Algebra 3-4 Unit 4 Rational 4... · Rational expression: a fraction that has polynomials in the...
Transcript of Algebra 3-4 Unit 4 Rational 4... · Rational expression: a fraction that has polynomials in the...
Name __________________________________________________________________ Period ____________
Algebra 3-4 Unit 4
Rational
4.1-2 I can multiply and divide rational expressions.
4.3-5 I can add and subtract rational expressions.
4.6-8 I can solve rational equations.
4.9-12 I can graph rational functions.
My goal for this unit: _____________________________________________________ ______________________________________________________________________ What I need to do to reach my goal: ________________________________________ ______________________________________________________________________ ______________________________________________________________________
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.1
Multiplying Rational Expressions
Rational expression: a fraction that has polynomials in the numerator and/or denominator.
To multiply two rational expressions: factor, cancel common factors, and simplify.
Remember, you cannot divide by 0 (ask siri why not) so values that makes the denominator 0 must be excluded.
Example Find the product and any excluded values. 2
2 2
2 8 1
1 6
x x x
x x x
− − −
− − −i
Step 1 Factor and multiply. ( 4)( 2)( 1)
( 1)( 1)( 3)( 2)
( 4)( 2)
x x x
x x x x
x x
− + −=
+ − − +
− +=
( 1)x −
( 1)( 1)x x+ − ( 3)( 2)x x− +
4
( 1)( 3)
x
x x
−=
+ −
Step 2 Cancel common factors.
Step 3 Write simplified product.
Step 4 Note excluded values. 2x ≠ − , 1x ≠ − , 1x ≠ , 3x ≠
Directions: Multiply and state all excluded values.
1. 3
6 6
10 3
x x
xi
2. 427 12
19 11
x x
xi
3. 214 41
15 49
x x
xi
4. 2
6 5
65xi
5. 3 21 7 49
9 7
x x
x x
+
+ +i
6. 26 54 7
9 6
x x x
x x
−
−i
7. 18 36 2
4 8 9 18
x
x x
−
− +i
8. ( )2
2
1056 11 15
15 11 56x x
x x+ −
− −i
9. 2
3 2
169 104 48 11
44 169 104 48
x x x
x x x
+ − −
+ −i
10. 2
2 2
3 5 5 62 39
15 34 15 5 50
x x x
x x x x
+ − −
+ + −i
11. ( ) ( ) ( )
( ) ( )
5 1 6 2
3 1 2
x x x
x x
+ − +
− +i
12. 2
2 3 2
5 3 18
3 18 2 16
x x
x x x x
+ −
+ − −i
13. 2 2
2
5 14 20
216
x x x x
xx
− − − −
+−i
14. 2
2 2
14 45 3
6 3 2 50
x x x
x x x
− +
− −i
15. 2 2
2
6 6 8
220
x x x x
xx x
− − − +
++ −i
16. 2
2 2 4
3 36
x x
xx x
− +
−+i
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.2
Multiplying and Dividing Rational Expressions
To divide two rational expressions: take the reciprocal of the divisor then multiply like yesterday.
Remember, you cannot divide by 0 (ask siri why not) so values that makes the denominator 0 must be excluded.
Example Find the quotient and any excluded values. 2 212 9 18
5 2 10
x x x x
x x
− − + +÷
+ +
Step 1 Rewrite as multiplication by
reciprocal of divisor.
2
2
12 2 10
5 9 18
( 4)( 3) 2( 5)
5 ( 6)( 3)
2( 4)( 3)( 5)
( 5)( 6)( 3)
2( 4)( 3)
x x x
x x x
x x x
x x x
x x x
x x x
x x
− − +=
+ + +
− + +=
+ + +
− + +=
+ + +
− +=
i
i
( 5)x +
( 5)x + ( 6) ( 3)x x+ +
2( 4)
( 6)
x
x
−=
+
Step 2 Factor
Step 3 Multiply
Step 4 Cancel common factors.
