END BEHAVIOR FOR POLYNOMIAL FUNCTIONS SUMMARY annx – x + > 0 even f(x) + f(x) +  > 0 odd f(x) – f(x) +  < 0 even f(x) – f(x) –  < 0 odd f(x) + f(x) –  GRAPHING POLYNOMIAL FUNCTIONS

39. Example • Find the x-intercepts and multiplicity of f(x) = 2(x+2)2(x-3) Solution: • x=-2 is a zero of multiplicity 2 or even • x=3 is a zero of multiplicity 1 or odd

40. Text Example Find all zeros of f(x)=-x4+4x3- 4x2. SolutionWe find the zeros of fby setting f(x)equal to 0. -x4+4x3- 4x2 = 0 We now have a polynomial equation. x4-4x3+ 4x2= 0 Multiply both sides by -1. (optional step) x2(x2- 4x+ 4) = 0 Factor out x2. x2(x- 2)2= 0 Factor completely. x2= 0 or (x- 2)2= 0 Set each factor equal to zero. x= 0 x= 2Solve for x.

41. Graphing Polynomial Functions –3 –2 –1 0 1 2 3 x The degree is odd and the leading coefficient is positive, so f(x) –7 3 3 –1 –3 3 23 f(x) . and f(x) – – + + as x as x Graphf(x) =x3 + x2 – 4x – 1. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.

42. Graphing Polynomial Functions x The degree is even and the leading coefficient is negative, so f(x) f(x) . and f(x) – – – + as x as x –3 –2 –1 0 1 2 3 –21 0 –1 0 3 –16 –105 Graphf(x) = –x4 – 2x3 + 2x2 + 4x. SOLUTION To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior.