Polynomial and Rational Functions Chapter 3
Quadratic Functions and Models Section 3.1
Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0)
-30 Quadratic Functions Example. Plot the graphs of f(x) = x 2, g(x) = 3x 2 and 30 20 10-10 -8-6 -4-2 2 4 6 8 10-10 -20
-30 Quadratic Functions Example. Plot the graphs of f(x) = x 2, g(x) = 3x 2 and 30 20 10-10 -8-6 -4-2 2 4 6 8 10-10 -20
Parabolas Parabola: The graph of a quadratic function If a > 0, the parabola opens up If a < 0, the parabola opens down Vertex: highest / lowest point of a parabola
Parabolas Axis of symmetry: Vertical line passing through the vertex
Parabolas Example. For the function f(x) = 3x 2 +12x 11 (a) Problem: Graph the function 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10
Parabolas Example. (cont.) (b) Problem: Find the vertex and axis of symmetry.
Parabolas Locations of vertex and axis of symmetry: Set Set Vertex is at: Axis of symmetry runs through vertex
Parabolas Example. For the parabola defined by f(x) = 2x 2 3x + 2 (a) Problem: Without graphing, locate the vertex. (b) Problem: Does the parabola open up or down?
x-intercepts of a Parabola For a quadratic function f(x) = ax 2 + bx + c: Discriminant is b 2 4ac. Number of x-intercepts depends on the discriminant. Positive discriminant: Two x-intercepts Negative discriminant: Zero x-intercepts Zero discriminant: One x-intercept (Vertex lies on x-axis)
x-intercepts of a Parabola
Graphing Quadratic Functions Example. For the function f(x) = 2x 2 + 8x + 4 (a) Problem: Find the vertex (b) Problem: Find the intercepts.
Graphing Quadratic Functions Example. (cont.) (c) Problem: Graph the function 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10
Graphing Quadratic Functions Example. (cont.) (d) Problem: Determine the domain and range of f. (e) Problem: Determine where f is increasing and decreasing.
Graphing Quadratic Functions Example. Problem: Determine the quadratic function whose vertex is (2, 3) and whose y-intercept is 11. 14 12 10 8 6 4 2-14 -12-10 -8-6 -4-2 2 4 6 8 10 12 14-2 -4-6 -8-10 -12-14
Graphing Quadratic Functions Method 1 for Graphing Complete the square in x to write the quadratic function in the form y = a(x h) 2 + k Graph the function using transformations
Graphing Quadratic Functions Method 2 for Graphing Determine the vertex Determine the axis of symmetry Determine the y-intercept f(0) Find the discriminant b 2 4ac. If b 2 4ac > 0, two x-intercepts If b 2 4ac = 0, one x-intercept (at the vertex) If b 2 4ac < 0, no x-intercepts.
Graphing Quadratic Functions Method 2 for Graphing Find an additional point Use the y-intercept and axis of symmetry. Plot the points and draw the graph
Graphing Quadratic Functions Example. For the quadratic function f(x) = 3x 2 12x + 7 (a) Problem: Determine whether f has a maximum or minimum value, then find it.
Graphing Quadratic Functions Example. (cont.) (b) Problem: Graph f 10 8 6 4 2-10 -8-6 -4-2 2 4 6 8 10-2 -4-6 -8-10
Quadratic Relations
Quadratic Relations Example. An engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon, M.
Quadratic Relations Speed, s Miles per Gallon, M 30 18 35 20 40 23 40 25 45 25 50 28 55 30 60 29 65 26 65 25 70 25
Quadratic Relations Example. (cont.) (a) Problem: Draw a scatter diagram of the data
Quadratic Relations Example. (cont.) (b) Problem: Find the quadratic function of best fit to these data.
Quadratic Relations Example. (cont.) (c) Problem: Use the function to determine the speed that maximizes miles per gallon.
Key Points Quadratic Functions Parabolas x-intercepts of a Parabola Graphing Quadratic Functions Quadratic Relations
Polynomial Functions and Models Section 3.2
Polynomial Functions Polynomial function: Function of the form f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 a n, a n 1,, a 1, a 0 real numbers n is a nonnegative integer (a n 0) Domain is the set of all real numbers Terminology Leading coefficient: a n Degree: n (largest power) Constant term: a 0
Polynomial Functions Degrees: Zero function: undefined degree Constant functions: degree 0. (Non-constant) linear functions: degree 1. Quadratic functions: degree 2.