Step 5 Write simplified product.
Step 6 Note excluded values. 6x ≠ − , 5x ≠ − , 3x ≠ −
Directions: Multiply or divide and state all excluded values.
1. 4 4
5 6
x x
x÷
2. 4 8
3 2
x xi
3. ( )
( ) ( )
6 2 2
1 10 10
x x
x x x
− −÷
− − −
4. ( )214 42
2 610
x xx
++ ÷
5. 227 9 3 8 3
10 10
x x x+ − −÷
6. 3 2
24 56 15 35
510 90
x x
x x
+ +÷
−
7. 3 2
2 20 2
14 3512 30
x
xx x
+÷
−−
8. 2
2 2
5 8 20 40
2 14 20 15 24
x x x
x x x x
− +
+ + −i
9. 215 12 7 14
40 32 3
x x x
x x
+ −
+i
10. ( )
2
5 13 5 30
13 14
x x
x x
+ −÷
+
11. 284 11 132
7 10 77 110
x x
x x
+÷
+ +
12. 2
2 2
2 10 12 1
22 42 20 11 100 100
x x
x x x x
+ −÷
− + + −
13. 77 11 21 3
3 42 3
x x
x
+ +÷
−
14. 2
2
4 2 2
22
x x
xx x
− +
+− −i
15. 2
2 2
6 16
3 3 2
x x x
x x x x
+ −
− − +i
16. 2
2 2
3 12 5
3 15 10 25
x x x
x x x x
− +÷
+ + +
17. A rectangular sandbox has a length of 3 24
2
x
x
+
+ feet and a width of
9
2 16
x
x
+
+ feet. Write a rational
expression that represents the area of the sandbox. Simplify the expression and state any excluded values.
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.3 (alt)
Add and Subtract Rational Expressions Like Denominators
In order to add or subtract fractions, there must be a common denominator.
Add or subtract the numerators
Keep the denominator the same
Simplify if possible
Example 1: 7
5
7
3
7
2=+
Example 2: 3333 6
3
12
26
12
2
12
5
y
yx
y
yx
y
yx
y
x +=
+=
++
Example 3: 3
2
)3)(53(
)53(2
)3)(53(
106
15143
4
15143
6622 −
=−−
−=
−−
−=
+−−
+−
−
xxx
x
xx
x
xxxx
x
Add or subtract and simplify where possible.
1. 3232 30
4
30
4
yx
yx
yx
yx −+
− 2.
255
2
255
3
−
++
− x
x
x
3. 106
6
106
2
+
−+
+ x
x
x 4.
128
1
128
3222
+−
−−
+−
−
xx
x
xx
x
5. 1620
4
1620
64
+
−+
+
−
x
x
x
x 6.
67
4
67
3222
+−
++
+−
−
xx
x
xx
x
7. 164
2
164
6
+
−−
+ x
x
x 8.
3103
5
3103
122
++
−+
++
−
xx
x
xx
x
9. xx
x
xx
x
3012
45
3012
122
+
++
+
+ 10.
20295
1
20295
222
+−
++
+− xx
x
xx
11. 96
4
96
522
+−+
+− xxxx 12.
1253
2
1253
422
−−
++
−−
−
xx
x
xx
x
13. 67
3
67
15322
++
−−
++
+
xx
x
xx
x 14.
xx
x
xx
x
204
5
204
522
+
−−
+
+
15. 60186
2
60186
522
−−
−−
−−
−
xx
x
xx
x 16.
6193
1
6193
2322
+−+
+−
−
xxxx
x
17. 2012
2
2012
6
−
−+
−
−
x
x
x
x 18.
2323 366
3
366
1
xx
x
xx
x
−
+−
−
+
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.3
Add and Subtract Rational Expressions
The addition or subtraction of rational expressions can be compared to the addition or subtraction of fractions.
