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

2.1 – The Graph of a Quadratic Polynomial Function –f(x)=a n x n + a n-1 x n-1 + … + a 1 x + a 0 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) = ax 2 + bx + c

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

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

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

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

Example Pg. 99 # 13 Identify the vertex and x-intercepts: h(x) = x 2 - 8x + 16

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

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.

Example Pg. 97 Example 5 Note both solutions.

Example Pg. 101 Example 55 Do this at home.

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

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

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.

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.

Real Zeros of Polynomial Fxns If f is a polynomial function and a is a real number, the following statements are equivalent. 1.x = a is a zero of the function f. 2.x = 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 f.

Repeated Zeros For a polynomial function, a factor of (x-a)k, k>1, yields a repeated zero x=a of multiplicity k. 1.If k is odd, the graph crosses the x-axis at x=a. 2.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

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

Activities (109) 1. Find all the real zeros of f(x) = 6x x x 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.

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

2.3 – Long Division of Polynomials

Example Use long division to divide the following:

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

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

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.

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

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.

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

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

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.

Example Pg. 125 Example 10 Describe the possible real zeros of f(x)=3x 3 - 5x 2 + 6x – 4.

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. 1.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. 2.Likewise, if c<0 and last row alternates in sign, c is a lower bound for the real zeros of f.

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

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

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.

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

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, i 2 = -1, i 3 = -i, i 4 = 1

Example Pg. 137 #s 5 and 9 Write the complex number in standard form sqrt(-16) 9. -5i + i 2

Solution Example Pg. 133 #s 5 and 9 Write the complex number in standard form i i

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

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

Solution Example Pg. 137 #19 Perform the addition or subtraction and write the result in standard form. 13i – i i

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.

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

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

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.

Example 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

Solution Example 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

2.5 – Conjugate Pairs Complex zeros come in pairs. –If f(x) has coefficients that are all real numbers and a+bi is a zero, then a-bi will be a zero as well as long as b does not equal zero.

2.5 – Factoring a Polynomial At this point you should be comfortable with the terms irrationals and reals and be able to move back and forth between the two. Read the study tip on pg. 142.

Example pg. 142 Example 5 a, b and c. Go over as a class.

Example pg. 143 Example 6 Look at both Algebraic and Graphical Solutions.

Activities (138) 1. Write as a product of linear factors: f(x) = x Find a third degree polynomial with integer coefficients that has 2, 3+i, 3-i as zeros.

2.6 – Rational Functions and Asymptotes Introduction to Rational Functions Horizontal and Vertical Asymptotes

2.6 – Introduction to Rational Functions A rational function is one that you can write in the form: f(x)=N(x)/D(x) where both numerator and denominator are polynomials and D(x) can’t be the zero polynomial.

Example pg. 146 Example 1 Find the domain of f(x) = 1/x and discuss the behavior of f near any excluded x- values. –This can be done via exploration (table) or graphically.

2.6 – Horizontal and Vertical Asymptotes The line x=a is a vertical asymptote of the graph of f if f(x)→∞ or f(x)→-∞ as x →a, either from the right or the left. The line y=b is a horizontal asymptote of the graph of f if f(x)→b as x→∞ or x→-∞.

2.6 – Horizontal and Vertical Asymptotes –For f(x) = N(x)/D(x) 1.The graph has vertical asymptotes at the zeros of D(x). 2.The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x). a)If n<m, the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote. b)If n = m, the graph of f has the line y =. Where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. c)If n>m, the graph of f has no horizontal aymptote.

Example Find all horizontal and vertical asymptotes of the graph of each rational function. a)f(x) = b) f(x) = For solution see page 148 in your text.

Example pg. 149 Example 4 For the function f, find the domain, the vertical asymptote, and the horizontal asymptote of f. f(x) =

Example pg. 150 Example 5 This doesn’t happen very often but it’s cool.

2.7 – Graphs of Rational Functions The Graph of a Rational Function Slant Asymptotes

2.7 – The Graph of a Rational Function 1.Simplify f, if possible. 2.Find and plot the y-intercept by evaluating f(0). 3.Find the zeros of the numerator, then plot the corresponding x-intercepts. 4.Find the zeros of the denominator, then plot the vertical asymptotes (dashed lines).

2.7 – The Graph of a Rational Function 5. Find and sketch the horizontal asymptote. 6.Plot at least one point between and one point beyond each x-intercept and vertical asymptote. 7.Use smooth curves to complete the graph between and beyond the vertical asymptotes.

2.7 – Slant Asymptotes Also called “oblique”. The graph of a function has one of these if the degree of the numerator is exactly one more than the degree of the denominator.

Example pg. 159 Example 5 We can practice sketching these functions and see a good example of oblique asymptotes.