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Lesson 8.5 Exercises, pages 680–688
A
3. For each graph of y = f(x), draw vertical lines to represent the
1
vertical asymptotes, if they exist, of the graph of y =
.
f(x)
a)
4
b)
y
x
2
y f(x)
0
4
4
x
2
y
2
4
The x-intercepts of the graph
of y ⴝ f(x) are 2 and 6.
1
So, the graph of y ⴝ
f(x)
has vertical asymptotes at
x ⴝ 2 and x ⴝ 6.
0
y f(x)
6
8
The x-intercept of the graph of
y ⴝ f(x) is ⴚ2. So, the graph of
1
yⴝ
has a vertical asymptote at
f(x)
x ⴝ ⴚ2.
4. For each graph in question 3, identify the values of x for which the
1
graph of y =
is above the x-axis, and for which it is below
f(x)
the x-axis.
a) Above the x-axis: 2<x<6
Below the x-axis:
x<2 or x>6
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b) Below the x-axis:
x ç ⺢, x ⴝ ⴚ2
Above the x-axis: never
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5. On each graph from question 3, sketch the graph of y =
a)
4
b)
y
y
2
y f (x)
1
f (x)
2
0
y
x
4
0
y
y f (x)
x
1
.
f(x)
1
f (x)
4
2
6
4
8
The x-axis is a horizontal
asymptote. Plot points where
the lines y ⴝ 1 and y ⴝ ⴚ1
intersect the graph. These
points are common to
both graphs. The graph of
y ⴝ f(x) has vertex (4, 4), so
point a4, b lies on y ⴝ
1
4
The x-axis is a horizontal
asymptote. Plot points where the
line y ⴝ ⴚ1 intersects the graph.
These points are common to both
graphs. Since the graph has one
vertical asymptote, it has Shape 2.
1
.
f(x)
Since the graph has 2 vertical
asymptotes, it has Shape 3.
6. Match each graph of y = f(x) to the corresponding graph of y =
1
.
f(x)
What features of the graphs did you use to make your decisions?
a)
4
b)
y
2
y
x
4
x
6
4
2
y f (x)
i)
2
2
0
2
4
4
6
y
4
2
0
y
2
ii)
1
f (x)
y f (x)
2
x
x
4
2
2
4
y
6
1
y
f(x)
0
2
The graph that corresponds to graph a is graph ii. The graph of y ⴝ f(x)
in part a has 2 x-intercepts, so the graph of its reciprocal function has
2 vertical asymptotes.
The graph that corresponds to graph b is graph i. The graph of y ⴝ f(x)
in part b has no x-intercepts, so the graph of its reciprocal function has
no vertical asymptotes.
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B
7. How many vertical asymptotes does the graph of each reciprocal
function have? Identify the equation of each vertical asymptote.
1
- (x + 3)2
a) y =
b) y =
(x ⴚ 2)(x ⴙ 6) ⴝ 0 when x ⴝ 2 or
x ⴝ ⴚ6. So, the graph of
ⴚ (x ⴙ 3)2 ⴝ 0 when
x ⴝ ⴚ3. So, the graph of
yⴝ
1
has
ⴚ (x ⴙ 3)2
yⴝ
1
has 2 vertical
(x ⴚ 2)(x ⴙ 6)
asymptotes, x ⴝ 2 and x ⴝ ⴚ6.
1 vertical asymptote,
x ⴝ ⴚ3.
c) y =
1
(x - 2)(x + 6)
1
x + x - 6
1
d) y =
2
x ⴙ x ⴚ 6 ⴝ (x ⴙ 3)(x ⴚ 2)
(x ⴙ 3)(x ⴚ 2) ⴝ 0 when
x ⴝ ⴚ3 or x ⴝ 2. So, the
2
1
graph of y ⴝ 2
has
x ⴙxⴚ6
2
- 3x - 9
ⴚ3x ⴚ 9 ⴝ ⴚ3(x 2 ⴙ 3)
Since x2 ⴙ 3 cannot be 0, the graph
2
of y ⴝ
1
has no vertical
ⴚ 3x2 ⴚ 9
asymptotes.
2 vertical asymptotes, x ⴝ ⴚ3
and x ⴝ 2.
8. On the graph of each quadratic function y = f(x), sketch a graph of
1
the reciprocal function y =
. Identify the vertical asymptotes, if
f(x)
they exist. Explain your strategies.
a)
The graph of y ⴝ f(x) has no x-intercepts,
y
x
4
y
2
1
f (x)
0
2
2
y f (x)
6
1
has no vertical
f(x)
asymptotes and has Shape 1.
