chapter 8 rational functions and...
TRANSCRIPT
Chapter 8
Rational Functions and
Relations
Section 8-1 Multiplying and Dividing Rational Expressions
• Like rational numbers, rational expressions are closed under addition, subtraction,
multiplication and division by a non-‐zero number.
• A rational expression is undefined by any value that makes the denominator equal to zero.
• Just like rational numbers, to simplify a rational expression you divide the numerator
and denominator by their greatest common factor. (GCF) Let’s practice……. 1. Simplify and state the values at which the expression is undefined:
�
3y y2 +10x + 21( )y + 7( ) y2 − 9( )
2. Multiply the expression.
�
8x21y3
⋅7y2
16x3
2b. Multiply the expression.
�
8x21y3
⋅7y2
16x3
3. Divide the expression
�
10xy2
21x2y3÷5y2
14x3
3b. Divide the expression
�
10xy2
21x2y3÷5y2
14x3
When a rational expression has more than one term in the numerator or denominator you may need to factor before you can simplify. Let’s practice……. 4. Simplify the expression, where is this expression undefined.
�
k − 3k +1
⋅1− k2
k2 − 4k + 3
5. Simplify the expression, where is this expression undefined.
�
2d + 6d2 + d − 2
÷d + 3
d2 + 3d + 2
Complex Fraction- Let’s practice……. 6. Simplify the complex fraction.
�
2a2b2
c48ab4
c3
7. Simplify the complex fraction.
�
a2 − b2
3c2 + 6ca + b3c
Section 8-2 Adding and Subtracting Rational Expressions
Least Common Multiple (LCM) Let’s practice: Find the least common multiple of each of the following sets. 1. 8, 12, and 18
2. 6, 12, and 15
3. 12a2b, 15abc, and 8b3c2 4.
�
x3 − x2 − 2x and
�
x2 − 4x + 4
5. Simplify:
�
5a2
6b+
914a2b2
6. Simplify
�
x +103x −15
−3x +156x − 30
7. Simplify:
�
x −1x2 − x − 6
+4
5x +10
Simplifying a complex fraction can be done with different LCD’s or the same LCD… Let’s Practice….
7a. Simplify:
�
1+2x
3y−4x
7b. Simplify:
�
1+2x
3y−4x
Section 8-3 Graphing Reciprocal Functions
Reciprocal Function - has the equation of form A reciprocal function is not defined for any value________________________________________________ The vertical asymptotes show where a function is undefined, while the horizontal asymptotes show the end behavior of a graph. Let’s practice….. Identify the asymptotes, domain, and range of each function. 1. Identify where x is not defined the vertical asymptote is ________ the horizontal asymptote is________ The domain is all real number except ___________ The range is all real numbers except____________
2. Identify where x is not defined the vertical asymptote is ________ the horizontal asymptote is________ The domain is all real number except ___________ The range is all real numbers except____________
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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Chapter 8 20 Glencoe Algebra 2
Skills PracticeGraphing Reciprocal Functions
Identify the asymptotes, domain, and range of each function.
1. 2.
Graph each function. State the domain and range.
3. f(x) = 1 ! x + 3
- 3 4. f(x) = -1 ! x + 5
- 6
f (x)
x
!2
!4
!6
!2!4!6
2
2
f (x)
x
!2
!4
!6
!2!4!6
2
2
5. f(x) = -1 ! x + 1
+ 3 6. f(x) = 1 ! x + 4
- 2
x
f (x)
!2
!2!4!6
2
4
6
2
x
f (x)
x
!2
!4
!6
!2!4!6
2
2
x
f (x)
!2
!4
!2!4
4
2
2 4
f (x) = 1x - 1
y
x!2
2
4
6
2
-1f (x) = x + 4
asymptotes: x = 1, y = 0D = {x | x " 1}R = {f(x) | f(x) " 0}
asymptotes: x = 0, y = 4D = {x | x " 0}R = {f(x) | f(x) " 4}
D = {x | x " -3}R = {f(x) | f(x) " -3}
D = {x | x " -5}R = {f(x) | f(x) " -6}
D = {x | x " -1}R = {f(x) | f(x) " 3}
D = {x | x " -4}R = {f(x) | f(x) " -2}
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Chapter 8 21 Glencoe Algebra 2
Identify the asymptotes, domain, and range of each function.
1. f(x) = 1 ! x - 1
- 3 2. f(x) = 1 ! x + 1
+3 3. f(x) = -3 ! x - 2
+ 5
x-2
-4
-6
-2
2 4
f (x)
x-2-4
2
4
6
2
f (x)
x-2
2
4
6
2 4 6
f (x)
Graph each function. State the domain and range.
4. f(x) = 1 ! x + 1
- 5 5. f(x) = -1 ! x - 3
- 4 6. f(x) = 3 ! x - 2
+ 4
x-2-4
-4
-6
-2
2
f (x)
x-2
-4
-6
-2
2 4
f (x)
x2
2
4
6
4 6-2
f (x)
7. RACE Kate enters a 120-mile bicycle race. Her basic rate is 10 miles per hour, but Kate will average x miles per hour faster than that. Write and graph an equation relating x (Kate’s speed beyond 10 miles per hour) to the time it would take to complete the race. If she wanted to finish the race in 4 hours instead of 5 hours, how much faster should she travel?
