Algebra 2 Ch.9 Notes Page 67 P Rational Functions and Their Graphs

Rational Function P(x) and Q(x) are polynomial functions. The domain of f(x) is all reals except where Q(x) = 0 f(x) = P(x)/Q(x) f(x) = 2x + 1 x x = +/- 3 is not a part of the domain

Points of Discontinuity y = -2x x y = 1 x y = (x+2)(x-1) x + 1 No Discontinuity Discontinuity at +/- 2 Discontinuity at -1

Finding Points of Discontinuity y = 1/(x 2 + 2x + 1) What makes the denominator = 0 ? Solve by factoring or using the quadratic formula y = (x+1)/(x 2 + 1)

Asymptotes and Holes in the Graphs y = (x - 2)(x + 1) (x - 2) y = (x + 1) (x - 1)(x + 2) y = (x - 2) (x - 2)(x - 1)

Describe the Asymptotes and Holes y = (x - 2) (x - 2) 2 y = (x - 3)(x + 4) (x - 3)(x - 3)(x + 4) Vert Asym at x = 2 No HOLE Vert Asym at x = 3 HOLE at x = -4

Finding Horizontal Asymptotes y = 3x + 5 x - 2 Divide the Numerator by the Denominator Rewrite the function The graph is a translation The Horizontal Asymptote is at y = 3 y = 2x x y = -2x + 6 x - 1

Properties of Horizontal Asymptotes A Rational Function has at most one Horizontal Asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at y = 0. If the degrees are equal, the graph has a horizontal asymptote at y = a/b. a = leading coefficient Numerator b = leading coefficient of Denominator. y = x 2 /x y = x/x 2 y = 4x 2 /2x 2

Sketching Graphs of Rational Functions y = x + 2 (x + 3)(x - 4) Degree of Denominator Greater Horizontal Asymptote at y = 0 Vertical Asymptote at x = -3 and x = 4 X-Intercept is at -2 When x > 4, y is Positive (Approaches x-axis from the Top) When x < -3, y is Negative (Approaches x-axis from the Bottom)

HW # P505 #1-6,10-13,19-21,25,27,28 Please put your name and class period at the top of the homework. Also include the homework number.