Horizontal asymptote: y ⴝ 0
Points (0, ⴚ4), (ⴚ1, ⴚ5), and (1, ⴚ5) lie on
y ⴝ f(x), so points (0, ⴚ0.25), (ⴚ1, ⴚ0.2),
and (1, ⴚ0.2) lie on y ⴝ
b)
1
.
f(x)
The graph of y ⴝ f(x) has 2 x-intercepts, so the
y
graph of y ⴝ
4
y f (x)
x
4
1
y
f (x)
so the graph of y ⴝ
1
has 2 vertical asymptotes,
f(x)
x ⴝ ⴚ3 and x ⴝ ⴚ1, and has Shape 3.
Horizontal asymptote: y ⴝ 0
Plot points where the lines y ⴝ 1 and y ⴝ ⴚ1
intersect the graph of y ⴝ f(x). These points are
common to both graphs. The graph of y ⴝ f(x)
has vertex (ⴚ2, ⴚ 2), so point (ⴚ2, ⴚ0.5)
lies on y ⴝ
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1
.
f(x)
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c)
d)
y
8
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y
y f (x)
8
6
6
1
y
f (x)
4
4
2
0
2
4
y f (x)
2
x
0
2
4
and (5, 0.25) lie on y ⴝ
the quadratic function y = f(x).
b)
y
y f (x)
x
2
y
1
.
f(x)
1
, sketch a graph of
f(x)
1
f (x)
0
4
y
y f (x)
2
2
4
6
8
The graph has one vertical
asymptote, x ⴝ ⴚ5, so the graph
of y ⴝ f(x) has vertex (ⴚ 5, 0).
The line y ⴝ ⴚ1 intersects the
graph at 2 points that are
common to both graphs.
38
1
f(x)
has no vertical asymptotes and has
Shape 1.
Horizontal asymptote: y ⴝ 0
Points (3, 2), (1, 4), and (5, 4) lie on
y ⴝ f(x), so points (3, 0.5), (1, 0.25),
9. On the graph of each reciprocal function y =
1
y
f (x)
x
8
x-intercepts, so the graph of y ⴝ
1
has 1 vertical
f(x)
10 8
6
The graph of y ⴝ f(x) has no
asymptote, x ⴝ 3, and has
Shape 2.
Horizontal asymptote: y ⴝ 0
Plot points where the line
y ⴝ 1 intersects the graph of
y ⴝ f(x). These points are
common to both graphs.
a)
1
f (x)
6
The graph of y ⴝ f(x) has 1
x-intercept, so the graph of
yⴝ
y
x
4
2
2
2
4
1
(0, )
4
4
The graph has 2 vertical
asymptotes, x ⴝ ⴚ1 and x ⴝ 1,
so the graph of y ⴝ f(x) has
x-intercepts ⴚ1 and 1.
The point a0, ⴚ b is on the line
1
4
of symmetry, so (0, ⴚ4) is the
vertex of y ⴝ f(x).
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c)
y
y f (x)
1
f (x)
10 8
6
4
2
2
1.5
1
0.5
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d)
y
6
y
4
x
2
0
y f (x)
y
0
The graph has no vertical
asymptote, so the graph of
y ⴝ f(x) has no x-intercept.
The point (ⴚ6, 1) is on the line of
symmetry, so (ⴚ 6, 1) is the vertex
of y ⴝ f(x). Points (ⴚ7, 0.5) and
(ⴚ5, 0.5) lie on y ⴝ f(x), so
points (ⴚ7, 2) and (ⴚ 5, 2) lie on
yⴝ
1
f (x)
2
x
4
6
8
The graph has one vertical
asymptote, x ⴝ 5, so the
graph of y ⴝ f(x) has vertex
(5, 0). The line y ⴝ 1
intersects the graph at 2
points that are common to
both graphs.
1
.
f(x)
10. The graphs of the reciprocal functions y =
shown below.
i)
ii)
y
x
0
2
4
2
6
y
1
f (x)
y f(x )
1
1
and y =
are
f(x)
g(x)
y g(x)
y
4
1
g(x)
y
2
x
6
4
2
2
2
a) Identify whether each function y = f(x) and y = g(x) is linear or
quadratic. How did you decide?
i) y ⴝ f(x) is quadratic. I recognize
the shape of the graph as that of
the reciprocal of a quadratic
function with 1 x-intercept.
ii) y ⴝ g(x) is linear. I recognize
the shape of the graph as
that of the reciprocal of a
linear function.
b) On the graph of each reciprocal function, sketch a graph of the
related linear or quadratic function. What strategy did you use
each time?
i) The graph has one vertical
asymptote, x ⴝ 4, so the graph
of y ⴝ f(x) has vertex (4, 0).
