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Graphs of Polynomial Functions

Graphs of Polynomial Functions. Pre-Calculus Day 13. Plan for the Day. Quiz – Radicals Review Homework - Quadratics 2.1 Page 116 1-8,29-33 (odd) algebraic and graphing calculator, 37-45 (odd), 77, 79, 83 Polynomials Homework. What You Should Learn.

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Graphs of Polynomial Functions

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  1. Graphs of Polynomial Functions Pre-Calculus Day 13

  2. Plan for the Day • Quiz – Radicals • Review Homework - Quadratics • 2.1 Page 116 1-8,29-33 (odd) algebraic and graphing calculator, 37-45 (odd),77, 79, 83 • Polynomials • Homework

  3. What You Should Learn • Determine key features of a polynomial graph • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. • Find and use zeros of polynomial functions as sketching aids. • Find a polynomial equation given the zeros of the function.

  4. Polynomials • What do you remember about polynomials?? • What would be key points of a polynomial? • Remember this …

  5. y y y f(x) = x3 – 5x2+4x + 4 x x x Graphs of Polynomial Functions Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. continuous not continuous continuous smooth not smooth polynomial not polynomial not polynomial Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

  6. Polynomial Function A polynomial functionis a function of the form where n is a nonnegative integer and a1, a2, a3, …an are real numbers. The polynomial function has a leading coefficient an and degreen. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree -2 5 1 3 14 0

  7. Classification of a Polynomial n = 0 constant Y = 3 linear n = 1 Y = 5x + 4 quadratic n = 2 Y = 2x2 + 3x - 2 cubic n = 3 Y = 5x3 + 3x2 – x + 9 quartic Y = 3x4 – 2x3 + 8x2 – 6x + 5 n = 4 n = 5 Y = -2x5+3x4–x3+3x2–2x+6 quintic

  8. Graphs of Polynomial Functions The polynomial functions that have the simplest graphs are monomials of the form f(x) = xn, where n is an integer greater than zero.

  9. y y x x Power Functions Polynomial functions of the form f(x) = xn, n 1 are called power functions. f(x) = x5 f(x) = x4 f(x) = x3 f(x) = x2 If n is odd, their graphs resemble the graph of f(x) = x3. If n is even, their graphs resemble the graph of f(x) = x2. Moreover, the greater the value of n,the flatter the graph near the origin

  10. The Leading Coefficient Test Polynomial functions have a domain of all real numbers. Graphs eventually rise or fall without bound as x moves to the right. Whether the graph of a polynomial function eventually rises or falls can be determined by the function’s degree (even or odd) and by its leading coefficient, as indicated in the Leading Coefficient Test.

  11. an positive y y x x an negative Leading Coefficient Test As x grows positively or negatively without bound, the value f(x) of the polynomial function f(x) = anxn + an – 1xn – 1 + … + a1x + a0 (an 0) grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree nis odd or even. neven nodd

  12. Find the left and right behavior of the polynomial.

  13. Zeros of Polynomial Functions It can be shown that for a polynomial function f of degree n,the following statements are true. 1. The function f has, at most, n real zeros. 2. The graph of f has, at most, n – 1 turning points. (Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa.) Finding the zeros of polynomial functions is one of the most important problems in algebra.

  14. Given the polynomials below, answer the following • What is the degree? • What is its leading coefficient? • How many “turns”(relative maximums or minimums) could it have (maximum)? • How many real zeros could it have (maximum)? • How would you describe the left and right behavior of the graph of the equation? • What are its intercepts (y for all, x for 1 & 2 only)? Equations:

  15. Zeros of Polynomial Functions • There is a strong interplay between graphical and algebraic approaches to this problem. • Sometimes you can use information about the graph of a function to help find its zeros, and in other cases you can use information about the zeros of a function to help sketch its graph. • Finding zeros of polynomial functions is closely related to factoring and finding x-intercepts.

  16. Real Zeros of Polynomial Functions Ify = f(x) is a polynomial function and a is a real number then the following statements are equivalent. 1. a is a zero of f. 2. a is a solution of the polynomial equation f(x) = 0. 3. x – a is a factor of the polynomial f(x). 4. (a, 0) is an x-intercept of the graph of y = f(x). A real number a is a zero of a function y = f(x)if and only if f(a) = 0. A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa. A polynomial function of degree n has at most n–1turning points and at most nzeros.