As with fractions, in order to add or subtract rational expressions, the denominators must be the same. The least
common multiple of the polynomials is the least common denominator (LCD).
Add Fractions Add Rational Expressions
3 1
4 6+
2
5 1
2
x
x x
++
Find the LCD. 12 2x2
Rename each term using
the LCD. 9 2
12 12+ 2 2
5 2 2
2 2
x x
x x
++
Add the numerators. 11
12 2
7 2
2
x
x
+
Directions: Find the least common multiple for each pair.
1. 2x6 and 6x
2. 10xy and 5x4y3
3. (x − 8)(x + 1) and (x + 1)
4. x2 − 5x + 6 and x − 3
5. x2 − 2x − 8 and x2 − 16
6. x2 + 2x − 15 and x2 − 7x + 12
Directions: Write the given expression as an equivalent rational expression that has the given denominator.
7. Expression: 1
2
x
x
+
+
Denominator: 2x2 + 4x
8. Expression: 5
2 3
x
x −
Denominator: 4x2 − 9
9. Expression: 24
4 8
x
x
−
−
Denominator: (2 − x)(x + 3)
10. Expression: 2
2
16
12
x
x x
−
− −
Denominator: (x + 3)(x + 4)(x − 4)
11. Expression: 183
1522
2
−+
−+
xx
xx
Denominator: (x + 6)(x − 3)(x + 3)
12. Expression: 2
122
2
−+
−+
xx
xx
Denominator: (x + 2)(x − 1)(x + 1)
To add or subtract rational expressions, they must have common denominators.
Example Add 2 1
3 1
x x
x x
+ −+
− +.
Step 1 Multiply each rational expression by a
common factor to get equivalent fractions
with a common denominator.
2 1
3 1
x x
x x
+ −+
− +
22 1 ( 2)( 1) 3 2
3 1 ( 3)( 1) ( 3)( 1)
x x x x x x
x x x x x x
+ + + + + += =
− + − + − +i
21 3 ( 1)( 3) 4 3
1 3 ( 1)( 3) ( 3)( 1)
x x x x x x
x x x x x x
− − − − − += =
+ − + − − +i
Step 2 Add the fractions. 2 2 23 2 4 3 2 5
( 3)( 1) ( 1)( 3) ( 1)( 3)
x x x x x x
x x x x x x
+ + − + − ++ =
− + + − + −
Step 3 Give excluded values that make the
denominator 0. 1x ≠ − , 3x ≠
Directions: Add or subtract. Identify any x-values for which the expression is undefined.
13.
2 3 4 5
4 4
x x
x x
− −+
+ +
14.
12 3 2
2 5 2 5
x x
x x
+ −−
− −
15. 2
1 1
x x
x x+
+ +
16. 4 1 2 7
4 4
x x
x x
+ ++
− −
17. 5 1 3
3 2 6
x x
x x
−+
+ +
18. 6 2
2
x
x x−
+
Common Denominators
Multiply
by
1x +
Multiply by
3
3
x
x
−
−
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.4
Add and Subtract Rational Expressions (Day 2)
Directions: Add or subtract. Identify any x-values for which the expression is undefined.
1. 2
7 2
43
x
xx−
+
2.
1
4 3 1
x x
x x
++
− +
3.
3 3
4 2 4 2
x x
x x
++
− +
4.
3 1
2 5
x
x x
++
+ −
5.
4 7
( 2)
x x
x x x
++
−
6.
8 2
3 1
x x
x x
+ −−
− −
7.
3 4
6 ( 5)( 6)
x
x x x−
+ − +
8. 2
3 1
5 7 10x x x−
− − +
9. 2
4 2
412
x x
xx x
++
−− −
10.
