Algebra 2Ch.9 Notes Page 67P67 9-3 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 + 1x2 - 9

x = +/- 3 is not a part of the domain

Points of Discontinuity

y = -2x x2 + 1

y = 1 x2 - 4

y = (x+2)(x-1) x + 1

No Discontinuity Discontinuity at +/- 2 Discontinuity at -1

Finding Points of Discontinuity

y = 1/(x2 + 2x + 1)

What makes the denominator = 0 ?Solve by factoring or using the quadratic formula

y = (x+1)/(x2 + 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 = 2No HOLE

Vert Asym at x = 3HOLE at x = -4

Finding Horizontal Asymptotes

y = 3x + 5 x - 2

Divide the Numerator by the DenominatorRewrite the functionThe graph is a translationThe Horizontal Asymptote is at y = 3

y = 2x2 + 5 x2 + 1

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 Numeratorb = leading coefficient of Denominator.

y = x2/x

y = x/x2

y = 4x2/2x2

Sketching Graphs of Rational Functions

y = x + 2 (x + 3)(x - 4)

Degree of Denominator GreaterHorizontal 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 #739-3 P505 #1-6,10-13,19-21,25,27,28

Please put your name and class period at the top ofthe homework. Also include the homework number.