Rational Functions. By: Kaushik Sriram, Roshan Kuntamukkala, and Sheshanth Vijayakumar

Transcription

1 Rational Functions By: Kaushik Sriram, Roshan Kuntamukkala, and Sheshanth Vijayakumar

2 What are Rational Functions? Dictionary Definition: In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials. Are you confused? Let s break it down: Rational functions are like ratios of two polynomials Ex. Rational Functions They are set up as fractions, with a numerator and denominator, which are both polynomials. Example:

3 Types of Variation Direct Variation Y varies directly with X K is always the constant of variation, also referred to as the constant of proportionality As X increases, Y increases and as X decreases, Y decreases Inverse Variation Y varies inversely with X This is a Rational Function As X increases Y decreases and as X decreases Y increases

4 More Types of Variation Joint Variation One variable varies jointly with more than one variable Combined Variation Variable varies directly with some variables and varies inversely with others

5 Simplifying Rational Expressions Steps 1. Factor both the numerator and denominator of the polynomial. 2. Look for common factors and cancel them out 3. Make sure to exclude any canceled X s and X s that make the denominator equal to zero in your solution!

6 Practice Example Simplify (8x^2+10x-3)/(2x^-7x-15) 1. By Factoring we get (2x+3)(4x-1)/(x-5)(2x+3) 2. Because we have (2x+3) on both the top and bottom, we can cancel these. Remember when canceling an expression to always exclude it from the solution! 3. We then get our final answer, (4x-1)/(x-5) where X cannot be equal to 5 or -3/2 Simplify (2x^2+x)/(2x^3+7x^2+3x) Now try this one on your own! Solution: 1/(x+3) where X cannot be 0,-3,and -½

7 Adding and Subtracting Rational Functions Steps 1. Factor your Denominators of your rational functions and find the LCM/LCD. The easiest way to do this is to multiply your denominators but sometimes you can find a simpler LCD. 2. Next, change the numerators of the rational fractions to reflect the change in denominator 3. Do the computation and then put your answer over the LCD. 4. Take out X values that make the denominator zero throughout the expression!

8 Multiplying and Dividing Rational Functions Steps 1. Factor all numerators and denominators 2. Cancel all of your common factors. These zeroes must be removed from your final solution as well as X values that make the denominator zero. 3. Multiply the denominators and numerators together or leave the solution in factored form. 4. In division, remember to multiply by the reciprocal, then perform these same steps.

9 Solving Rational Equations To solve Rational Equations, make the proportion (a/b)=(c/d) and then cross multiply Be cautious of Extraneous Solutions as the denominator cannot be zero. Let s watch a video from PatrickJMT as an example!

10 Holes and VA/HA Horizontal Asymptote: 1. If the degree of the numerator is less than that of the denominator, then the graph has y = 0, or the X-axis, as the horizontal asymptote. 2. If the degrees of the numerator and the denominator are same, then the graph has Y=B, where B is the ratio of the leading coefficients ot the numerator and denominator. 3. If the degree of the numerator is greater than that of the denominator, then the graph does not have horizontal asymptotes. Holes: Holes are found in cancelled zeroes of the rational function. In a cancelled expression (X-A) The hole would be at X=A. Vertical Asymptote: Vertical Asymptotes are the zeroes found in the denominator after finding holes.

11 Review Questions 1. If y varies inversely as x, and y = 32 when x = 3, find x when y = If y varies jointly as x and z, and y = 33 when x = 9 and z = 12, find y when x = 16 and z = (x-2)/(x+1)+(3/x) Answers: 1.96/ /9 3.(X^2+X+3)/(X^2+X) (1/6k^2)=(1/3k^2)-(1/k) /6 6.