2
2
3 1 2
63 18
x x
xx x
− +−
−− −
11. 2
5 10
3 9
x x
x x
− −+
+ −
12. 2
2
32 15
x x
xx x
++
+− −
13. 2 2
4 1
2 6
x x
x x x x
+ −−
− − + −
14. 2
6 2
97 18
x x
xx x
+−
−− −
15. 2 2
3 5
6 8 15
x
x x x x−
− − − +
16. 2
3
43 4
x
xx x+
−− −
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.5
Add and Subtract Rational Expressions (Day 3)
Directions: Add or subtract. Identify any x-values for which the expression is undefined.
1. 2
2
11
x
xx−
+−
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. The electric potential generated by a certain arrangement of electric charges is given by4 1
e e
x x+
− +,
where e is the fundamental unit of electric charge and x measures the location where the potential is being
measured. Express the electric potential as a rational expression.
12. A ferry shuttles from Seattle to Vancouver Island and back. Because of head winds, the return trip is slower than the trip to the island. The average speed of the ferry, in miles per hour, is given by the
expression: 2
.
50 60
d
d d+
What is the average speed of the ferry?
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.6
Solve Rational Equations (Day 1)
Multiply through by the LCD.
11
3
32 +=
− x
x
x
x
LCD is (2x − 3)(x + 11)
11
3)11)(32(
32)11)(32(
++−=
−+−
x
xxx
x
xxx
11
3)11)(32(
32)11)(32(
++−=
−+−
x
xxx
x
xxx
)3)(32())(11( xxxx −=+ (Yeah! No more denominators!)
x2 + 11x = 6x2 − 9x
5x2 − 20x = 0
5x(x − 4) = 0
5x = 0 x − 4 = 0
x = 0 x = 4
Directions: Solve for the variable using the method of cross multiplication. Check each answer.
1. 2
4 2
x x +=
2. 9
4 4
x
x=
3.
2 3 1
7 6
x x
x
+ +=
4. 4
5 9
x
x
−=
−
5. 2
3 7
x
x
−=
+
6. 2
23
x
x
+=
−
Substitute solutions in the original
problem to make sure they are
not extraneous solutions:
110
)0(3
3)0(2
0
+=
−
0 = 0
114
)4(3
3)4(2
4
+=
−
15
12
5
4=
5
4
5
4=
7. 2 4 2
5
x
x x
+=
8. 2 6
6 2
x x
x x
−=
− +
9. 25
3
+=
+
−
x
x
x
x
10. xx
x 1
22
−=
−
11. 2
102
−
+
x
x =
x
4
12. 2
4 1
24 xx=
−−
13. 4 2 8
4 4
x x
x x
+=
− −
14. 2
5 10
1 2
x x
x x x
−=
+ − −
15. 6
2 2 1
x x
x x
−=
+ −
16. 2
2
2 3 15
5 9 20
x x x
x x x
−=
− − +
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.7
Solve Rational Equations (Day 2)
Multiply through by the LCD.
Example Solve the rational equation algebraically. 2
1 2
2 4 6 8
x
x x x x+ =
− − − +
Step 1 Multiply each term by the LCD. 1 2
( 2)( 4) ( 2)( 4) ( 2)( 4)2 4 ( 2)( 4)
xx x x x x x
x x x x− − + − − = − −
− − − −
Step 2 Cancel common factors.
2
x
x −( 2)x −
1( 4)
4x
x− +
−( 2) ( 4)x x− −
2
( 2)( 4)x x=
− −( 2)( 4)x x− −
( 4) ( 2) 2x x x− + − =
Step 3 Simplify and solve the remaining equation.
2
2
4 2 2
3 4 0
( 4)( 1) 0
x x x
x x
x x
− + − =
− − =
− + =
4x = or 1x = −
Step 4 Check for extraneous solutions that are excluded values. 4x = is an excluded value.
1x = − is the solution.
Directions: Solve for the variable using the method of cross multiplication. Check each answer.
