Unit 1 Transformations & Conics - Weebly
Transcript of Unit 1 Transformations & Conics - Weebly
1
Unit 1 Transformations
General Outcome: • Develop algebraic and graphical reasoning through the study of relations.
Specific Outcomes:
1.1 Demonstrate an understanding of the effects of horizontal and vertical translations on
the graphs of functions and their related equations:
▪ ( )y f x h= −
▪ ( )y k f x− =
1.2 Demonstrate an understanding of the effects of horizontal and vertical stretches on the
graphs of functions and their related equations:
▪ ( )y af x=
▪ ( )y f bx=
1.3 Apply translations and stretches to the graphs and equations of functions:
1.4 Demonstrate an understanding of the effects of reflections on the graphs of functions
and their related equations, including reflections through the:
▪ x-axis ( )y f x= −
▪ y-axis ( )y f x= −
▪ line y x= 1( ) or ( )y f x x f y−= =
1.5 Demonstrate an understanding of inverse of relations.
Topics:
• Function Notation Page 2
• Translations (Outcome 1.1) Page 7
• Reflections (Outcomes 1.4 and 1.5) Page 18
• Stretches (Outcome 1.2) Page 29
• Combining Transformations (Outcomes 1.3 and 1.4) Page 39
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Unit 1 Transformations
Function Notation:
( )f x → pronounced function “f” at x
→ means the function itself is
called “f” and the variable used
within the function is “x”
Ex) If 2( ) 5 8f x x x= + − , find the following.
a) (3)f b) ( 6)f − c) ( )12
f
3
Ex) If 2( ) 4 5f x x x= + + and ( ) 1 2g x x= + − find the
following.
a) (1) (8)f g− b) 3 ( 2)g x +
c) ( )(3)f g d) 2 ( 3) 11f x− − +
Ex) If ( )f x x= , write the following in function
notation.
a) 4x − b) 2x c) 3 2x + +
4
Ex) If 3( )f x x= , write the following in function
notation.
a) 3( 1)x + b) 32 3x + c) 37( 1) 4x− − +
Types of Functions:
Linear Quadratic
Basic Equation: Basic Equation:
5
Cubic Absolute Value
Basic Equation: Basic Equation:
Radical Rational
Basic Equation: Basic Equation:
6
Function Notation Assignment:
1) If 3( ) 5f x x= − , find the following in simplest form.
a) ( 1)f − b) 4 ( )f x c) (4 )f x
d) ( ) 5f x + e) ( 5)f x + f) ( )f x−
2) If ( )f x x= , write the following in terms of function f.
a) 1x − b) 3x + c) 2 1x − d) 3 x−
3) If 2( )f x x= , write the following in terms of function f.
a) 2 3x + b)
2( 3)x + c) 23x d)
2(3 )x
e) 24 7x − f) ( )24 7x − g)
42 1x− − h) 23( 2)x− − −
7
Vertical Translations:
The graph of 2y x= is shown below.
Sketch the graphs of 2 3y x= + and 2 4y x= − .
2y x=
Rule: Function Notation:
Mapping Notation:
8
Horizontal Translations:
The graph of 2y x= is shown below.
Sketch the graphs of 2( 3)y x= − and 2( 4)y x= + .
2y x=
Rule: Function Notation:
Mapping Notation:
9
In General if ( )y f x a b= + + then
0a the graph moves __________
0a the graph moves __________
0b the graph moves __________
0b the graph moves __________
Mapping Notation: ______________________
Ex) Describe how the graph of 4y x= − can be
obtained from the graph of y x= .
Ex) Describe how the graph of 4 3y x= + can be
obtained from the graph of 4y x= .
10
Ex) Describe in words and using mapping notation how
the graph of 6( 1) 5y x= − − can be obtained from the
graph of 6 3y x= − .
Ex) Write the equation of the graph produced when
the graph of y x= is translated 4 units to the
left and 7 units down.
Ex) Write an equation for each function represented
by the thick line
a) y x= b) 1y
x=
11
Ex) What vertical translation needs to be applied to
the graph of y x= so that is passes through the
point ( )16, 7 ?
Ex) What horizontal translation needs to be applied
to the graph of y x= so that is passes through
the point ( )17, 8 ?
