All quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas

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1 Chapter Three: Polynomial and Rational Functions 3.1: Quadratic Functions Definition: Let a, b, and c be real numbers with a 0. The function f (x) = ax 2 + bx + c is called a quadratic function. All quadratic functions have graphs that are U -shaped and are called parabolas. Let s look at some parabolas f (x) = x 2 4x + 5 g (x) = (x 2) Standard Form of a Quadratic Function f (x) = a(x h) 2 + k, a 0 The graph of f is a parabola whose axis of symmetry 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. Sketch the graph and identify the vertex of: f (x) = 2x 2 + 8x + 7 From J. P. Wood, Spring

2 Write f (x) = -x 2 + 6x 8 in standard form and sketch graph For f (x) = ax 2 + bx + c, the vertex is:v(h, k) = (-, f(- )) To graph any quadratic function: 1. Determine from the lead coefficient a if the parabola opens upward or downward. 2. Determine the vertex of the parabola. 3. Find any x- or y-intercepts. 4. Plot ordered pairs and connect smoothly Graph: f (x) = (x + 3) and g (x) = -x 2 2x + 1 b 2a b 2a Maximum and Minimum Values of Quadratics 1. If a > 0, the minimum functional value is f (- ) or k 2. If a < 0, the maximum functional value is f (- ) or k b 2a b 2a From J. P. Wood, Spring

3 3.2: Polynomial Functions and Their Graphs Definition of a 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 defined by f (x) = a n x n + a n-1 x n-1 + a 2 x 2 + a 1 x + a 0 is called a polynomial function of degree n. The leading number a n, the coefficient of the variable to the highest power, is called the leading coefficient Graphs of degree two or higher are smooth and continuous Graphs of polynomials do not have breaks or sharp corners From J. P. Wood, Spring

4 End Behavior of Polynomial Functions Leading Coefficient Test: As x moves without bound to the left or to the right, the graph of the function f (x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0 eventually rises or falls in the following manner: When n is odd: When n is even: Using the Lead Coefficient Test From J. P. Wood, Spring

5 Zeroes of Polynomial Functions: x-values for which f (x) = 0 These values of x are called zeroes or roots or solutions to the equation. For a polynomial of degree n, the graph has at most n 1 turning points (graphs change from increasing to decreasing or vice versa. Also, the function has at most n zeroes. Real Zeroes of Polynomial Functions Find all the real zeroes of: f (x) = x 3 x 2 2x Multiplicity of Zeroes: f (x) = -x 4 + 4x 3 4x 2 Notice each factor occurs twice If the same factor, x r, occurs k times, but not k + 1 times, we call r a zero with multiplicity k Notice that k, an integer, can be odd or even 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. From J. P. Wood, Spring

6 Find the zeroes and multiplicities of: f (x) = ½(x + 1)(2x 3) 2 The Intermediate Value Theorem Let f be a polynomial with real coefficients. If f (a) and f (b) have opposite signs, then there is at least one value c between a and b for which f (c) = 0. f (x) has at least one real root between a and b. a b Strategy for Graphing Polynomial Functions (p.336) 1. Use Lead coefficient Test to determine graph s end behavior 2. Find x-intercepts and multiplicities of roots, if any. 3. Find the y-intercept 4. Use symmetry, if applicable, to draw the graph. 5. Check number of turns on graph is not more than n 1. Graph: f (x) = x 4 2x From J. P. Wood, Spring

7 3.5: Rational Functions and Their Graphs A rational function is a function that can be written in form: N(x) D(x) f (x) = D(x) 0 The Domain of a rational function is the set of real numbers except the x- values that make the denominator zero. Find the domain of the following functions: a. b. c. 2x + 1 Consider f (x) = x 1 [-6, 6]1; [-4, 6]1 Definition of Vertical and Horizontal Asymptotes 1. A line x = a is a vertical asymptote of the graph of f if: f (x) or f (x) - as x a + or as x a 2. A line y = b is a horizontal asymptote of the graph of f if: f (x) b as x + From J. P. Wood, Spring

8 To find Asymptotes of Rational Functions: Let N(x) a n x n + a n-1 x n a 1 x + a 0 f (x) = D(x) = b m x m + b m-1 x m b 1 x + b 0 Where N(x) and D(x) have no common factors The graph of f (x) has vertical asymptotes at the zeroes of D(x) The vertical asymptotes are the lines x = a, where a is a zero of the denominator. The graph of f (x) has one or no horizontal asymptote determined by comparing the degrees of N(x) and D(x). If n < m, then f (x) has horizontal asymptote of y = 0 If n = m, then f (x) has horizontal asymptote of y = If n > m, then f (x) has no horizontal asymptote. a n b m Find the following horizontal asymptotes: a. b. c. From J. P. Wood, Spring