1. 9 2 2
1x
x x
++ =
2. 3 5
24 4x x
− =+ +
3. 6 2
43 3x x
− =− −
4. 3 1 4
12 x x
+ = +
Multiply by LCD
( 2)( 4)x x− −
1 2
2 4 ( 2)( 4)
x
x x x x+ =
− − − − Factor the
denominator2
( 2)( 4)x x− −
5. 1 2 4
3 3
x
x x x
−− =
6. 2 2
5 5 5 1
4 4
x
xx x x x
−− =
− −
7. 2 7 10 1
4x x
xx x
− ++ = +
8. 2
6 71
x x− + =
9. 2
2
8 121
3 2 13 21
x x x
x x x
− ++ =
+ + +
10. 2 27 10 13 40
5 30 6 5 30
x x x x x
x x x
+ + − ++ =
− − −
11. 1
2 4x
+ =
12. 3
2xx
+ =
13. 5 4
36 x
+ =
14. 12
4 3x
+ =
15. 1
3 2
x xx
x
++ =
+
16. 2 = 41
3−
+x
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.8
Solve Rational Equations (Day 3)
Directions: Solve each rational equation. https://my.hrw.com/content/hmof/math/hsm/common/video/video.html#videoId=ref:RW_HSM_ALG2_009_en
1. 4
4
82
)4(42 +
=−+
−
xxx
x
2. 128
4
26
12
+−=
−+
− xxx
x
x
3. 2 1 1
3 3
x
x x x
−+ =
− −
4. 3 5
3 5 5
x x x
x x x
+ ++ =
− − −
5. Fran can clean the garage in 3 hours, but it takes
Angie 4 hours to do the same job. How long would
it take them to clean the garage working together?
6. Kent can paint a certain room in 6 hours, but
Kendra needs 4 hours to paint the same room.
How long would it take them working together?
7. An artist is designing a picture frame whose
length l and width w satisfy the Golden Ratio,
which is .w l
l l w=
+ If the length of the frame is
24 inches, what is the width of the frame?
8. John can pick a bushel of peaches in 30 minutes.
His little sister can pick a bushel of peaches in 45
minutes. How long will it take them to pick a
bushel of peaches working together?
9. Marco can build a lap top twice as fast as Cliff.
Working together, it takes them 5 hours. How
long would it have taken Marco working alone?
10. Kiyoshi can paint a certain fence in 3 hours by
himself. If Red helps, the job takes only 2 hours.
How long would it take Red to paint the fence by
himself?
11. Every week, Linda must stuff 1000 envelopes.
She can do the job by herself in 6 hours. If Laura
helps, they get the job done in 5 hrs and 30 mins.
How long would it take Laura to do the job by
herself?
12. Mr. McGregor has discovered that a large dog can
destroy his entire garden in 2 hours and that a small
boy can do the same job in 1 hour. How long
would it take the large dog and the small boy
working together to destroy Mr. McGregor’s
garden?
13. Edgar can blow the leaves off the sidewalks
around the capitol building in 2 hours using a
gasoline-powered blower. Ellen can do the same
job in 8 hours using a broom. How long would it
take them working together?
14. It takes a computer 8 days to print all of the
personalized letters for a national sweepstakes. A
new computer is purchased that can do the same
job in 5 days. How long would it take to do the job
with both computers working on it?
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.9
Graphing Rational Functions (Day 1)
Graph of 1
yx
=
Vertical Asymptote: 0x =
Horizontal Asymptote: y = 0
How to graph a rational function: 21
1+
−=
xy
Find intercepts if there are
any (for x-intercepts y = 0;
for y-intercepts x = 0)
x-intercept: y = 0
21
10 +
−=
x
1
12
−=−
x
−2(x − 1) = 1
−2x + 2 = 1
−2x = −1
x = 0.5
y-intercept: x = 0
210
1+
−=y
21
1+
−=y
y = −1 + 2
y = 1
Asymptotes (a line that approaches a given curve but does not meet it):
Horizontal and slant:
• If numerator power = denominator power; asymptote at y = b
a where a and b
are leading coefficients.
• If power in denominator is larger; asymptote at y = 0.
