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Precalculus – MAT 129

Precalculus – MAT 129. Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF. Chapter Two. Polynomial and Rational Functions. 2.1 – Quadratic Functions. The Graph of a Quadratic Function The Standard Form of a Quadratic Function Finding Minimum and Maximum Values.

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Precalculus – MAT 129

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  1. Precalculus – MAT 129 Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF

  2. Chapter Two Polynomial and Rational Functions

  3. 2.1 – Quadratic Functions • The Graph of a Quadratic Function • The Standard Form of a Quadratic Function • Finding Minimum and Maximum Values

  4. 2.1 – The Graph of a Quadratic • Polynomial Function • f(x)=anxn + an-1xn-1 + … + a1x + a0 • Called a polynomial function of x with degree n. • We have already talked about constant functions and linear functions (degree 0 and degree 1) • Quadratic Function • f(x) = ax2 + bx + c

  5. Parabola • Quadratic Functions graph in a u-shape called a parabola. • Symmetric about the axis of the parabola • Vertex at the point where the axis intersects the parabola. • See beige box on pg. 93

  6. 2.1 – Standard Form of Quadratic • The equation of the quadratic written in standard form or vertex form is written in the form below: f(x) = a(x-h)2 + k

  7. Completing the Square • Often to go from the polynomial form to the standard form of the quadratic equation one must use the process of completing the square. • See example 2 on pg. 95

  8. Finding x-intercepts • There are three ways that you can use to find the x-intercepts • Graphically: use the trace function and approx. • Algebraically • factor • Use quadratic equation

  9. Example 1.2.1 Pg. 99 # 13 Identify the vertex and x-intercepts: h(x) = x2 - 8x + 16

  10. Solution - Ex. 1.2.1 Completing the square we get: h(x) = (x – 4)2 – 0 This gives a vertex of (4,0). This is also the x-intercept.

  11. 2.1 – Finding Minimum and Maximum Values • If a > 0, f has a minimum at –b/2a. • If a< 0, f has a maximum at –b/2a.

  12. Example 2.2.1 Pg. 97 Example 5 Note both solutions.

  13. Example 3.2.1 Pg. 101 Example 55 Do this at home.

  14. 2.2 – Polynomial Fxns of Higher Degree • Graphs of Polynomial Functions • The Leading Coefficient Test • Zeros of Polynomial Functions • The Intermediate Value Theorem

  15. 2.2 – Graphs of Polynomial Functions • Polynomial graphs are continuous • They do not have breaks or sharp turns.

  16. 2.2 – The Leading Coefficient Test • From the leading coefficient of a polynomial equation you can tell what the graph should look like. • Best summarized in the blue box on pg. 105 • We will use this when we are talking about zeros.

  17. 2.2 – Zeros of Polynomial Fxns • It can be shown tht for a polynomial function f of degree n, the following statements are true. • The function f has at most n real zeros. • The graph of f has at most (n-1) relative minima or maxima.

  18. Real Zeros of Polynomial Fxns 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 f. • x = a is a solution of the polynomial equation f(x) = 0. • (x-a) is a factor of the polynomial f(x). • (a,0) is an x-intercept of the graph of f.

  19. Repeated Zeros For a polynomial function, a factor of (x-a)k, k>1, yields a repeated zero x=a of multiplicity k. • If k is odd, the graph crosses the x-axis at x=a. • If k is even, the graph touches the x-axis (but does not cross the x-axis) at x=a. See Figure 2.23 on p. 107 and study tip on pg. 108

  20. Polynomial Functions Read technology tip on pg. 109. Do the partner activity in the exploration on pg. 109.

  21. Activities (109) 1. Find all the real zeros of f(x) = 6x4 -33x3 - 18x2. 2. Determine the right-hand and left-hand behavior of the function above. 3. Find a polynomial function of degree 3 that has zeros of 0, 2, and -1/3.

  22. 2.3 – Real Zeros of Polynomial Functions • Long Division of Polynomials • Synthetic Division • The Remainder and Factor Theorems • The Rational Zero Test • Other Tests for Zeros of Polynomials

  23. 2.3 – Long Division of Polynomials

  24. Example 1.2.3 Use long division to divide the following:

  25. 2.3 – Synthetic Division • A shortcut for dividing polynomials when dividing by divisors of the form x-k. • See blue box on pg. 119.

