PCMath12
Name:
UNIT 5 LEARNING GUIDE – RATIONAL EXPRESSIONS
INSTRUCTIONS:
Using a pencil, complete the following questions as you work through the related lessons.
Show ALL of your work as is explained in the lessons. Do your best and always ask questions if
there is anything that you don’t understand.
5.1 RATIONAL EXPRESSIONS
1. Determine whether each of the following are rational expressions. If the expression is not
rational, explain why.
a)
4x + 2
3
b) x 2 - x - 6
c)
4m - n
m + 2n
d)
2x - x
x
e)
-4
3
x +1
f)
2x - 1
3x + 7
2. Determine what should replace Ä to make the expressions in each pair equivalent.
a)
d)
5 Ä
,
6 42
3
Ä
,
, x ¹ 0,3
x + 2 x ( x + 2)
2021-11-09
b)
2 Ä
,
, a¹0
3 30a
e)
Page 1 of 25
c)
m + 1 4m + 4
,
, m¹3
m-3
Ä
a
Ä
,
, x ¹ ±5
a + 5 ( a + 5)( a - 5)
PCMath12
2021-11-09
Page 2 of 25
PCMath12
3. Identify the non-permissible values for the variables in each rational expression.
1
5x
a -b
a)
b) 2
c)
6w
x +3
b -1
d)
m+5
m2 - 1
e)
g)
2x +1
5
h)
j)
3x
3x + 6 x
k)
m)
x+4
x 2 + 16
n)
2
f)
6x - 5
2x ( x + 4)
i)
4 ( x + 1)
( x + 2 )( x + 1)
x ( x + 3)
2
x + 7 x + 12
l)
x +1
3x + 8 x + 5
n+5
x 2 - 25
o)
x2 -1
x3 - 8
3
( x + 2)( x - 1)
2
x2 -1
x3 + 1
4. Make an equivalent rational expression for each of the following and state the non-permissible
values.
a)
!(#$%)
#$'
'(#$()
b) (#$()()*+)
2021-11-09
Page 3 of 25
PCMath12
5.2 CHARACTERISTICS
1. Given the rational function: y =
!
"#$
+k
a) Determine the equation of the horizontal asymptote.
b) Determine the equation of the vertical asymptote.
c) Describe the possible effects‘a’ can have on the graph?
2. For each of the following graphs, explicitly state the equations of both asymptotes. Use the
!
asymptotes to write the full equation in the form: y = "#$ + k. Show your work for finding "a"
a)
2021-11-09
b)
Page 4 of 25
PCMath12
c)
d)
e)
f)
2021-11-09
Page 5 of 25
PCMath12
5.3 GRAPHS
1. Identify any Horizontal Asymptotes, Vertical Asymptotes and Removable Discontinuities
%
a) y = "&'
."#-
c) y = ) ! #'/
"! &%)&0
e) y = "! #,"#-1
2021-11-09
()&*)()#,)
b) y = ()#,)()&-)
"! &'
d) y = "! &"#0
"! #.)#,
f) y = ."! #%"#,
Page 6 of 25
PCMath12
2. Sketch the graph of each of the following rational functions. Explicitly state the values asked
for below and show how you determined those values.
a)
b)
c)
d)
e)
f)
g)
the equations of any horizontal asymptotes
the equations of any vertical asymptotes
any non-permissible values as a coordinate (other than asymptotes)
any x-intercepts
any y-intercepts
the domain
the range
i.
y = "#, + 2
2021-11-09
-
Page 7 of 25
PCMath12
,"
ii. y = "&'
#."
