Chapter 2 Functions and Graphs Section 4 Polynomial and Rational Functions This NEW version of Lesson 2-4B doesn’t cover slant asymptotes.

Learning Objectives for Section 2.4 Polynomial and Rational Functions The student will be able to graph and identify properties of rational functions. The student will be able to solve applications of polynomial and rational functions. Barnett/Ziegler/Byleen Business Calculus 12e

Rational Functions A rational function f(x) is a quotient of two polynomials, n(x) and d(x), for all x such that d(x) is not equal to zero. Example: Let n(x) = x + 5 and d(x) = x – 2, then f(x) = is a rational function whose domain is (−∞,2)∪(2,∞) Barnett/Ziegler/Byleen Business Calculus 12e

Examples of Graphs hole hole

Graphing Rational Functions Determine the domain. Find the locations of holes. Determine the equations of its vertical asymptotes. Determine the equations of its horizontal asymptotes. Find x and y intercepts. Barnett/Ziegler/Byleen Business Calculus 12e

Step 1: Factor & State Domain 𝐷:(−∞,−2)∪(−2,∞) 𝐷:(−∞,2)∪(2,3)∪(3,∞)

Step 2: Find Holes If any factors can be cancelled out, then this indicates that there are holes in the graph. To find the x-coordinate of the hole, set the factor equal to zero and solve for x. Find the y-coordinate by substituting the x-coordinate back into the “reduced” function.

Example (Step 2) - Find the Holes Reduced function x + 6 = 0 x = -6 f(-6) = -6 + 5 = -1 Hole at (-6, -1) Reduced function 𝑓 1.5 = 1 1.5−2 2x – 3 = 0 x = 1.5 Hole at (1.5, -2) = 1 −0.5 =−2

Step 3: Find Vertical Asymptotes Using the “reduced” rational function, determine if there are any values of x that will make the denominator zero. If there are, this indicates that there are vertical asymptotes. You get the equations of these asymptotes by solving the denominator.

Example (Step 3) - Find the Vertical Asymptotes Vertical asymptote is the line x=-4 Vertical asymptotes are the lines x=-3 and x=5

Step 4: Find Horizontal Asymptotes Using the “reduced” rational function 𝑦= 𝑛(𝑥) 𝑑(𝑥) Case 1: If degree of n(x) < degree of d(x) then y = 0 is the horizontal asymptote. (i.e. the x-axis) Case 2: If degree of n(x) = degree of d(x) then y = a/b is the horizontal asymptote, where a is the leading coefficient of n(x) and b is the leading coefficient of d(x). Case 3: If degree of n(x) > degree of d(x) there is no horizontal asymptote.

Example (Step 4) Horizontal Asymptotes deg n(x) < deg d(x) deg n(x) = deg d(x) Horizontal asymptote is y=0 (x-axis) Horizontal asymptote is y=3/4 f x = 2 𝑥 2 +1 𝑥−2 deg n(x) = deg d(x) deg n(x) > deg d(x) Horizontal asymptote is y=-3 No Horizontal asymptote

Drawing the Graph of a Rational Function State the domain. Graph holes as open points Draw asymptotes as dashed lines. Find and graph x and y intercepts. Use your graphing calculator to see how the graph approaches the asymptotes.

Graph 1 𝐷𝑜𝑚𝑎𝑖𝑛:𝑥2, 3 −∞,2 ∪ 2,3 ∪ 3,∞ 1 𝑥−2 =0  no x-intercepts Hole at (3, 1) Vertical Asymptote: x=2 𝐷𝑜𝑚𝑎𝑖𝑛:𝑥2, 3 −∞,2 ∪ 2,3 ∪ 3,∞ Horizontal Asymptote: y=0 f(0) = - ½  y-int = - ½ 1 𝑥−2 =0  no x-intercepts

Average Cost Let 𝐶 𝑥 be a company’s cost function which determines the total cost of making x items. The average cost function is 𝐶 (𝑥) which determines the average cost per item. 𝐶 (𝑥) is always a rational function. The average cost of making x items approaches the horizontal asymptote value. 𝐶 𝑥 = 𝐶(𝑥) 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e

Application of Rational Functions H.I.C. makes surfboards at a fixed cost of $300 per day. On a given day when 20 boards are made, the total costs are $5100. Assuming the total cost per day, C(x), is linearly related to the total output per day, x, write an equation for C(x). Find the average cost function. Sketch a graph of the average cost function, including any asymptotes for 1 ≤ x ≤ 30. What does the average cost per board approach as production increases? Barnett/Ziegler/Byleen Business Calculus 12e

Application of Rational Functions H.I.C. makes surfboards at a fixed cost of $300 per day. On a given day when 20 boards are made, the total costs are $5100. Assuming the total cost per day, C(x), is linearly related to the total output per day, x, write an equation for C(x). y = mx + b (#items, total cost) = (20, 5100) 5100 = m(20) + 300 m = 240 So, C(x) = 240x + 300 Barnett/Ziegler/Byleen Business Calculus 12e

Application of Rational Functions Continued… C(x) = 240x + 300 The average cost per board for an output of x boards is found by 𝐶 𝑥 = 𝐶(𝑥) 𝑥 . Find the average cost function. 𝐶 𝑥 = 240𝑥+300 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e

Application of Rational Functions 𝐶 𝑥 = 240𝑥+300 𝑥 Continued... Sketch a graph of the average cost function, including any asymptotes for 1 ≤ x ≤ 30. vertical asym: x = 0 (y-axis) horizontal asym: y = 240 y1 = (240x + 300)/x Window settings x: [1, 30] y: [0, 500] Barnett/Ziegler/Byleen Business Calculus 12e

Application of Rational Functions Continued… What does the average cost per board approach as production increases? As x increases, the graph approaches the horizontal asymptote y = 240. So the average cost per surfboard approaches $240. Barnett/Ziegler/Byleen Business Calculus 12e