  26. Example 2.2.3 Pg. 127 # 17 and 21 Do on the board.

  27. 2.3 – The Remainder and Factor Theorems • The Remainder Theorem • If a polynomial f(x) is divided by x-k, the remainder is r = f(k). • The Factor Theorem • A polynomial f(x) has a factor (x-k) if and only f(k)=0.

  28. Example 3.2.3 Pgs. 120 - 121 • Example 5 • Use the Remainder Theorem to evaluate at x = -2. • Example 6 • Note both solutions. • I have an additional example (p. 128 #45).

  29. 2.3 – The Rational Zero Test • To use this test you make a list of all possible rational roots. • Divide factors of the constant by factors of the leading coefficient.

  30. Example 4.2.3 Use the Rational Zero Test to find all the possible rational zeros of

  31. 2.3 – Other Tests for Zeros of Polynomials • Descartes’s Rule of Signs • Upper and Lower Bound Rules

  32. Descartes’s Rule of Signs • The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. • The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.

  33. Example 5.2.3 Pg. 125 Example 10 Describe the possible real zeros of f(x)=3x3 - 5x2 + 6x – 4.

  34. Upper and Lower Bound Rules Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is defined by x-c, using synthetic division. • If c>0 and each # in the last row is either positive or zero, c is an upper bound for the real zeros of f. • Likewise, if c<0 and last row alternates in sign, c is a lower bound for the real zeros of f.

  35. Example 6.2.3 Pg. 126 Example 11 Find real zeros of: f(x)=6x3 - 4x2 + 3x – 2.

  36. 2.4 – Complex Numbers • The Imaginary Unit i • Operations with Complex Numbers • Complex Conjugates • Fractal and the Mandelbrot Set

  37. 2.4 – The Imaginary Unit i • Read from beginning of pg. 131 to the blue box. • If a and b are real numbers the number a+bi is a complex number in standard form. • A number of the form bi where b≠0 is called a pure imaginary number.

  38. 2.4 – The Imaginary Unit i • Do the exploration on page 133. • Fill in the chart for the powers of i.

  39. 2.4 – Operations with Complex Numbers • Addition: • (a+bi) + (c+di) = (a+c) + (b+d)i • Subtraction: • (a+bi) - (c+di) = (a-c) + (b-d)i • Multiplication: • i = i , i2 = -1, i3 = -i , i4 = 1

  40. Example 1.2.4 Pg. 137 #s 5 and 9 Write the complex number in standard form. 5. 5 + sqrt(-16) 9. -5i + i2

  41. Solution Example 1.2.4 Pg. 133 #s 5 and 9 Write the complex number in standard form. 5. 4 + 5i 9. -1-5i

  42. Example 2.2.4 Pg. 132 and 133 Example 1 a and d. Example 2 a and d.

  43. Example 3.2.4 Pg. 137 #19 Perform the addition or subtraction and write the result in standard form. 13i – (14 – 7i)

  44. Solution Example 3.2.4 Pg. 137 #19 Perform the addition or subtraction and write the result in standard form. 13i – 14 + 7i -14 + 20i

  45. 2.4 – Complex Conjugates • Note that the product of two complex numbers can be a real number when complex conjugates are multiplied. • They are of the form: (a+bi)(a-bi) • When dividing complex numbers you will need to multiply top and bottom of the fraction by the conjugate of the denominator to write in standard form.

  46. Example 4.2.4 Pg. 137 #48 Write the quotient in standard form. 48. 3/(1-i) =3/2 + (3/2)i

  47. 2.5 – The Fundamental Theorem of Algebra • The Fundamental Theorem of Algebra • Conjugate Pairs • Factoring a Polynomial

  48. 2.5 – The Fundamental Theorem of Algebra • If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system.

  49. Example 1.2.5 pg. 140 Example 3 Write f(x) as the product of linear factors and list all the zeros of f. • Check graph • Synthetically divide • Factor

  50. Solution Example 1.2.5 pg. 136 Example 3 f(x)=(x-1)(x-1)(x+2)(x-2i)(x+2i) Gives the zeros: x=1, x=1, x=-2, x=2i, x=-2i

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