Chapter 2: Polynomial and Rational Functions

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1 Chapter 2: Polynomial and Rational Functions Section 2.1 Quadratic Functions Date: Example 1: Sketching the Graph of a Quadratic Function a) Graph f(x) = 3 1 x 2 and g(x) = x 2 on the same coordinate plane. b) Graph f(x) = 2x 2 and g(x) = x 2 on the same coordinate plane. The Standard Form of a Quadratic Function is f(x) = a(x-h) 2 + k The graph f is a parabola whose axis is the vertical line x = h and whose vertex is the point (h, k). If a > 0, the parabola opens upward, and if a < 0, the parabola opens downward. 1

2 Example 2: Writing a Quadratic Function in Standard Form Write f(x) = 2x 2 + 8x + 7 in standard form, sketch, and identify the vertex. Example 3: Write f(x) = -x 2 + 6x 8 in standard form, sketch, and identify the vertex. Example 4: Find an equation for the parabola whose vertex is (1,2) and that passes through the point (0,0), as shown. 2

3 Example 5: The Maximum Height of a Baseball A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f(x) = x 2 + x + 3 where f(x) is the height of the baseball (in feet) and x is the distance from home plate (in feet). What is the maximum height reached by the baseball. Example 6: Charitable Contributions According to a survey conducted by Independent Sector, the percent of income that Americans give to charities is related to the amount of income. For families with annual incomes of $100,000 or less, the percent can be modeled by P =.0014x x , 5 x 100 where x is the annual income in thousands of dollars. According to this model, what income level corresponds to the minimum percent of charitable contributions? 3

4 Section 2-3: Polynomial and Synthetic Division Long Division of Polynomials: Example 1: Long Division of Polynomials Use your calculator to graph: f (x) = 6x 3 19x 2 +16x 4 Notice that a zero of f occurs at x = 2. Since x = 2 is a zero of f, you know that (x 2) is a factor of f(x). Divide 6x 3 19x 2 +16x 4 by x 2, and use the result to factor the polynomial completely. The Division Algorithm: f (x) = d(x)q(x) + r(x) where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If the remainder r(x) is zero, d(x) divides evenly into f(x). The Division Algorithm can also be written as Example #2: Long Division of Polynomials Divide x 3 1 by x 1. 4

5 Example #3: Long Division of Polynomials Divide 2x x 3 5x 2 + 3x 2 by x 2 + 2x 3. Synthetic Division: Synthetic Division a shortcut for long division of polynomials Note: Synthetic division works only for divisors of the form x k. REMEMBER: x + k = x (-k) Example #4: Using Synthetic Division Use synthetic division to divide x 4 10x 2 2x + 4 by x

6 The Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder is r = f(k). Example #5: Using the Remainder Theorem Use the Remainder Theorem to evaluate the following function at x = 2. f (x) = 3x 3 + 8x 2 + 5x 7 The Factor Theorem: A polynomial f(x) has a factor (x k) if and only if f(k) = 0. Example #6: Factoring a Polynomial Show that (x 2) and (x + 3) are factors of f (x) = 2x 4 + 7x 3 4 x 2 27x 18. Summary: The remainder r, obtained in the synthetic division of f(x) by x k, provides the following information: 1.) The remainder r gives the value of f at x = k. That is, r = f(k). 2.) If r = 0, (x k) is a factor of f(x). 3.) If r = 0, (k, 0) is an x-intercept of the graph of f. 6

7 The imaginary unit i = 1 where i 2 = -1. Section 2.5 Complex Numbers By adding real numbers to multiples of the imaginary unit, you obtain complex numbers. In standard form, a complex number is a + bi. To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. Example 1: a) (3 i) + (2 + 3i) = b) 2i + (-4 2i) = c) 3 (-2 + 3i) + (-5 + i) = The Associative, Commutative, and Distributive Properties also apply to complex numbers. Example 2: a) (i)(-3i) = b) (2 i)(4 + 3i) 7

8 c) (3 + 2i)(3 2i) d) (3 + 2i) 2 Example 3: Writing Complex Numbers in Standard Form a) 3 12 b) c) ( ) 2 Example 4: Complex Solutions of a Quadratic Equation Solve 3x 2 2x + 5 = 0 8

Remember: The domain of a rational function of x includes all real numbers except x-values that make the

9 Section 2-7: Rational Functions INTRODUCTION Rational Function can be written in the form f (x) = p(x) q(x) where p(x) and q(x) are polynomials. Remember: The domain of a rational function of x includes all real numbers except x-values that make the denominator zero! Example 1: Finding the Domain of a Rational Function Find the domain of f (x) = 1 and discuss the behavior of f near any excluded x-values. Graph this x function using your calculator. 9

10 HORIZONTAL AND VERTICAL ASYMPTOTES Definition of Horizontal and Vertical Asymptotes: 1.) The line x = a is a vertical asymptote of the graph of f if as x a f (x) or f (x), either from the right or from the left. 2.) The line y = b is a horizontal asymptote of the graph of f if as x f (x) b or x. The following graphs show the horizontal and vertical asymptotes of three rational functions. 10

11 Asymptotes of a Rational Function: Let f be the rational function given by f (x) = p(x) q(x) = a n x n + a n 1 x n 1 + +a 1 x + a 0 b m x m + b m 1 x m 1 + +b 1 x + b 0 where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x). 2. The graph of f has one or no horizontal asymptote determined as follows. a. If n < m, the graph of f has the x-axis (y = 0) as a horizontal asymptote. b. If n = m, the graph of f has the line y = a n b m as a horizontal asymptote. c. If n > m, the graph of f has no horizontal asymptote. Example #2: Finding Horizontal Asymptotes Find the horizontal asymptotes of the following functions. a.) f (x) = 2x 3x 2 +1 b.) f (x) = 2x2 3x

12 c.) f (x) = 2x3 3x 2 +1 SKETCHING THE GRAPH OF A RATIONAL FUNCTION Guidelines for Graphing Rational Functions: Let f (x) = p(x) q(x), where p(x) and q(x) are polynomials with no common factors. 1. Find and plot the y-intercept (if any) by evaluating f (0). 2. Find the zeros of the numerator (if any) by solving the equation p(x) = 0. Then plot the corresponding x-intercepts. 3. Find the zeros of the denominator (if any) by solving the equation q(x) = 0. Then sketch the corresponding vertical asymptotes. 4. Find and sketch the horizontal asymptote (if any) by using the rule. 5. Plot at least one point between and at least one point beyond each x-intercept and vertical asymptote. 6. Use smooth curves to complete the graph between and beyond the vertical asymptotes. 12

13 Example #3: Sketching the Graph of a Rational Function Sketch the following functions. a.) g(x) = 3 x 2 b.) f (x) = 2x 1 x c.) f (x) = x x 2 x 2 13

14 d.) f (x) = 2 ( x 2 9) x 2 4 SLANT ASYMPTOTES If the degree of the numerator of a rational function is exactly one more than the degree of the denominator, the graph of the function has a slant asymptote. To find the equation of a slant asymptote, use long division. Ex: Find the slant asymptote of the following function, and sketch. f (x) = x 2 x x +1 Example #7: A Rational Function with a Slant Asymptote Sketch the graph of f (x) = x 2 x 2 x 1 14

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)