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

2. 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

3. 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

4. Examples of Graphs
hole
hole

5. 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

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

7. 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.

8. Example (Step 2) - Find the Holes
Reduced function
x + 6 = 0
x = -6
f(-6) =
= -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

9. 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.

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

11. 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.

12. 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

13. 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.

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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