Rational Functions and Their Graphs Davenport Math III

Rational Functions and Their Graphs Davenport Math III

Why Should You Learn This? Rational functions are used to model and solve many problems in the business world. n Some examples of real-world scenarios are: n ¨ Average speed over a distance (traffic engineers) ¨ Concentration of a mixture (chemist) ¨ Average sales over time (sales manager) ¨ Average costs over time (CFO’s)

Introduction to Rational Functions n What is a rational number? A number that can be expressed as a fraction: n What is an irrational number? A number that cannot be expressed as a fraction: n A rational function has the form

Parent Function n The parent function is n The graph of the parent rational function looks like… n The graph is not continuous and has asymptotes

Transformations The parent function n How does this move? n

Transformations n n The parent function How does this move?

Transformations n n The parent function And what about this?

Transformations n The parent function n How does this move?

Transformations

Domain Find the domain of Think: what numbers can I put in for x? ? Denominator can’t equal 0 (it is undefined there)

You Do: Domain Find the domain of Denominator can’t equal 0

You Do: Domain Find the domain of Denominator can’t equal 0

Vertical Asymptotes At the value(s) for which the domain is undefined, there will be one or more vertical asymptotes. List the vertical asymptotes for the problems below. none

Vertical Asymptotes The figure below shows the graph of The equation of the vertical asymptote is

Vertical Asymptotes Definition: The line x = a is a vertical asymptote of the graph of f(x) if or as x approaches “a” either from the left or from the right. Look at the table of values for

Vertical Asymptotes x f(x) -3 -1 -1 1 -2. 5 -2 -1. 5 2 -2. 1 -10 -1. 9 10 -2. 01 -100 -1. 99 100 -2. 001 -1000 -1. 999 1000 As x -2 approaches____ from the _______, left As x -2 approaches____ right from the _______, f(x) approaches _______. Therefore, by definition, there is a vertical asymptote at

Vertical Asymptotes - 4 Describe what is happening to x and determine if a vertical asymptote exists, given the following information: x -4 f(x) -1. 333 -3. 5 -2. 545 x -2 -2. 5 f(x) 1 2. 2222 11. 837 Therefore, a vertical-2. 9 asymptote -12. 16 -2. 99 119. 84 -3. 01 -120. 2 occurs at x =-2. 999 -3. 1199. 8 -3. 001 -1200 -3. 1 As x -3 approaches____ left from the _______, f(x) approaches _______. As x -3 approaches____ right from the _______, f(x) approaches _______.

Vertical Asymptotes Set denominator = 0; solve for x n Substitute x-values into numerator. The values for which the numerator ≠ 0 are the vertical asymptotes n

Example n § § § What is the domain? x ≠ 2 so What is the vertical asymptote? x = 2 (Set denominator = 0, plug back into numerator, if it ≠ 0, then it’s a vertical asymptote)

You Do § Domain: x 2 + x – 2 = 0 § § (x + 2)(x - 1) = 0, so x ≠ -2, 1 Vertical Asymptote: x 2 + x – 2 = 0 § § § (x + 2)(x - 1) = 0 Neither makes the numerator = 0, so x = -2, x = 1

The graph of a rational function NEVER crosses a vertical asymptote. Why? n Look at the last example: Since the domain is , and the vertical asymptotes are x = 2, -1, that means that if the function crosses the vertical asymptote, then for some y-value, x would have to equal 2 or -1, which would make the denominator = 0!

Points of Discontinuity (Holes) Set denominator = 0. Solve for x n Substitute x-values into numerator. You want to keep the x-values that make the numerator = 0 (a zero is a hole) n To find the y-coordinate that goes with that x: factor numerator and denominator, cancel like factors, substitute x-value in. n

Example n n Function: Solve denom. n Factor and cancel n Plug in -2: Hole is

Asymptotes n Some things to note: ¨ Horizontal asymptotes describe the behavior at the ends of a function. They do not tell us anything about the function’s behavior for small values of x. Therefore, if a graph has a horizontal asymptote, it may cross the horizontal asymptote many times between its ends, but the graph must level off at one or both ends. ¨ The graph of a rational function may or may not cross a horizontal asymptote. ¨ The graph of a rational function NEVER crosses a vertical asymptote. Why?

Horizontal Asymptotes Definition: The line y = b is a horizontal asymptote if as or Look at the table of values for

Horizontal Asymptotes x f(x) 1 . 3333 -1 1 10 . 08333 -10 -0. 125 100 . 0098 -100 -0. 0102 1000 . 0009 -1000 -0. 001 0 y→_____ as x→____ 0 as y→____ x→____ Therefore, by definition, there is a horizontal asymptote at y = 0.

Examples Horizontal Asymptote at y = 0 What similarities do you see between problems? The degree of the denominator is larger than the degree of the numerator.

Examples Horizontal Asymptote at y = 2 Horizontal Asymptote at What similarities do you see between problems? The degree of the numerator is the same as the degree or the denominator.

Examples No Horizontal Asymptote What similarities do you see between problems? The degree of the numerator is larger than the degree of the denominator.

Asymptotes: Summary 1. The graph of f has vertical asymptotes at the zeros _____ of q(x). 2. The graph of f has at most one horizontal asymptote, as follows: line y = 0 a) If n < d, then the ______ is a horizontal asymptote. b) If n = d, then the line ______ is a horizontal asymptote (leading coef. over leading coef. ) no c) If n > d, then the graph of f has ______ horizontal asymptote.

You Do Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: x = -1 Horizontal Asymptote: y = 2

You Do Again Find all vertical and horizontal asymptotes of the following function Vertical Asymptote: none Horizontal Asymptote: y = 0

Oblique/Slant Asymptotes The graph of a rational function has a slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator. Long division is used to find slant asymptotes. The only time you have an oblique asymptote is when there is no horizontal asymptote. You cannot have both. When doing long division, we do not care about the remainder.

Example Find all asymptotes. Vertical x=1 Horizontal Slant none y=x

Example n Find all asymptotes: Vertical asymptote at x = 1 y=x+1 n > d by exactly one, so no horizontal asymptote, but there is an oblique asymptote.

Solving and Interpreting a Given Scenario

The Average Cost of Producing a Wheelchair n A company that manufactures wheelchairs has costs given by the function C(x) = 400 x + 500, 000, where the x is the number of wheelchairs produced per month and C(x) is measured in dollars. The average cost per wheelchair for the company is given by …

Original: C(x) = 400 x + 500, 000 n a. § C(x) = 400 x + 500, 000 x Find the interpret C(1000), C(10, 000), C(100, 000). C(1000) = 900; the average cost of producing 1000 wheelchairs per month is $900.

C(x) = 400 x + 500, 000 x § § Find the interpret C(10, 000) = 450; the average cost of producing 10, 000 wheelchairs per month is $450. Find the interpret C(100, 000) = 405; the average cost of producing 100, 000 wheelchairs per month is $405.

C(x) = 400 x + 500, 000 x b. What is the horizontal asymptote for the average cost function? n Since n = d (in degree) then y = 400 n Describe what this represents for the company.

C(x) = 400 x + 500, 000 x The horizontal asymptote means that the more wheelchairs produced per month, the closer the average cost comes to $400. Lower prices take place with higher production levels, posing potential problems for small businesses.