12
Translations Assignment:
1) Given the graph of ( )y f x= below, sketch the graph of the transformed
function.
a) ( ) ( ) 3g x f x= + b) ( ) ( 2)g x f x= −
c) ( ) ( 4)g x f x= + d) ( ) ( ) 2g x f x= −
13
2) Given the function ( ) 2 3f x x= − + and ( ) 2 1g x x= + + , the
transformations that will transform ( )y f x= to become ( )y g x= are a
translation of
A] 4 units left and 2 units down
B] 4 units right and 2 units up
C] 1 unit left and 3 units up
D] 2 units left and 4 units down
3) Describe, using mapping notation, how the graphs of the following functions
can be obtained from the graph of ( )y f x= .
a) ( 10)y f x= + b) 6 ( )y f x+ =
c) ( 7) 4y f x= − + d) 3 ( 1)y f x− = −
4) Determine the equation of each transformed function.
a) 1
( )f xx
= is translated 5 units to the left and 4 units up.
b) 2( )f x x= is translated 8 units to the right and 6 units up.
14
c) ( )f x x= is translated 10 units to the right and 8 units down.
d) ( )y f x= is translated 7 units to the left and 12 units down.
5) Given the graph of ( )y f x= below, describe the transformations that can be
applied to the graph of ( )f x to obtain the transformed function, then sketch
the transformed graph.
a) ( ) ( 4) 3r x f x= + − b) ( ) ( 2) 4s x f x= − −
15
c) ( ) ( 2) 5t x f x= − + d) ( ) ( 3) 2v x f x= + +
6) The transformation of the function 3( )f x x= is described by the mapping
notation ( ) ( ), 4, 9x y x y→ − + . Describe the transformation on ( )y f x= .
7) What vertical translation is applied to the graph of 2y x= if the translated
image passes through the point ( )4, 19 ?
16
8) What horizontal translation is applied to the graph of 3y x= if the translated
image passes through the point ( )19, 125− ?
9) Determine the equation for each function represented by the thick line graph.
a) b)
10) Explain how the graph of ( ) 4g x x= + could be a vertical translation of 4
units up or a horizontal translation of 4 units to the left of the graph ( )f x x= .
17
11) The graph of the function 2y x= is translated so that the translated graph has
zeros of 7 and 1.
a) Determine the equation of the transformed graph.
b) Describe the translations of the graph 2y x= .
12) Describe how the graph of 1
yx
= could be transformed into the graph of
13
5y
x− =
−.
13) The roots of the quadratic equation 2 12 0x x− − = are 3− and 4. Determine
the roots of the equation 2( 5) ( 5) 12 0x x− − − − = .
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Reflections:
Reflection over the x-axis:
The graph of 2( ) 10 25y f x x x= = − + is shown
below.
Write an equation that Sketch the graph of
represents ( )y f x= − ( )y f x= − on the grid
below. (Use your
calculator.)
Rule: Function Notation:
Mapping Notation:
19
Reflection over the y-axis:
The graph of 2( ) 10 25y f x x x= = − + is shown below.
Write an equation that Sketch the graph of
represents ( )y f x= − ( )y f x= − on the grid
below. (Use your
calculator.)
Rule: Function Notation:
Mapping Notation:
20
Reflection over the line y x= :
The graph of 2( ) ( 5)y f x x= = − is shown below.
Write an equation that Sketch the graph of
represents 1( )y f x−= 1( )y f x−= on the grid
or ( )x f y= (inverse) below. (Use your
calculator.)
Rule: Function Notation:
Mapping Notation:
**When to use: 1( )y f x−= ( )x f y=
21
In general:
If ( )y f x= is transformed into
1( )y f x−=
( )y f x= − ( )y f x= − or
( )x f y=
The graph is The graph is The graph is
reflected reflected reflected
over the over the over the
________ ________ ________
Ex) In each case below sketch the indicated
reflection given the graph of ( )y f x= .
a) ( )y f x= − b) ( )x f y= c) ( )y f x= −
22
Ex) Sketch the reflection indicated for each case
below given the graph of ( )y f x= .
a) ( )y f x= − b) ( )y f x= − c) ( )x f y=
Ex) If the graph of ( )y f x= is indicated by the thin
line graph, write an equation that represents
each transformation (thick line graph) in terms
of ( )f x .
a) b) c)
23
Ex) If 2
6( )
3f x
x=
+, write an equation for each of
the following.
a) ( )y f x= − b) ( )y f x= − c) ( )x f y=
Sketch the graphs for each of the following.