PracticeGraphing Reciprocal Functions
x
t
46
2
8101214
642 10 148 12
x = 1; f(x) = -3 x = -1; f(x) = 3 x = 2; f(x) = 5D = {x | x " 1} D = {x | x " -1} D = {x | x " 2}R = {f(x) | f(x) " -3} R = {f(x) | f(x) " 3} R = {f(x) | f(x) " 5}
D = {x | x " -1} D = {x | x " 3} D = {x | x " 2}R = {f(x) | f(x) " -5} R = {f(x) | f(x) " -4} R = {f(x) | f(x) " 4}
t = 120 # x + 10
;
6 miles per hour
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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Chapter 8 22 Glencoe Algebra 2
Word Problem PracticeGraphing Reciprocal Functions
1. VACATION The Porter family takes a trip and rents a car. The rental costs $125 plus $0.30 per mile.
a. Write the equation that relates the cost per mile to the number of miles traveled.
b. Explain any limitations to the range or domain in this situation.
2. PLANES A plane is scheduled to leave Dallas for an 800-mile flight to Chicago’s O’Hare airport at time t = 0. The departure is delayed for two hours. Write two equations that represent the planes’ speed, r, on the vertical axis as a function of travel time, t, on the horizontal axis. Graph the equations below. How do the two curves relate?
3. BIOLOGY A rabbit population follows the function P(t) = 40!
t + 2+ 10, with P(t)
equal to the rabbit population after t months. Eventually, what happens to the rabbit population?
4. COMPUTERS To make computers, a company must pay $5000 for rent and overhead and $435 per computer for parts.
a. Write the equation relating average cost to make a computer to how many computers are being made.
b. Graph the function you found in part a.
c. What is the minimum number of computers the company needs to make so that the average cost is less than $685?t
r
200300
100
400500600700
321 5 74 6
n
C
200300
100
400500600700
302010 50 7040 60
The rabbit population stabilizes at 10 rabbits.
They cannot travel zero miles or a negative number of miles.
C = 125 ! m + 0.3
The second graph is the ! rst graph translated 2 units to the right.
r = 800 ! t - 2
r = 800 ! t ;
C = 5000 ! n + 435
20
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Chapter 8 23 Glencoe Algebra 2
EnrichmentQueue Lengths
An engineer is planning a portion of the interstate highway system, and has to decide how many toll gates to construct at a given place on the road. It is expected that, on the average, one car will pass the given place every 6 seconds. It is also assumed that the average time to collect a toll is 5 seconds.
Let U = average service time
!!! average time between customer arrivals
.
Thus, in this case, U = 5 ! 6 .
A line of waiting customers is called a “queue.” A queue length of one means one person is being served; a queue length of two means one person is being served and one person is waiting to be served; and so on.
According to queue theory, the average queue length (assuming random arrival and service time) is U !
(1 - U) .
1. What is the average queue length if only one toll gate is installed in the situation described above? 5
2. If two toll gates are installed, the average time between customer arrivals at each gate should double. Find the new value for U and the new average queue length. U = 5 !
12 ; queue length: 5 !
7
3. If three toll gates are installed, find the values for U and the average queue length. U = 5 !
18 ; queue length: 5 !
13
4. If n toll gates are installed, write expressions for U and the average queue length. U = 5 !
6n ; queue
length: 5 ! 6n - 5
5. In a different part of the expressway, the average time between customer arrivals is predicted to be 2 seconds while the average service time remains 5 seconds. If 4 toll gates were built, what would be the average queue length? 5 !
3
6. What will happen on the expressway if the value of U is greater than or equal to one? The queue length would increase without bound.
7. In a third part of the expressway, it is expected that one car will pass the proposed toll gate site each second while the average service time remains 5 seconds. Find the number of toll gates needed for an average queue length of less than 2. 8 or more gates
8. What other things can be done to reduce queue length besides adding toll gates? Can you think of other applications for queue theory? Decrease service time with automatic tellers. Grocery store checkout lines, ticket lines for concerts or sports events
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Section 8-5 Variation Functions
Read pages 562 – 565. Study the examples. Direct Variation: Constant of variation: Example: If y varies as x and y = 15 when x = 5, find y when x = 7. What is the constant of variation? Joint variation: Example: It is stated that y varies jointly as x and z. If y = 12 when z = 8 and x = 3
using this same joint variation find y when x = 10 and z = 5.
What is the constant of variation? Inverse variation: Example: If r varies inversely as t and r = 6 when t = 2, find r when t =7.
Example: The intensity, I, of light (in ft-‐candles) received from a source varies inversely as the square of the distance d (in ft), from the source. If the light intensity is 2 ft-‐candles at 13 feet, find the light intensity at 19 feet, to the nearest hundredth.
Combined variations – Example: Suppose f varies directly as g, and f varies inversely as h. Find g when f = 6 and h = 5, if g = 18 when h = 3 and f = 5.
Section 8-6 Solving Rational Equations and Inequalities
Rational Equation:
1. Find the least common denominator (LCD) 2. Multiply each term by the common denominator or make common denominators. 3. Simplify the fractions. 4. Solve the equation. 5. Check your answer.
Let’s practice….
1.
�
5y − 2
+ 2 =176 2.
�
524
+23− x
=14
3.
�
7n3n + 3
−5
4n + 4=12
4.
�
x − 2x + 2
+1
x − 2>x − 4x − 2
1. You have 16 ounces of a 10% brine (salt and water) solution. How much of an 80% salt solution will you need to have to make your brine solution 30% brine?
2. Jason’s takes 5 hours to travel 24 kilometers downstream and the same distance back in his boat. If the current in the river is 2 kilometers per hour, what is the speed of Jason’s boat in still water?
3. Willy and Lou mow lawns together. Willy can mow all the lawns by himself and
complete the job in 20 hours. If Lou does the lawns himself he can complete the job alone in 15 hours. How long will it take them to complete the job when they work together?