iii. y = "#'−1
2021-11-09
Page 8 of 25
PCMath12
iv. y =
"! &,"
"! #'
"! #'
v. y = "! #%"&0
2021-11-09
Page 9 of 25
PCMath12
vi. y =
."! #%"#,
"! #'"&,
vii. y =
2021-11-09
"! &0"#.*
"! #."#,
Page 10 of 25
PCMath12
5.4 EQUATIONS
1. Write an equation for each of the following graphs, Begin with a factored equation and
expand for your final answer
a)
c)
2021-11-09
b)
d)
Page 11 of 25
PCMath12
e)
f)
g)
h)
2021-11-09
Page 12 of 25
PCMath12
2. Write the equation of a possible rational function given the following information. Begin with
a factored equation and expand for your final answer.
a) Vertical asymptotes x = −1, 2 and x–intercepts at x = −3 and x = 4
b) Vertical asymptote x = 2, discontinuous point (−3, 1), x–intercept at x = −5 and horizontal
asymptote at y = 2.
c) Discontinuous at point (−2, −1), x–intercept at x = -3/2, y-intercept at y=3 and no
horizontal or vertical asymptotes.
2021-11-09
Page 13 of 25
PCMath12
5.5 CONIC SECTIONS
1. From each of following diagrams identify the conic formed and state how each is formed by
the intersection of a plane and the double-napped cone.
a.
b.
c.
2. A plane intersects a double-napped cone to form a circle. Assume the plane moves parallel to
its original position. Describe what happens to the circle that is formed when the plane
a) moves further away from the vertex
b) is at the vertex of the double-napped cone.
c) moves closer to the vertex
2021-11-09
Page 14 of 25
PCMath12
5.6 CONIC EQUATIONS
1. Graph the following circles on the same grid & label.
a) 𝑥 . + 𝑦 . = 1
b) 𝑥 . + 𝑦 . = 9
c) 𝑥 . + 𝑦 . = 36
2. State the center and the radius for each of the following circles.
a. 𝑥 . + 𝑦 . = 5
b. (𝑥 − 3). + (𝑦 + 5). = 49
-0
c. (𝑥 + 6). + 𝑦 . = .%
3. Write the equation of a circle with a center at (0,0) and a radius of
20 .
4. Write the equation of a circle with a center at (-5,3) and a radius of
9
.
2
2021-11-09
Page 15 of 25
PCMath12
5. Write the equation of a circle with a center at (1,-2) and a diameter that is 8 units long.
6. Write the equation of a circle where the endpoints of the diameter are at (6,7) and (-6, -7).
7. Graph the following ellipses on the same grid & label.
a)
)!
'
+ 𝑦. = 1
b) 𝑥 . +
c)
)!
/
2!
'
=1
2!
+ 3- = 1
d) 2𝑥 . + 8𝑦 . = 32
8. For each of the following ellipses, State the center and the distance from the centre to the
vertex and co-vertex. State the coordinates of the vertices
a.
b.
)!
'
+
2!
/
()#')!
.%
=1
+
(2&0)!
'/
=1
c. 4(𝑥 + 6). + 25𝑦 . = 100
2021-11-09
Page 16 of 25
PCMath12
9. Write the equation for each ellipse
a)
b)
c)
d)
e) An ellipse with a center at (1,-2) that is 8 units wide and 10 units tall
10. Convert the following ellipse to standard form: 4 x 2 + 9y 2 = 36
2021-11-09
Page 17 of 25
PCMath12
11. Graph the following parabolas and label the co-ordinates of the vertex
a) 𝑦 = (𝑥 − 2). − 3
b) 𝑦 . = 2𝑥
-
c) 𝑥 . = . 𝑦
d) (𝑦 − 2). = (𝑥 + 5)
12. a) Write the equation of a parabola that has a vertex at (5,-4), opens to the right, and passes
through the point (6,-5)
b) Write the equations for a Parabola with vertex at (-3, 2), horizontal axis of symmetry, and
passing through the point (5, 6).
2021-11-09
Page 18 of 25
PCMath12
5.7 TRANSFORMATIONS
1. Explain how an ellipse is different from a circle in terms of transformations.
2. Write the equation of the conic section after the given transformation
a)
b)
()#')!
/
)!
'
−
−
(2#-)!
-11
(2#,)!