Polynomial Functions Example. Determine which of the following are polynomial functions? For those that are, find the degree. (a) Problem: f(x) = 3x + 6x 2 (b) Problem: g(x) = 13x 3 + 5 + 9x 4 (c) Problem: h(x) = 14 (d) Problem:
Polynomial Functions Graph of a polynomial function will be smooth and continuous. Smooth: no sharp corners or cusps. Continuous: no gaps or holes.
Power Functions Power function of degree n: Function of the form f(x) = ax n a 0 a real number n>0 is an integer.
Power Functions The graph depends on whether n is even or odd.
Power Functions Properties of f(x) = ax n Symmetry: If n is even, f is even. If n is odd, f is odd. Domain: All real numbers. Range: If n is even, All nonnegative real numbers If n is odd, All real numbers.
Power Functions Properties of f(x) = ax n Points on graph: If n is even: (0, 0), (1, 1) and ( 1, 1) If n is odd: (0, 0), (1, 1) and ( 1, 1) Shape: As n increases Graph becomes more vertical if x >1 More horizontal near origin
Graphing Using Transformations Example. Problem: Graph f(x) = (x 1) 4 4 2-4 -2 2 4-2 -4
Graphing Using Transformations Example. Problem: Graph f(x) = x 5 + 2 4 2-4 -2 2 4-2 -4
Zeros of a Polynomial Zero or root of a polynomial f: r a real number for which f(r) = 0 r is an x-intercept of the graph of f. (x r) is a factor of f.
Zeros of a Polynomial
Zeros of a Polynomial Example. Problem: Find a polynomial of degree 3 whose zeros are 4, 2 and 3. 40 30 20 10-10 -5 5 10-10 -20-30 -40
Zeros of a Polynomial Repeated or multiple zero or root of f: Same factor (x r) appears more than once Zero of multiplicity m: (x r) m is a factor of f and (x r) m+1 isn t.
Zeros of a Polynomial Example. Problem: For the polynomial, list all zeros and their multiplicities. f(x) = 2(x 2)(x + 1) 3 (x 3) 4
Zeros of a Polynomial Example. For the polynomial f(x) = x 3 (x 3) 2 (x + 2) (a) Problem: Graph the polynomial 40 20-4 -2 2 4-20 -40
Zeros of a Polynomial Example. (cont.) (b) Problem: Find the zeros and their multiplicities
Multiplicity Role of multiplicity: r a zero of even multiplicity: f(x) does not change sign at r Graph touches the x-axis at r, but does not cross 40 20-4 -2 2 4-20 -40
Multiplicity Role of multiplicity: r a zero of odd multiplicity: f(x) changes sign at r Graph crosses x-axis at r 40 20-4 -2 2 4-20 -40
Turning points: Turning Points Points where graph changes from increasing to decreasing function or vice versa Turning points correspond to local extrema. Theorem. If f is a polynomial function of degree n, then f has at most n 1 turning points.
End Behavior Theorem. [End Behavior] For large values of x, either positive or negative, that is, for large x, the graph of the polynomial f(x) = a n x n + a n 1 x n 1 + L + a 1 x + a 0 resembles the graph of the power function y = a n x n
End Behavior End behavior of: f(x) = a n x n + a n 1 x n 1 + L + a 1 x + a 0
Analyzing Polynomial Graphs Example. For the polynomial: f(x) =12x 3 2x 4 2x 5 (a) Problem: Find the degree. (b) Problem: Determine the end behavior. (Find the power function that the graph of f resembles for large values of x.)
Analyzing Polynomial Graphs Example. (cont.) (c) Problem: Find the x-intercept(s), if any (d) Problem: Find the y-intercept. (e) Problem: Does the graph cross or touch the x-axis at each x-intercept:
Analyzing Polynomial Graphs Example. (cont.) (f) Problem: Graph f using a graphing utility 80 60 40 20-4 -2 2 4-20 -40-60 -80
Analyzing Polynomial Graphs Example. (cont.) (g) Problem: Determine the number of turning points on the graph of f. Approximate the turning points to 2 decimal places. (h) Problem: Find the domain
Analyzing Polynomial Graphs Example. (cont.) (i) Problem: Find the range (j) Problem: Find where f is increasing (k) Problem: Find where f is decreasing
Cubic Relations
Cubic Relations Example. The following data represent the average number of miles driven (in thousands) annually by vans, pickups, and sports utility vehicles for the years 1993-2001, where x = 1 represents 1993, x = 2 represents 1994, and so on.