• If power in numerator is larger; no horizontal asymptote.
o Slant asymptote (numerator ÷ denominator; ignore remainder)
Vertical:
• Asymptote at x = a where a is any value that makes the denominator = 0
Holes:
• A factor in numerator and denominator. Cancels out but leaves a hole at this
point on the graph.
Denominator power is
larger; horizontal
asymptote at y = 0 then
translate up 2 so y = 2
Vertical asymptote when
denominator would be 0
therefore at x = 1
Domain: all real numbers except x-values that make
the denominator 0.
Domain: (-∞, 1) ∪ (1, ∞)
Range: given: ( ) = +−
af x k
x h; range is all y-values
except y ≠ k.
Range: (-∞, 2) ∪ (2, ∞)
f(x) = k shows vertical
h shows horizontal
Using the graph of f(x) = x
1 as a guide, graph each transformation. Fill in each blank.
1. 2
1)(
−=
xxg
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
2. 3
1)( +=
xxh
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
3. ( ) = +−
15
3g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
4. ( ) = −+
11
8g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.10
Graphing Rational Functions (Day 2)
Directions: Fill in each blank and use that information to sketch the graph.
1. ( ) = +−
12
3f x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
2. g(x) = 14.
( 2)x+
− −
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
3. 5
13
yx
= −+
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
4. ( ) = ++
17
5g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
5. ( ) =+
2
4g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
6. 2
64
yx
−= +
−
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
7. ( ) −+
17
3f x
x====
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
8. 2
35
yx
= −−
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.11
Graphing Rational Functions (Day 3)
Directions: Fill in each blank and use that information to sketch the graph.
1. ( )+ −
=+
2 4 5
1
x xf x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
2. 1
34)(
2
+
+−=
x
xxxf
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
3. 1
7yx
−= +
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
4.
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
( ) = −−
15
6f x
x
5.
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
6. 1
( ) 3.3 6
f xx
= −+
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
7. ( ) = −−
4 1
9 4g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
8.
( ) = −
+
112
2
3
g x
x
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
( ) = ++
11
4f x
x
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.12
Graphing Rational Functions (Day 4)
Directions: Fill in each blank and use that information to sketch the graph.
1. ( )2 2
.x
f xx
+=
Describe the transformation: ___________________
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
2. 2
2
12( ) .
3 4
x xf x
x x
+ −+ −+ −+ −====
+ −+ −+ −+ −
Asymptotes: ________________________________
Hole: ______________________________________
Domain: ___________________________________
Range: ____________________________________
Intercepts: __________________________________
3. Write the functions in the form ( )a
f x kx h
= += += += +−−−−
by using the graph.
4. Write the functions in the form ( )a
f x kx h
= += += += +−−−−
by using the graph.
5. Write the functions in the form ( )a
f x kx h
= += += += +−−−−
by using the graph.
6. Write the functions in the form ( )a
f x kx h
= += += += +−−−−
by using the graph.
7. The number n of daily visitors to a new store can be modeled by the function ( )+
=250 1000x
nx
, where x
is the number of days the store has been open.
a. What is the horizontal asymptote of this function and what does it represent? _______________________
_____________________________________________________________________________________
b. To the nearest integer, how many visitors can be expected on day 30? ____________________________
8. The annual transportation costs, C, incurred by a company follow the formula 2500
,C ss
= + where C is
in thousands of dollars and s is the average speed the company’s trucks are driven, in miles per hour. Use
your graphing calculator to find the speed at which cost is at a minimum.
9. Li Ming rowed 2 miles upstream and then 3 miles downstream. His average speed rowing in still water is
2 miles per hour. The model that describes the situation is 2 3 10
( )2 2 (2 )(2 )
sf s
s s s s
− += + =
− + − + where s is
the average speed of the current and f(s) is the total amount of time that Li Ming rowed. Sketch a graph of
the function and use it to determine the average speed of the current if Li Ming rowed for 3 hours.
