Warm-up Solve the following rational equation.

Set Equation to ZERO Next Slide

Problem Continued MUST CHECK ANSWERS x = -4 does not work

Rational Function Discontinuities Section 2-6 Rational Function Discontinuities

Objectives I can identify Graph Discontinuities Vertical Asymptotes Horizontal Asymptotes Slant Asymptotes Holes I can find “x” and “y” intercepts

Rational Functions A rational function is any ratio of two polynomials, where denominator cannot be ZERO! Examples:

Asymptotes Asymptotes are the boundary lines that a rational function approaches, but never crosses. We draw these as Dashed Lines on our graphs. There are three types of asymptotes: Vertical Horizontal (Graph can cross these) Oblique (Slant)

Vertical Asymptotes Vertical Asymptotes exist where the denominator would be zero. They are graphed as Vertical Dashed Lines There can be more than one! To find them, set the denominator equal to zero and solve for “x”

Example #1 Find the vertical asymptotes for the following function: Set the denominator equal to zero x – 1 = 0, so x = 1 This graph has a vertical asymptote at x = 1

y-axis Vertical Asymptote at x = 1 x-axis 9 8 7 6 5 4 3 2 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 x-axis -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9

Finding Asymptotes VERTICAL ASYMPTOTES There will be a vertical asymptote at any “x” value, so anywhere that would make the denominator = 0 So there are vertical asymptotes at x = 4 and x = -1. VERTICAL ASYMPTOTES Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

Other Examples: Find the vertical asymptotes for the following functions:

To find Vertical Asymptote(s) 1) Set reduced denominator = 0 Solve for x = #. Your answer is written as a line.

Horizontal Asymptotes Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary. To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator See next slide:

Horizontal Asymptote (HA) Given Rational Function: Compare DEGREE of Numerator to Denominator If N < D , then y = 0 is the HA If N > D, then the graph has NO HA If N = D, then the HA is

HORIZONTAL ASYMPTOTES We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes. 1 < 2 degree of top = 1 If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0. 1 degree of bottom = 2

HORIZONTAL ASYMPTOTES The leading coefficient is the number in front of the highest powered x term. If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at: y = leading coefficient of top leading coefficient of bottom degree of top = 2 1 degree of bottom = 2 horizontal asymptote at:

OBLIQUE ASYMPTOTES If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder. degree of top = 3 degree of bottom = 2 Oblique asymptote at y = x + 5

Other Examples: Find the horizontal asymptote for the following functions:

To find Horizontal Asymptote(s) 1) Compare DEGREE of numerator and denominator Num BIGGER then NO HA Num SMALLER then y = 0 Degree is SAME then

Oblique Asymptotes (OA) Slant asymptotes exist when the degree of the numerator is one larger than the denominator. Cannot have both a HA and SA To find the SA, divide the Numerator by the Denominator. The results is a line y = mx + b that is the SA.

Example of SA -2

To find Oblique Asymptote(s) 1) DEGREE of Numerator must be ONE bigger than Denominator Divide with Synthetic or Long Division Don’t use the Remainder Get y = mx + b

Holes A hole exists when the same factor exists in both the numerator and denominator of the rational expression and the factor is eliminated when you reduce!

Example of Hole Discontinuity

HOLES To Find Holes 1) Factor. 2) Reduce. A hole is formed when a factor is eliminated from the denominator. Set eliminated factor = 0 and solve for x. 5) Find the y-value of the hole by substituting the x-value into reduced form and solve for y. 6) Your answer is written as a point. (x, y)

To find x- intercept(s) Set reduced numerator = 0 2) Solve for x. 3) Answer is written as a point. (#, 0)

To find y- intercept 1) Substitute 0 in for all x’s in reduced form. Solve for y. Answer is a point. (0, #)

Homework SOARING 21ST CENTURY MATHEMATICS CHAPTER 2 PAGE 33