Cubic Relations Year, x Average Miles Driven, M 1993, 1 12.4 1994, 2 12.2 1995, 3 12.0 1996, 4 11.8 1997, 5 12.1 1998, 6 12.2 1999, 7 12.0 2000, 8 11.7 2001, 9 11.1
Cubic Relations Example. (cont.) (a) Problem: Draw a scatter diagram of the data using x as the independent variable and M as the dependent variable.
Cubic Relations Example. (cont.) (b) Problem: Find the cubic function of best fit and graph it
Key Points Polynomial Functions Power Functions Graphing Using Transformations Zeros of a Polynomial Multiplicity Turning Points End Behavior Analyzing Polynomial Graphs Cubic Relations
The Real Zeros of a Polynomial Function Section 3.6
Division Algorithm Theorem. [Division Algorithm] If f(x) and g(x) denote polynomial functions and if g(x) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q(x) and r(x) such that where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x).
Division Algorithm Division algorithm f(x) is the dividend q(x) is the quotient g(x) is the divisor r(x) is the remainder
Remainder Theorem First-degree divisor Has form g(x) = x c Remainder r(x) Either the zero polynomial or a polynomial of degree 0, Either way a number R. Becomes f(x) = (x c)q(x) + R Substitute x = c Becomes f(c) = R
Remainder Theorem Theorem. [Remainder Theorem] Let f be a polynomial function. If f(x) is divided by x c, the remainder is f(c).
Remainder Theorem Example. Find the remainder if f(x) = x 3 + 3x 2 + 2x 6 is divided by: (a) Problem: x + 2 (b) Problem: x 1
Factor Theorem Theorem. [Factor Theorem] Let f be a polynomial function. Then x c is a factor of f(x) if and only if f(c) = 0. If f(c) = 0, then x c is a factor of f(x). If x c is a factor of f(x), then f(c) = 0.
Factor Theorem Example. Determine whether the function f(x) = 2x 3 x 2 + 4x + 3 has the given factor: (a) Problem: x + 1 (b) Problem: x 1
Number of Real Zeros Theorem. [Number of Real Zeros] A polynomial function of degree n, n 1, has at most n real zeros.
Rational Zeros Theorem Theorem. [Rational Zeros Theorem] Let f be a polynomial function of degree 1 or higher of the form f(x) = a n x n + a n 1 x n 1 + L + a 1 x + a 0 a n 0, a 0 0, where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of a 0 and q must be a factor of a n.
Rational Zeros Theorem Example. Problem: List the potential rational zeros of f(x) = 3x 3 + 8x 2 7x 12
Finding Zeros of a Polynomial Determine the maximum number of zeros. Degree of the polynomial If the polynomial has integer coefficients: Use the Rational Zeros Theorem to find potential rational zeros Using a graphing utility, graph the function.
Finding Zeros of a Polynomial Test values Test a potential rational zero Each time a zero is found, repeat on the depressed equation.
Finding Zeros of a Polynomial Example. Problem: Find the rational zeros of the polynomial in the last example. f(x) = 3x 3 + 8x 2 7x 12
Finding Zeros of a Polynomial Example. Problem: Find the real zeros of f(x) = 2x 4 + 13x 3 + 29x 2 + 27x + 9 and write f in factored form
Factoring Polynomials Irreducible quadratic: Cannot be factored over the real numbers Theorem. Every polynomial function (with real coefficients) can be uniquely factored into a product of linear factors and irreducible quadratic factors Corollary. A polynomial function (with real coefficients) of odd degree has at least one real zero
Factoring Polynomials Example. Problem: Factor f(x)=2x 5 9x 4 + 20x 3 40x 2 + 48x 16
Bounds on Zeros Bound on the zeros of a polynomial Positive number M Every zero lies between M and M.
Bounds on Zeros Theorem. [Bounds on Zeros] Let f denote a polynomial whose leading coefficient is 1. f(x) = x n + a n 1 x n 1 + L + a 1 x + a 0 A bound M on the zeros of f is the smaller of the two numbers Max{1, a 0 + a 1 + L + a n-1 }, 1 + Max{ a 0, a 1,, a n-1 }
Bounds on Zeros Example. Find a bound to the zeros of each polynomial. (a) Problem: f(x) = x 5 + 6x 3 7x 2 + 8x 10 (b) Problem: g(x) = 3x 5 4x 4 + 2x 3 + x 2 +5
Intermediate Value Theorem Theorem. [Intermediate Value Theorem] Let f denote a continuous function. If a < b and if f(a) and f(b) are of opposite sign, then f has at least one zero between a and b.
