Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions, including complex fractions. A rational expression is To simplify a rational expression, x 2 6x 9 x 2 9 3x 2 18x 27 x 3 Multiplying rational expressions is similar to multiplying rational numbers. 3 4x 2 4x3 21 14 4x 5 a b c ac d bd, b, d 0. 28 4a 3 4a5 21 3 49a 4 To multiply one rational expression by another, multiply as with fractions. You can simplify before or after multiplying but it is usually easier to simplify first. 2x 2 3x 2 2 15x3 5x 2 4x 2 1 x 2 4x 5 x 2 3x 2 x 2 4 x 2 30 1
To divide rational expression: a b c d a b d c ad,b,c,d 0. bc 4x 8 5x 20 x2 30 x 2 4x (x 3) 2 x 5 x 2 9 x 2 85 Complex fractions 3x 2 4y 12x 8y 2 x y x y y x y x c. x 2 4x 3 x 2 6x 8 x 2 9x 18 x 2 7x 10 d. 1 12 27x 2 9x 2 2
Section 8.4 Adding and Subracting Rational Expressions OBJECTIVE: Add and subtract rational expressions. Adding two rational expressions with the same denominator Rational Numbers 1 7 3 7 1 3 7 4 7 Rational Expressions 3 x 5 2 x 3 5 8 2 x 2 x 2 2x 3 6x 5 3 2 5 2x 2x 1 To add rational expressions with unlike denominators, LCD (Least Common Denominator) is 3x 2 x 12 x2 1 4 4 x 5 5 x 3 c. x x 5 50 x 2 25 3
d. 4 x 2 3 x 2 x 2 3 e. x y y 1 Write each expression as a single rational expression in simplest form. 5x 5x 2 10 7 7 x x x 7 2x x 2 49 4
Section 8.2 - Rational Functions and Their Graphs OBJECTIVES: Identify and evaluate rational function. Graph a rational function, find the domain, write equations for its asymptotes, and identify any holes in its graph. DEFINITIONS: Polynomial: is a monomial or a sum of terms that are monomials. The exponents of the variables are whole numbers ex. 3x 4 5x 2 2 Rational expression: is the quotient of two polynomials, P where P and Q are polynomials. Q Rational function: is a function defined by a rational expression. R(x) P, P and Q are polynomials Q Domain of a function: is the set of all real numbers except those real numbers that make the denominator equal to zero. Vertical Asymptotes: If (x-a) is a factor in the denominator if a rational function but not a factor of it s numerator, then x=a is a vertical asymptote of the graph of the function. Horizontal Asymptotes: Let P be a rational function. Q If the degree P < degree Q, then y=0 (x-axis) is a horizontal asymptote. If the degree P = degree Q, and a, b are the leading coefficients, then y a is a horizontal asymptote b If the degree P > degree Q, then there is no horizontal asymptote. Hole in the graph: if (x-b) is a factor of the denominator and the numerator of a rational function, then there is a hole in the graph at x=b unless x=b is a vertical asymptote. Determine whether each function is a rational function. If not a rational function, state why not. Variable exponents are whole numbers. The variable can not be under a radical sign ( ), inside an absolute value expression, or in the exponent, etc. 3 x 1 2 x 1 2 x x x 2 7x 12 x 1 c. d. x x 2 2 x 2 e. 2 2x 2 3 Finding the domain of a rational function: [Set the denominator = 0, then solve for x. These are the values of the domain that are excluded from the domain, x ]. f (x) x x g(x) c. h(x) x 5 ()(2x 5) x 2 25 5
What is an asymptote? Find the vertical asymptote(s), if any: [The graph will never cross a vertical asymptote.] f (x) 3 x y 3x x 1 c. f (x) 2x x 2 4 d. g(x) 2x 1 5 Find the horizontal asymptote(s), if any: [The graph sometimes crosses a horizontal asymptote.] f (x) The degree of a polynomial is. x 7 x 2 9x 20 y 4x2 7x 12 2x 2 4 c. y x 4 2x 2 x 5 Find hole(s) in the graph, if any: f (x) x2 x 2 y x 3 x 2 85 c. g(x) (x 3)(x 2)() (x 3)() Find the domain of each rational function. Identify all asymptotes and holes in the graph of each rational function. Then graph. y 4x x 2 y x 2 4x 21 #35. Write a rational function with the given asymptotes and holes. Asym at x = 2 and y = 0 whole at x=3 6
Section 8.5 Solving Rational Equations and Inequalities Objectives: Solve a rational equation or inequality by using algebra, a table, or a graph. Solve problems by using a rational equation or inequality. Definitions: A rational equation is An extraneous solution is A rational inequality is When we have an equation, we can eliminate the fraction. We multiply each side of the equation by the Least Common Denominator (LCD). Always remember to check your answers because sometimes we get an extraneous solution. 2y 1 4y 4 6 5 x 2 1 c. 1 t 1 3 8 3t 7
d. b b 3 b b 2 10 b 2 b 6 e. x 4 x 2 2 x 2 17 x 2 4 f. 3z z 1 2z z 6 5z2 15z 20 z 2 7z 6 g. t 2 2t 3 t 2 2t 3 21 4t 2 9 8