The line y ⴝ ⴚ1 intersects the
graph at 2 points that are
common to both graphs.
ii) Vertical asymptote is about
x ⴝ ⴚ1.8, so graph of
y ⴝ g(x) has x-intercept
ⴚ1.8. Mark points at y ⴝ 1
and y ⴝ ⴚ1 on graph of
yⴝ
1
, then draw a line
g(x)
through these points for the
graph of y ⴝ g(x).
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11. Graph each pair of functions on the same grid.
Explain your strategies.
a) y = 4(x + 2)2 and y =
1
4(x + 2)2
The graph of y ⴝ 4(x ⴙ 2)2 opens up,
has vertex ( ⴚ2, 0), and x-intercept ⴚ2.
The graph of y ⴝ
y
y 4(x 2)2
1
has vertical
4(x ⴙ 2)2
4
asymptote, x ⴝ ⴚ2 and horizontal
1
y
4(x 2)2
asymptote y ⴝ 0. Plot points where the
4
line y ⴝ 1 intersects the graph of
y ⴝ 4(x ⴙ 2)2. These points are common
to both graphs. The graph of the
reciprocal function has Shape 2.
b) y = -2x2 - 3 and y =
2
x
2
0
2
1
2
- 2x - 3
The graph of y ⴝ ⴚ2x2 ⴚ 3 opens down,
with vertex (0, ⴚ 3), so the graph has no
x-intercepts. The graph of y ⴝ
y
1
ⴚ2x2 ⴚ 3
2
1
y
2x 2 3
4
x
2
2
0
4
2
has no vertical asymptotes; the horizontal
asymptote is y ⴝ 0. Points (0, ⴚ3), (1, ⴚ5),
and (ⴚ 1, ⴚ5) lie on y ⴝ ⴚ2x2 ⴚ 3.
y 2x 2 3
So, points a0, ⴚ b , (1, ⴚ 0.2), and
1
3
(ⴚ1, ⴚ 0.2) lie on y ⴝ
1
. The graph
ⴚ 2x2 ⴚ 3
of the reciprocal function has Shape 1.
c) y = 2x2 - 4x - 6 and y =
1
2x2 - 4x - 6
The graph of y ⴝ 2x2 ⴚ 4x ⴚ 6,
or y ⴝ 2(x ⴚ 3)(x ⴙ 1) opens up,
has x-intercepts 3 and ⴚ1, and
vertex (1, ⴚ 8). The graph of
yⴝ
1
has vertical
2x2 ⴚ 4x ⴚ 6
y
1
2x 2 4x 6
1
8
2
x
0
6
asymptotes x ⴝ 3 and x ⴝ ⴚ1,
and a horizontal asymptote y ⴝ 0.
Plot points where the lines y ⴝ 1
and y ⴝ ⴚ1 intersect the graph
of y ⴝ 2(x ⴚ 3)(x ⴙ 1). These
points are common to both graphs.
Point (1, ⴚ8) lies on y ⴝ 2x2 ⴚ 4x ⴚ 6,
so point a1, ⴚ b lies on y ⴝ
y 2x 2 4x 6
y
4
2
2
4
6
8
1
.
2x2 ⴚ 4x ⴚ 6
The graph of the reciprocal function has
Shape 3.
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12. A quadratic function and its reciprocal are graphed on a grid. The
horizontal lines y = 1 and y = - 1 do not intersect either graph.
Sketch a possible graph of the quadratic function and its reciprocal.
Explain your strategy.
If the lines y ⴝ 1 and y ⴝ ⴚ1 do not
intersect the graph of a quadratic
function, the vertex of the graph either
lies above the line y ⴝ 1 and opens up,
or it lies below the line y ⴝ ⴚ1 and
opens down. In either case, the graph of
the quadratic function has no x-intercepts
and the graph of the corresponding
reciprocal function has no vertical
asymptotes. Here is the graph of
y ⴝ x2 ⴙ 3 and y ⴝ
6
y
4
1
x
4
2
y x2 3
3
2
2
y1
0
2
4
y 1
2
4
1
.
x2 ⴙ 3
13. A reciprocal function has the form y =
1
, a Z 0, b Z 0.
ax2 + b
How can you determine the number of asymptotes of the graph
of y =
1
when:
ax2 + b
a) both a and b are positive?
Look at the quadratic function y ⴝ ax2 ⴙ b. When both a and b are
positive, the graph opens up and its vertex is above the x-axis.
So, the graph of y ⴝ ax2 ⴙ b has no x-intercepts and the graph of
yⴝ
1
has no vertical asymptotes.
ax2 ⴙ b
b) both a and b are negative?
Look at the quadratic function y ⴝ ax2 ⴙ b. When both a and b are
negative, the graph opens down and its vertex is below the x-axis.