,0
= −1 is translated up 3 and left 6
= 1 is compressed horizontally by a factor of 2
c) (𝑥 + 2). + (𝑦 − 3). = 25 is expanded vertically by a factor of 2
d) (𝑦 − 3). = 2(𝑥 − 5) is reflected in the y axis
2021-11-09
Page 19 of 25
PCMath12
3. Draw the graph of the conic section after the given transformation
a) translation left 2 down 4
b) reflection in the x-axis
c) vertical expansion by a factor of 4
d) horizontal compression by a factor of 1/2
e) Vertical compression by a factor of 3
f) vertical compression by a factor of -4
2021-11-09
Page 20 of 25
PCMath12
4. Write the equation of the conic section after the given transformations
a)
()#')!
-0
b) 𝑥 . +
+
(2)!
'
(2#,)!
/
= 1 is
= 1 is
expanded vertically by a factor of 2
horizontally reflected in the y-axis,
c) (𝑥 − 5). + (𝑦 − 3). = 25 is
2021-11-09
translated 5 left and down 4
Page 21 of 25
PCMath12
5. For each graph, apply the stated transformations and write the resulting equation
a)
Vertical expansion by a factor of 2
Equation:__________________________
b)
Translated left 4 and up 2
Equation:__________________________
c)
Vertical compression by 1/2
Equation:__________________________
2021-11-09
Page 22 of 25
PCMath12
ANSWER KEY
5.1 Rational Expressions
1. a) Rational b) Rational c) Rational d) Not Rational → √𝑥 is not rational
e) Rational f) Not Rational → 2" is not necessarily rational
2. a) 35
b) 20a c)4m – 12
d) 3x
e) a(a – 5)
3. a) 𝑤 ≠ 0 b) 𝑥 ∈ ℝ
c) 𝑏 ≠ 1d) 𝑚 ≠ −1, 1
e) 𝑥 ≠ 2
f) 𝑥 ≠ 0, −4
#$
g) 𝑥 ∈ ℝ
h) 𝑥 ≠ −2, 1
i) 𝑥 ≠ −1, −2 j) 𝑥 ≠ 0, −2
k) 𝑥 ≠ −3, −4 l) 𝑥 ≠ % , −1
m) 𝑥 ∈ ℝ
n) 𝑥 ≠ ±5
o) 𝑥 ≠ −1
4. answers will vary
5.2 Characteristics
1.
2.
a) y = k
b) x = h
&
a) y = − '
c) the a can cause an exp/comp of y as well as a reflection in the x-axis
&
b) y = '()
&
c) y = − '#& + 1
&
d) y = '() + 4
)
*
e) y = '(& − 3
f) y = '#% + 2
5.3 Graphs
2.
i. a) y = 2
ii. a) y = 3
b) x = 3
b) x = −4
iii. a) y = −3 b) x = 4
c) None
c) None
d) x = 2.5
d) x = 0
e) y = 1.66
e) y = 0
f) x ≠ 3
f) x ≠ −4
g) y ≠ 2
g) y ≠ 3
c) None
d) x = 1.33
e) y = −1
f) x ≠ 4
g) y ≠ −3
d) x = 0, −3 e) y = 0
f) x ≠ 2, −2
g) y ≠ 1
iv. a) y = 1
b) x = 2, −2
c) None
v. a) y = 1
b) x = 3
c) (2, −4) d) x = −2
e) y = −0.66
f) x ≠ 2, 3
g) y ≠ 1, −4
vi. a) y = 2
b) x = 1
c) (2,5) d) x = −0.5
e) y = −1
f) x ≠ 1, 2
g) y ≠ 2, 5
i.
ii.
iii.
iv.
v.
vi.
2021-11-09
Page 23 of 25
PCMath12
5.4 Equations
1
!