9 Using Transformations to Graph Rational Functions 1 1 Use the graph of f( x) = to graph g( x) x = ( x 2) + From J. P. Wood, Spring

10 Guidelines for Graphing Rational Functions Let f (x) = N(x) D(x) where N(x) and D(x) are polynomials with no common factors 1. Determine if the graph of f has symmetry. 2. Find and plot the y-intercept (if any) by evaluating f (0). 3. Find the zeroes of the numerator (if any) by solving N(x) = 0. Plot the corresponding x-intercepts. 4. Find the zeroes of the denominator (if any) by solving D(x) = 0. Sketch the corresponding vertical asymptotes. 5. Find and sketch the horizontal asymptote (if any) using the rules. 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. Sketch the graph of f (x) = 2x 1 x From J. P. Wood, Spring

11 f (x) = 2(x 2 9) x 2 4 Slant Asymptotes: Remember if the degree of N(x) is greater than the degree of D(x) we have no horizontal asymptote. However, if the degree of the numerator is exactly one greater than the degree of the denominator, the graph of the function has a slant asymptote. f (x) = x 2 x x + 1 Long Division From J. P. Wood, Spring

12 3.6: Polynomial and Rational Inequalities A polynomial inequality is any inequality that is in form: f (x) < 0 f (x) > 0 f (x) < 0 f (x) > 0 Five Step Process for solving (p 385): 1. Express inequality as f (x) < 0 2. Solve for f (x) = 0 to obtain boundary points 3. Graph boundary points (critical #s) on the # line 4. Test #s within boundaries/partitions for solution set 5. Write solution set Solve the following inequalities and draw solutions on a number line Solving a Rational Inequality From J. P. Wood, Spring

13 3.7: Modeling Using Variation Variation formulas show how one quantity changes in relation to other quantities. Quantities can vary directly, inversely, or jointly. C = 0.02PP d Direct Variation: If a situation is described by an equation of form: y = kx where k is a nonzero constant, we say that y varies directly as x or y is directly proportional to x. k is the constant of variation or the constant of oportionality Solving Variation Problems 1. Write an equation that models the problem. 2. Substitute given values into the equation and solve for k. 3. Substitute the value k into the equation in step 1 above. 4. Use equation in step 3 to answer the problem s question. Solving a direct variation problem: The volume of blood, B, in a person s body varies directly as body weight, W. A person who weighs 160lbs has approximately 5 quarts of blood. Estimate the number of quarts of blood in a person who weighs 200lbs. From J. P. Wood, Spring

14 Direct Variation with Powers: y varies directly as the n th power of x if there exists some nonzero constant k such that: y = kx n We can also say that y is directly proportional to the n th power of x Solving a direct variation problem with powers The distance, s, that a body falls from rest varies directly as the square of the time, t, of the fall. If a skydiver falls 64ft in 2 seconds, how far will she fall in 4.5 seconds? Inverse Variation If a situation is described by an equation of k form: y = where k is a nonzero constant, we say x that y varies inversely as x or y is inversely proportional to x. The number k is the constant of variation Solving an inverse variation problem In general, if the temperature remains constant, the pressure, P, of a gas in a container varies inversely as the volume, V, of the container. The pressure of a sample gas in a container whose volume is 8in 3 is 12 lb/in 2. If the sample gas expands to a volume of 22in 3, what is the new pressure of the gas? From J. P. Wood, Spring

15 Combined Variation: In combined variation, direct and inverse variation occur at the same time Example: A company has determined that the monthly sales, S, of its product varies directly as its advertising budget, A, and inversely as the selling price of the product, P. When $60,000 is spent on advertising and the selling price is $40/unit, the monthly sales are 12,000 units. Determine monthly sales of units if advertising is increased by $10,000. Joint Variation: a variation in which a variable varies directly as a product of two or more variables. mm 1 2 F = G d 2 The force of gravitation, F, between two bodies varies jointly as the product of their masses, m 1 and m 2, and inversely as the square of the distance between them, d. (G is a gravitational constant) From J. P. Wood, Spring