Topic 11
Review
TOPIC VOCABULARY
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Check Your Understanding
Choose the correct term to complete each sentence.
1. When the numerator and denominator of a rational
expression are polynomials with no common factors,
the rational expression is in ? .
2. If a quantity varies directly with one quantity and
inversely with another, it is a(n) ? .
3. A(n) ? has a fraction in its numerator, denominator,
or both.
4. If a is a zero of the polynomial denominator of a rational
function, the function has a(n) ? at x = a.
5. A(n) ? of the graph of a rational function is one of the
continuous pieces of its graph.
11-1 Inverse Variation
Quick Review
Exercises
k
x
An equation in two variables of the form y = or xy = k,
where k ≠ 0, is an inverse variation with a constant of
variation k. Joint variation describes when one quantity
varies directly with two or more other quantities.
Example
Suppose that x and y vary inversely, and x = 10 when
y = 15. Write a function that models the inverse variation.
Find y when x = 6.
k
y=x
k
15 = 10
, so k = 150.
The inverse variation is y = 150
x .
When x = 6, y = 150
6 = 25.
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Topic 11 5HYLHZ
6. Suppose that x and y vary inversely, and x = 30 when
y = 2. Find y when x = 5.
Write a direct or inverse variation equation for each
relation.
7.
x
y
3
4
8
24
18
9
8.
x
y
5
7
9
30
42
54
Write the function that models each relationship. Find z
when x = 4 and y = 8.
9. z varies jointly with x and y. When x = 2 and y = 2,
z = 7.
10. z varies directly with x and inversely with y. When
x = 5 and y = 2, z = 10.
11-2 Attributes and Transformations of Reciprocal Funtions
Quick Review
Exercises
The graph of a reciprocal function has two parts called
branches. The graph of y = x a- c + d is a translation of
y = ax by c units horizontally and d units vertically. It has
a vertical asymptote at x = c and a horizontal asymptote
at y = d.
Graph each equation. Identify the x- and y-intercepts
and the asymptotes of the graph.
11. y = 1x
12. y = -22
x
-1
13. y = x - 4
Example
3
Graph the equation y = x −
2 + 1. Identify the x- and
y-intercepts and the asymptotes of the graph.
2
14. y = x + 3 - 1
Write an equation for the translation of y = 4x that has
the given asymptotes.
c = 2, so the vertical asymptote is x = 2.
d = 1, so the horizontal asymptote is y = 1.
Translate y = 3x two units to the right and one unit up.
When y = 0, x = -1.
y
4
The x-intercept is ( -1, 0).
When x = 0, y = - 12.
⫺8 ⫺4
The y-intercept is (0, - 12).
O
4
15. x = 0, y = 3
16. x = 2, y = 2
17. x = -3, y = -4
x
⫺4
18. x = 4, y = -3
11-3 Asymptotes of Rational Functions
Quick Review
The rational function f (x) =
Exercises
P(x)
Q(x)
has a point of
discontinuity for each real zero of Q (x).
If P (x) and Q (x) have
r no common factors, then f (x) has a vertical asymptote
when Q (x) = 0.
r a common real zero a, then there is a hole or a vertical
asymptote at x = a.
Example
Find any points of discontinuity for the graph of the
rational function y = x 2.5
+ 7 . Describe any vertical or
horizontal asymptotes and any holes.
There is a vertical asymptote at x = -7 and a horizontal
asymptote at y = 0.
Find any points of discontinuity for each rational
function. Sketch the graph. Describe any vertical or
horizontal asymptotes and any holes.
19. y =
x-1
(x + 2)(x - 1)
20. y =
x3 - 1
x2 - 1
21. y =
2x 2 + 3
x2 + 2
22. The start-up cost of a company is $150,000. It costs
$.17 to manufacture each headset. Graph the function
that represents the average cost of a headset. How
many must be manufactured to result in a cost of less
than $5 per headset?
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11-4 Rational Expressions
Quick Review
Exercises
A rational expression is in simplest form when its
numerator and denominator are polynomials that have
no common factors.
Simplify each rational expression. State any restrictions
on the variable.
23.
x 2 + 10x + 25
x 2 + 9x + 20
Example
24.
Simplify the rational expression. State any restrictions on
the variable.
x 2 - 2x - 24
x 2 + 7x + 12
25.
4x 2 - 2x
2x
,
x 2 + 5x + 4 x 2 + 2x + 1
#
2x2 + 7x + 3
x2 − 16
x−4
x2 + 8x + 15
(2x + 1)(x + 3) (x - 4)(x + 4)
=
(x + 3)(x + 5)
x-4
#
=
# xx -- 61
2
26. What is the ratio of the volume of a sphere to its
surface area?
(2x + 1)(x + 4)
, x ≠ -5, x ≠ -3, and x ≠ 4
x+5
11-5 Adding and Subtracting Rational Expressions
Quick Review
Exercises
To add or subtract rational expressions with different
denominators, write each expression with the LCD. A
fraction that has a fraction in its numerator or denominator
or in both is called a complex fraction. Sometimes you can
simplify a complex fraction by multiplying the numerator
and denominator by the LCD of all the rational expressions.
Simplify the sum or difference. State any restrictions
on the variable.
Example
Simplify the complex fraction.
) # xy
) # xy
1 #
#
x xy + 3 xy
=5
#
#
y xy + 4 xy
(
(
1
1
x+3
x+3
=
5
5
y+4
y+4
y + 3xy
= 5x + 4xy
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Topic 11 Review
1
x+3
5
y+4
3x
6
+
x2 - 4 x + 2
28. 2 1 - 2 2
x - 1 x + 3x
27.
Simplify the complex fraction. State any restrictions on
the variable.
29.
30.
2 - 2x
1
3-x
1
x+y
4
11-6 Solving Rational Equations
Quick Review
Exercises
Solving a rational equation often requires multiplying
each side by an algebraic expression. This may introduce
extraneous solutions—solutions that solve the derived
equation but not the original equation. Check all possible
solutions in the original equation.
Solve each equation. Check your solutions.
5
31. 1x = x - 4
2
1
-6
32. x + 3 - x =
x(x + 3)
1
x
18
33. 2 + 6 = x
Example
Solve the equation. Check your solution.
1
2
1
2x − 5x = 2
(1
2
)
(1 )
10x 2x - 5x = 10x 2
34. You travel 10 mi on your bicycle in the same amount
of time it takes your friend to travel 8 mi on his
bicycle. If your friend rides his bike 2 mi/h slower
than you ride your bike, find the rate at which
each of you is traveling.
5 - 4 = 5x
x = 15
Check
1
()
2 15
-
2
()
5 15
5
1
=2-2=2 ✔
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