So, the graph of y ⴝ ax2 ⴙ b has no x-intercepts and the graph of
yⴝ
1
has no vertical asymptotes.
ax2 ⴙ b
c) a and b have opposite signs?
Look at the quadratic function y ⴝ ax2 ⴙ b. When a and b have
opposite signs, the graph either opens up with its vertex below the
x-axis, or it opens down with its vertex above the x-axis. So, the graph
of y ⴝ ax2 ⴙ b has 2 x-intercepts and the graph of y ⴝ
2 vertical asymptotes.
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has
ax ⴙ b
2
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C
1
has one vertical
px2 + (2p + 1)x + p
1
asymptote. Show that p = - 4 .
14. The reciprocal function y =
If the reciprocal function has one vertical asymptote, then the related
quadratic function has one x-intercept; that is, the quadratic equation
px2 ⴙ (2p ⴙ 1)x ⴙ p ⴝ 0 has equal roots.
The equation has equal roots when b2 ⴚ 4ac ⴝ 0.
Substitute: b ⴝ 2p ⴙ 1, a ⴝ p, c ⴝ p
(2p ⴙ 1)2 ⴚ 4(p)(p) ⴝ 0
4p2 ⴙ 4p ⴙ 1 ⴚ 4p2 ⴝ 0
4p ⴙ 1 ⴝ 0
p ⴝⴚ
1
4
15. Determine the values of k for which the reciprocal function
1
has:
y = 2
x + kx + 4
a) no vertical asymptotes
The reciprocal function has no vertical asymptotes, so the related
quadratic function has no x-intercepts; that is, x2 ⴙ kx ⴙ 4 ⴝ 0 has no
real roots. The equation has no real roots when b2 ⴚ 4ac<0.
Substitute: a ⴝ 1, b ⴝ k, c ⴝ 4
k2 ⴚ 4(1)(4)<0
k2 ⴚ 16<0
k2<16
ⴚ4<k<4
b) one vertical asymptote
The reciprocal function has one vertical asymptote, so the related
quadratic function has one x-intercept; that is, x2 ⴙ kx ⴙ 4 ⴝ 0 has
equal roots. The equation has equal roots when b2 ⴚ 4ac ⴝ 0.
Substitute: a ⴝ 1, b ⴝ k, c ⴝ 4
k2 ⴚ 4(1)(4) ⴝ 0
k2 ⴚ 16 ⴝ 0
k2 ⴝ 16
k ⴝ 4 or k ⴝ ⴚ4
c) two vertical asymptotes
The reciprocal function has two vertical asymptotes, so the related
quadratic function has two x-intercepts; that is, x2 ⴙ kx ⴙ 4 ⴝ 0
has 2 real roots. The equation has 2 real roots when b2 ⴚ 4ac>0.
Substitute: a ⴝ 1, b ⴝ k, c ⴝ 4
k2 ⴚ 4(1)(4)>0
k2 ⴚ 16>0
k2>16
k<ⴚ4 or k>4
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16. Determine the equation of each quadratic function y = f(x) you
graphed in question 9. Describe your strategies.
a) The graph has vertex (ⴚ5, 0) and passes through the point (ⴚ6, ⴚ1).
So, the equation has the form: y ⴝ a(x ⴙ 5)2
Substitute: x ⴝ ⴚ6, y ⴝ ⴚ1
ⴚ1 ⴝ a(ⴚ 6 ⴙ 5)2
ⴚ1 ⴝ a
The equation of the quadratic function is y ⴝ ⴚ(x ⴙ 5)2.
b) The graph has vertex (0, ⴚ4) and passes through the point (1, 0).
So, the equation has the form: y ⴝ ax2 ⴚ 4
Substitute: x ⴝ 1, y ⴝ 0
0 ⴝ a(1)2 ⴚ 4
4ⴝa
The equation of the quadratic function is y ⴝ 4x2 ⴚ 4.
c) The graph has vertex (ⴚ6, 1) and passes through the point (ⴚ 5, 2).
So, the equation has the form: y ⴝ a(x ⴙ 6)2 ⴙ 1
Substitute: x ⴝ ⴚ5, y ⴝ 2
2 ⴝ a( ⴚ5 ⴙ 6)2 ⴙ 1
1ⴝa
The equation of the quadratic function is y ⴝ (x ⴙ 6)2 ⴙ 1.
d) The graph has vertex (5, 0) and passes through the point (4, 1).
So, the equation has the form: y ⴝ a(x ⴚ 5)2
Substitute: x ⴝ 4, y ⴝ 1
1 ⴝ a(4 ⴚ 5)2
1ⴝa
The equation of the quadratic function is y ⴝ (x ⴚ 5)2.
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