(3, -2)
Name _________________________________________________________________ Period _____________
Algebra 3-4 Unit 4.13
Are You Ready for Unit 4 Assessment
I can add, subtract, multiply, and divide rational expressions.
1. Multiply and state the excluded values:
23
2
549
6
162
8018
xx
x
x
xx
+
+•
+
++
2. Divide and state the excluded values:2 2
2
7 18 2
2 82 32
x x x x
xx
− − + −÷
−−
3. Divide and state the excluded values:
168
54
16
42
2
2++
−+÷
−
+
xx
xx
x
x
4. Divide and state the excluded values:
86
20
6
1582
2
2
2
+−
−+÷
−−
+−
xx
xx
xx
xx
5. Multiply the rational expresssions and state the excluded values.
186
93
3
9 22
+
+•
−
−
x
xx
x
x
6. Divide and simplify:
20
183
6
1272
2
2
2
−−
−+÷
−−
+−
xx
xx
xx
xx
7. What is the least common denominator
(LCD) of the expressions 20
32
−− xx and
82
42
−+ xx?
8. What is the least common denominator
(LCD) of the expressions 352
22
−− xx
x and
152
12
−+
+
xx
x?
9.
10. Evaluate the problem and identify the mistake. Fully explain the error then simplify the problem correctly.
123
2
209
32 +
−++
+
xxx
x
Step 1: )4(3
2
)5)(4(
3
+−
++
+
xxx
x
Step 2: )5)(4(3
)5(2
)5)(4(3
)3(3
++
+−
++
+
xx
x
xx
x
Step 3: )5)(4(3
102
)5)(4(3
93
++
+−
++
+
xx
x
xx
x
Step 4: )5)(4(3
19
++
+
xx
x
11. Add: 209
36
209
322
++
++
++ xx
x
xx
12. Add: 158
4
158
2422
2
++
++
++
++
xx
x
xx
xx
13. Simplify: 103
3522
2
−−
−+
xx
xx
14. Simplify: 56
982
2
++
−−
xx
xx
I can solve rational equations.
15. Solve: 7
15
1
3
+=
+
+
xx
x
16. Solve: 13
22
7
1
−
−=
−
x
xx
17. Solve: 24
331
−
+=
+
x
x
x
x
18. Solve: 1
42
2
2
+
+=
−
+
x
x
x
x
19. Solve: x
x
xx
1
2
11
2
−+
+=
20. Solve: 1
61522 +
−−
+=
+
+
x
x
xxxx
x
21. Solve: xx 2
1
2
132
=+−
22. Solve: 1
3
1
2
+=
− xx
23. Solve: 4
1
2
9
22
5 −=−
− xx
24. Solve: 2 1 1
3 3
x
x x x
−+ =
− −
25. Jan and Stan both work at the library. It takes Jan 42 minutes to put a cart of books away. The same cart would take Stan 58 minutes. How long will it take if they work together?
26. Working together it took Sara and Cara took 4 hours to complete a task. If it would have taken Sara 7 hours to complete it by herself, how long would it have taken Cara by herself?
I can graph rational functions.
27. Graph the function f(x) = 32
4−
+x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
28. Graph the function f(x) = 53
1+
−
−
x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
29. Graph the function f(x) = 3
14
+
+
x
x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
30. Graph the function f(x) = 5
7
−
+
x
x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
31. Graph the function f(x) = 25
52
−
+
x
x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
32. Graph the function f(x) = 12
52
−−
+
xx
x using
technology and fill in the blanks. Vertical asymptote: ______________________ Horizontal asymptote: ____________________
33. Graph the function f(x) = 152
22
−+
+
xx
x using
technology and fill in the blanks. Domain: ______________________________ Range: _______________________________
34. Graph the function f(x) = 76
52
−+
+
xx
x using
technology and fill in the blanks. Domain: ______________________________ Range: _______________________________