Intermediate Value Theorem Example. Problem: Show that f(x) = x 5 x 4 + 7x 3 7x 2 18x + 18 has a zero between 1.4 and 1.5. Approximate it to two decimal places.
Key Points Division Algorithm Remainder Theorem Factor Theorem Number of Real Zeros Rational Zeros Theorem Finding Zeros of a Polynomial Factoring Polynomials Bounds on Zeros Intermediate Value Theorem
Complex Zeros; Fundamental Theorem of Algebra Section 3.7
Complex Polynomial Functions Complex polynomial function: Function of the form f(x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 a n, a n 1,, a 1, a 0 are all complex numbers, a n 0, n is a nonnegative integer x is a complex variable. Leading coefficient of f: a n Complex zero: A complex number r with f(r) = 0.
Complex Arithmetic See Appendix A.6. Imaginary unit: Number i with i 2 = 1. Complex number: Number of the form z = a + bi a and b real numbers. a is the real part of z b is the imaginary part of z Can add, subtract, multiply Can also divide (we won t)
Complex Arithmetic Conjugate of the complex number a + bi Number a bi Written Properties:
Complex Arithmetic Example. Suppose z = 5 + 2i and w = 2 3i. (a) Problem: Find z + w (b) Problem: Find z w (c) Problem: Find zw (d) Problem: Find
Fundamental Theorem of Algebra Theorem. [Fundamental Theorem of Algebra] Every complex polynomial function f(x) of degree n 1 has at least one complex zero.
Fundamental Theorem of Algebra Theorem. Every complex polynomial function f(x) of degree n 1 can be factored into n linear factors (not necessarily distinct) of the form f(x) = a n (x r 1 )(x r 2 ) L (x r n ) where a n, r 1, r 2,, r n are complex numbers. That is, every complex polynomial function f(x) of degree n 1 has exactly n (not necessarily distinct) zeros.
Conjugate Pairs Theorem Theorem. [Conjugate Pairs Theorem] Let f(x) be a polynomial whose coefficients are real numbers. If a + bi is a zero of f, then the complex conjugate a bi is also a zero of f.
Conjugate Pairs Theorem Example. A polynomial of degree 5 whose coefficients are real numbers has the zeros 2, 3i and 2 + 4i. Problem: Find the remaining two zeros.
Conjugate Pairs Theorem Example. Problem: Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 2, 1 and 4 + i.
Conjugate Pairs Theorem Example. Problem: Find the complex zeros of the polynomial function f(x) = x 4 + 2x 3 + x 2 8x 20
Key Points Complex Polynomial Functions Complex Arithmetic Fundamental Theorem of Algebra Conjugate Pairs Theorem
Properties of Rational Functions Section 3.3
Rational Functions Rational function: Function of the form p and q are polynomials, q is not the zero polynomial. Domain: Set of all real numbers except where q(x) = 0
Rational Functions is in lowest terms: The polynomials p and q have no common factors x-intercepts of R: Zeros of the numerator p when R is in lowest terms
Rational Functions Example. For the rational function (a) Problem: Find the domain (b) Problem: Find the x-intercepts (c) Problem: Find the y-intercepts
Graph of Graphing Rational Functions 10 7.5 5 2.5-10 -5 5 10-2.5-5 -7.5-10
Graphing Rational Functions As x approaches 0, is unbounded in the positive direction. Write f(x) Read f(x) approaches infinity Also: May write f(x) as x 0 May read: f(x) approaches infinity as x approaches 0
Graphing Rational Functions Example. For Problem: Use transformations to graph f. 4 2-6 -4-2 2 4-2 -4
Asymptotes Horizontal asymptotes: Let R denote a function. Let x or as x, If the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R.
Asymptotes Vertical asymptotes: Let x c If the values R(x), then the line x = c is a vertical asymptote of the graph of R.
Asymptotes Asymptotes: Oblique asymptote: Neither horizontal nor vertical Graphs and asymptotes: Graph of R never intersects a vertical asymptote. Graph of R can intersect a horizontal or oblique asymptote (but doesn t have to)
Asymptotes A rational function can have: Any number of vertical asymptotes. 1 horizontal and 0 oblique asymptote 0 horizontal and 1 oblique asymptotes 0 horizontal and 0 oblique asymptotes There are no other possibilities
Vertical Asymptotes Theorem. [Locating Vertical Asymptotes] A rational function in lowest terms, will have a vertical asymptote x = r if r is a real zero of the denominator q.
