3.5 Other Algebraic Functions

1. 3.5 Other Algebraic Functions Polynomials and rational functions are smaller groups of Algebraic Functions Another group of Algebraic Functions are Rational Power Functions. A rational power function is a function where the exponent is not an integer. n is an integer greater than1 m and n have no common factors.

2. Rational functions can be written different ways: When the rational powers are positive the graph increases. When the rational powers are negative the graph decreases. If m is greater than n, the graph will approach infinity quickly. If m is less than n, the graph will approach infinity slowly. y y x x m<n m>n

3. Ex 1: Sketch the following graphs: Solution: a.) this graph will be shifted 2 units left and 1 unit down. The y-coordinates will be multiplied by a factor of 2. (vertical elongation) The parent function is the square root function. y Since m<n, the graph will increase slowly. 2 x -1

4. b.) The graph will be shifted ¾ of a unit to the right. y All of the output is multiplied by 8 (vertical elongation). The graph is not a line! x

5. Ex 2: Sketch the graph of Use a sign graph to find the domain and to see how the graph approaches the vertical asymptote. Solution: (x - 2) - - - - - - - - - - - - - - - 0 + + + (x + 2) - - - - 0 + + + + + + + + + + + fcn + + + - - - - - - - - - 0 + + + ________________________ -2 0 2 vert asymptote: x = -2 Horiz. Asymptote: y = 1 y x-int: x = 2 y-int: none x

6. Ex 3:Sketch the graph of Notice that the root is in the denominator of the rational function. When this occurs, you will have two horizontal asymptotes. Recall: when you take the square root of a value you get two solutions: one positive and one negative. Denominator cannot be zero. The radicand cannot be zero nor negative. x ≠ -2, 1 (x + 2) - - - 0 + + + + + + + + + + + (x – 1) - - - - - - - - - - - 0 + + + + + + fcn + + - - - - - - 0 + + + + + + _________________________ -2 -1 0 1 2 vert. asmyptotes: x = -2, 1 x-int: x = 2 y-int: none

7. The horizontal asymptotes are the tricky ones!!!  Now distribute the power! We are only concerned with the first term here. Since the degrees of both polynomials is 1, we have horizontal asymptotes at the ratio of their leading coefficients. We have two because the denominator of this function was a square root. Horizontal asymptotes:y = ±1

8. (x – 2) - - - - - - - - - - - - - - 0 + + + + (x + 2) - - - 0 + + + + + + + + + + + (x – 1) - - - - - - - - - - - 0 + + + + + + fcn - - - + + + + + 0 - 0 + + + + _________________________ -2 -1 0 1 2 y x