  17. Repeated Zeros If k is the largest integer for which (x – a)k is a factor of f(x)and k > 1, then a is a repeated zero of multiplicityk. 1. If k is odd the graph of f(x) crosses the x-axis at (a, 0). 2. If k is even the graph of f(x) touches, but does not cross through, the x-axis at (a, 0). y x Example: Determine the multiplicity of the zeros of f(x) = (x – 2)3(x +1)4. Zero Multiplicity Behavior crosses x-axis at (2, 0) 3 odd 2 touches x-axis at (–1, 0) –1 4 even

  18. Example - Finding the Zeros of a Polynomial Function Find all real zeros of f(x) = –2x4 + 2x2. Then determine the number of turning points of the graph of the function.

  19. Example – Solution cont’d Solution: To find the real zeros of the function, set f(x) equal to zero and solve for x. –2x4 + 2x2 = 0 –2x2(x2 – 1) = 0 –2x2(x – 1)(x + 1) = 0 So, the real zeros are x = 0 (double root), x = 1, and x = –1. Because the function is a fourth-degree polynomial, the graph of f can have at most 4 – 1 = 3 turning points. Set f(x) equal to 0. Remove common monomial factor. Factor completely.

  20. Zeros of Polynomial Functions In the example, note that because the exponent is greater than 1, the factor –2x2 yields the repeated zero x = 0. Because the exponent is even, the graph touches the x-axis at x = 0.

  21. y Turning point x Turning point Turning point f(x) = x4– x3 – 2x2 Another example: Find all the real zeros and turning points of the graph of f(x) = x4– x3 – 2x2. Factor completely: f(x) = x4 – x3 – 2x2 = x2(x+1)(x – 2). The real zeros are x = –1,x= 0, and x = 2. These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0). The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.

  22. Zeros of Polynomial Functions This means that when the real zeros of a polynomial function are put in order, they divide the real number line into intervals in which the function has no sign changes. These resulting intervals are test intervals in which a representative x-value in the interval is chosen to determine if the value of the polynomial function is positive (the graph lies above the x-axis) or negative (the graph lies below the x-axis).

  23. as , y (2, 0) x (–2, 0) (0, 0) Example: Sketch the graph of f(x) = 4x2 – x4. f(x) = –x4 + 4x2 1. Write the polynomial function in standard form: The leading coefficient is negative and the degree is even. 2. Find the zeros of the polynomial by factoring. f(x) = –x4 + 4x2 = –x2(x2 – 4) = – x2(x + 2)(x –2) Zeros: x = –2, 2 multiplicity 1x = 0 multiplicity 2 x-intercepts: (–2, 0), (2, 0) crosses through (0, 0) touches only Example continued

  24. y (–1.5, 3.9 ) (1.5, 3.9) (0.5, 0.94) (–0.5, 0.94 ) x Example continued: Sketch the graph of f(x) = 4x2 – x4. 3. Since f(–x) = 4(–x)2 – (–x)4= 4x2 – x4=f(x), the graph is symmetrical about the y-axis. 4. Plot additional points and their reflections in the y-axis: (1.5, 3.9) and (–1.5, 3.9 ), (0.5, 0.94 )and (–0.5, 0.94) 5. Draw the graph.

  25. Find the Polynomial Given the zeros, find an equation (assume lowest degree): Zeros: 2, 3 Answer: (x – 2)(x – 3) = x2 – 5x +6 Zeros: 0 (multiplicity of 2), -2, 5 Answer: x2 (x + 2)(x – 5)= x2(x2 – 3x + 10)= x4 – 3x3 + 10x2 Zeros: 2, 3 (multiplicity of 2), -4(multiplicity of 3) – leave in factored form Answer: (x – 2)(x – 3)2 (x + 4)3

  26. Can you??? • Determine key features of a polynomial graph • Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. • Find and use zeros of polynomial functions as sketching aids. • Find a polynomial equation given the zeros of the function.

  27. Homework 8 • 2.2 Page 130 • 1-8 all (matching) • 13-18(left and right behavior), all • 27-41 odds (finding zeros-verify with a calculator) • 47-55 odds • Quiz next class – Complex Numbers

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