Vertical Asymptotes Example. Find the vertical asymptotes, if any, of the graph of each rational function. (a) Problem: (b) Problem:
Vertical Asymptotes Example. (cont.) (c) Problem: (d) Problem:
Horizontal and Oblique Asymptotes Describe the end behavior of a rational function. Proper rational function: Degree of the numerator is less than the degree of the denominator. Theorem. If a rational function R(x) is proper, then y = 0 is a horizontal asymptote of its graph.
Horizontal and Oblique Asymptotes Improper rational function R(x): one that is not proper. May be written where is proper. (Long division!)
Horizontal and Oblique Asymptotes If f(x) = b, (a constant) Line y = b is a horizontal asymptote If f(x) = ax + b, a 0, Line y = ax + b is an oblique asymptote In all other cases, the graph of R approaches the graph of f, and there are no horizontal or oblique asymptotes. This is all higher-degree polynomials
Horizontal and Oblique Asymptotes Example. Find the hoizontal or oblique asymptotes, if any, of the graph of each rational function. (a) Problem: (b) Problem:
Horizontal and Oblique Asymptotes Example. (cont.) (c) Problem: (d) Problem:
Key Points Rational Functions Graphing Rational Functions Vertical Asymptotes Horizontal and Oblique Asymptotes
The Graph of a Rational Function; Inverse and Joint Variation Section 3.4
Analyzing Rational Functions Find the domain of the rational function. Write R in lowest terms. Locate the intercepts of the graph. x-intercepts: Zeros of numerator of function in lowest terms. y-intercept: R(0), if 0 is in the domain. Test for symmetry Even, odd or neither.
Analyzing Rational Functions Locate the vertical asymptotes: Zeros of denominator of function in lowest terms. Locate horizontal or oblique asymptotes Graph R using a graphing utility. Use the results obtained to graph by hand
Analyzing Rational Functions Example. Problem: Analyze the graph of the rational function Domain: R in lowest terms: x-intercepts: y-intercept: Symmetry:
Analyzing Rational Functions Example. (cont.) (cont.) Vertical asymptotes: Horizontal asymptote: Oblique asymptote:
Analyzing Rational Functions Example. (cont.) (cont.) 4 2-4 -2 2 4-2 -4
Analyzing Rational Functions Example. Problem: Analyze the graph of the rational function Domain: R in lowest terms: x-intercepts: y-intercept: Symmetry:
Analyzing Rational Functions Example. (cont.) (cont.) Vertical asymptotes: Horizontal asymptote: Oblique asymptote:
Analyzing Rational Functions Example. (cont.) (cont.) 6 4 2-6 -4-2 2 4 6-2 -4-6
Variation Inverse variation: Let x and y denote 2 quantities. yvariesinverselywith x If there is a nonzero constant such that Also say: y is inversely proportional to x
Variation Joint or Combined Variation: Variable quantity Q proportional to the product of two or more other variables Say Qvariesjointlywith these quantities. Combinations of direct and/or inverse variation are combined variation.
Variation Example. Boyle s law states that for a fixed amount of gas kept at a fixed temperature, the pressure P and volume V are inversely proportional (while one increases, the other decreases).
Variation Example. According to Newton, the gravitational force between two objects varies jointly with the masses m 1 and m 2 of each object and inversely with the square of the distance r between the objects, hence
Key Points Analyzing Rational Functions Variation
Polynomial and Rational Inequalities Section 3.5
Solving Inequalities Algebraically Rewrite the inequality Left side: Polynomial or rational expression f. (Write rational expression as a single quotient) Right side: Zero Should have one of following forms f(x) > 0 f(x) 0 f(x) < 0 f(x) 0
Solving Inequalities Algebraically Determine where left side is 0 or undefined. Separate the real line into intervals based on answers to previous step.
Solving Inequalities Test Points: Algebraically Select a number in each interval Evaluate f at that number. If the value of f is positive, then f(x) > 0 for all numbers x in the interval. If the value of f is negative, then f(x) < 0 for all numbers x in the interval.
Solving Inequalities Algebraically Test Points (cont.) If the inequality is strict (< or >) Don t include values where x = 0 Don t include values where x is undefined. If the inequality is not strict ( or ) Include values where x = 0 Don t include values where x is undefined.
Example. Solving Inequalities Algebraically Problem: Solve the inequality x 5 16x
Key Points Solving Inequalities Algebraically