Mini-Lecture 5.3
The Graph of a Rational Function
Learning Objectives:
1. Analyze the Graph of a Rational Function (p. 358)
2. Solve Applied Problems Involving Rational Functions (p. 365)
Examples:
1. Follow Steps 1 through 8 on page 360 to analyze the graph of the function.
3
f ( x) 
( x  1)( x 2  4)
2. Graph the function using a graphing utility, then use MINIMUM to obtain the
4
minimum value, rounded to two decimal places. f ( x)  2 x 
x
3. Find a rational function that might have the given graph.
4. A company is planning to manufacture all-terrain vehicles (ATV’s). Fixed monthly
cost will be $200,000 and it will cost $2500 to produce each ATV.
a) Write the cost function, C, of producing x, ATV’s.
b) Write the average cost function, C , of producing x ATV’s.
c) Find C 1000  .
d) What is the horizontal asymptote for the function, C ?
Teaching Notes:
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Emphasize “Analyzing the Graph of a Rational Function’ in the book.
When using the graphing utility, show graphs in both the Connected mode and the
Dot mode.
Some students will try to skip steps and drawn inaccurate graphs. Encourage
them to be thorough and use graph paper.
Even though this is not a step in the book, when graphing encourage students to
use MAXIMUM and MINIMUM on the graphing utility.
Answer: 1 ) Step1: Domain= x x  1, x  2, x  2 , Step 2: R( x) 
3
( x  1)( x  2)( x  2)
,
Step3: no x-intercepts, y-intercept=-0.75, Step4: neither even nor odd, Step5: vertical asymptotesx=1, x=-2, and x=2, Step6: horizontal asymptote- y=0, Step7: graph below
2) See graph below-minimum of 5.66.
3) f ( x) 
b) C ( x ) 
( x  2)
( x  1)( x  3)
; 4) a. C ( x)  200, 00  2500 x ,
200, 000  2500 x
x
, c) $2700; d) y  2500