1. a) y = − "
b) y = x+2
" ! #+
f) y = " !("#) g) y = *
#
2. a) 𝑦 = **#+*+!0
+*+0
" +,* # +-*
*.!
b) 𝑦 = 0*.!1
*+0
x2 −2x
x−2
c) y = x−1
d) y = 𝑥2+𝑥−2
e) y =
𝑥2 −3𝑥+2
𝑥2 +𝑥−2
#
.*+/
h) y = * *+!
#
.2*./
c) 𝑦 = 0* *.0
5.5 Conic Sections
1. a) Circle – when a plane intersects a double-napped cone such that the plane is parallel to the axis or perpendicular
to the axis.
b) Parabola – when a plane intersects a double-napped cone such that the plane is parallel to the generator.
c) Ellipse – when a plane intersects a double-napped cone such that the plane is neither perpendicular nor parallel
to the axis and the angle of intersection is greater than the generator angle.
2. a) The radius of the circle increases so the circle is larger b) The radius of the circle becomes infinitely small so a
point is formed c) The radius of the circle decreases so the circle is smaller
5.6Conic Equations
2a. (0,0), √5 b. (3,-5), 7 c. (−6,0),
1.
3. 𝒙𝟐 + 𝒚𝟐 = 𝟐𝟎 4.
5.(𝒙 − 𝟏)
𝟐
7.
(𝒙 + 𝟓)𝟐 + (𝒚 − 𝟑)𝟐 =
𝟐
+ (𝒚 + 𝟐) = 𝟏𝟔
6.
𝟐
'
%
𝟏𝟖
𝟒
𝟐
𝒙 + 𝒚 = 𝟖𝟓
8.a) Center (0,0), distance from center to vertex 2,
distance from center to co-vertex 3, Vertices (-2,0) , (2,0) , (0,-3) , (0,3)
b) Center (4,-6), distance from center to vertex 5,
distance from center to co-vertex 7, Vertices(-1,-6),(9,-6),(4,1),(4,-13)
c) Center (6,0), distance from center to vertex 5,
distance from center to co-vertex 2, Vertices (-11,0),(-1,0),(-6,2),(-6,-2)
𝟐
𝟐
𝟐
𝒚
9. a) 𝒙𝟒 + 𝟐𝟓
=𝟏
𝟐
𝟐
𝟐
(𝒚(𝟐)
d) (𝒙#𝟑)
+ 𝟔𝟒 = 𝟏
𝟏𝟔
𝟐
𝟐
𝟐
𝒙
𝒚
b) 𝟒𝟗
+ =𝟏
𝟏𝟔
𝟐
(𝒚#𝟔)
c) (𝒙(𝟑)
+
=𝟏
𝟑𝟔
𝟒
𝟐
(𝒚(𝟐)
e) (𝒙#𝟏)
+ 𝟐𝟓 = 𝟏
𝟏𝟔
𝟐
10. 𝒙𝟗 + 𝒚𝟒 = 𝟏
11. a)
b)
12. a) (𝒚 + 𝟒)𝟐 = (𝒙 − 𝟓)
2021-11-09
c)
b) (𝒚 − 𝟐)𝟐 = 𝟐(𝒙 + 𝟑)
Page 24 of 25
d)
PCMath12
5.7 Transformations
1. An ellipse is a circle that has either been horizontally and vertically compressed or expanded.
2.a)
(𝑥+2)2
9
−
(𝑦−4)2
100
= −1
b) 𝑥 ) −
(@#%)!
%A
c)
=1
(𝑥+2)2
+
25
3.a)
b)
c)
d)
e)
f)
#
4.a) (𝑥 − 4)% + (𝑦)% = 16 b) 𝑥 0 + (4+5)
7
5.a)
("(&)!
+
+
(@#A)!
2021-11-09
&A
=1
=1
(𝑦−6)2
100
c) 𝑥 ) + (𝑦 + 1)) = 25
b) (𝑦 − 4)) = 𝑥 + 3
c)
("(*)!
)$
+
(@#).$)!
Page 25 of 25
).$!
=1
=1
d) (𝑦 − 3)) = −2(𝑥 + 5)