2
6( )
3y f x
x= =
+
( )y f x= − ( )y f x= − ( )x f y=
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Ex) Consider the function 2( ) ( 3) 5f x x= + − .
a) Graph the function ( )f x and its inverse.
b) Is the inverse of ( )f x a function? Explain how you
can tell.
c) If the inverse is not a function, how can the domain
of ( )f x be restricted so that the inverse of ( )f x is a
function?
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Reflections Assignment:
1) Sketch the reflection of each graph in the x-axis and then determine the
equation of the reflected image.
a) ( ) 3f x x= b) 2( ) 1g x x= + c) 1
( )h xx
=
2) Sketch the reflection of each graph in the y-axis and then determine the
equation of the reflected image.
a) ( ) 3f x x= b) 2( ) 1g x x= + c)
1( )h x
x=
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3) Use mapping notation to describe how the graph of each function can be found from
the graph of the function ( )y f x= .
a) ( )y f x= − b) ( )y f x= − − c) ( )x f y=
4) Determine the zeros of the function ( ) ( 4)( 3)f x x x= + − after each transformation.
a) ( )y f x= − b) ( )y f x= −
5) The graph of a function ( )y f x= is contained completely in the third quadrant.
Identify in which quadrant the graph will be in after each transformation.
a) ( )y f x= − b) ( )y f x= − c) ( )x f y= d) ( )y f x= − −
6) Given the graph of ( )y g x= below, identify the coordinates of the invariant point
after each transformation.
a) ( )y g x= − b) ( )x g y= c) ( )y g x= −
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7) Given the graph of each relation below, sketch the graph of its inverse.
a) b)
8) For each function below, determine the equation of its inverse.
a) ( ) 2 3f x x= + b) 3( ) ( 4) 7g x x= − +
9) For each case below, identify a restriction that can be placed on the domain of ( )f x
so that its inverse is also a function.
a) 2( ) ( 21) 16f x x= + − b)
2
1( ) 20
( 18)f x
x= +
−
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10) The function for converting the temperature from degrees Fahrenheit, x, to degrees
Celsius ,y, is 5
( 32)9
y x= − .
a) Determine the equivalent temperature in degrees Celsius for 90 F.
b) Determine the inverse of this function. What does it represent? What do the
variables represent?
c) Determine the equivalent temperature in degrees Fahrenheit for 32 C.
d) If both functions were to be graphed, what does the invariant point represent in
this situation?
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Stretches:
Vertical Stretches:
The graph of 2( ) 4y f x x= = − is shown below.
Write the equation Write the equation
of 3 ( )y f x= of 1 ( )2
y f x=
Sketch the graph Sketch the graph
of 3 ( )y f x= on the of 1 ( )2
y f x= on the
grid below. grid below.
Rule: Function Notation:
Mapping Notation:
30
Horizontal Stretches:
The graph of 2( ) 4y f x x= = − is shown below.
Write the equation Write the equation
of (4 )y f x= of ( )13
y f x=
Sketch the graph Sketch the graph
of (4 )y f x= on the of 1( )3
y f x= on the
grid below. grid below.
Rule: Function Notation:
Mapping Notation:
31
Ex) Sketch the graph of the indicated transformation
if the graph of ( )y f x= is given.
a) 3 ( )y f x= b) (2 )y f x=
Ex) Write the equation of the image of ( )y f x= after
each transformation.
a) a horizontal b) a vertical c) a horizontal
stretch by a stretch by a stretch by a
factor of 3 factor of 14
factor of 12
about the y-axis. about the x-axis. about the y-axis,
and a vertical
stretch by a
factor of 37
about the x-axis.
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Ex) How can the graph of 3 ( )y f x= be obtained
from the graph of ( )y f x= ?
Ex) What happens to the graph of ( )y f x=
( , ) (8 , )x y x y→ ?
Ex) Write the equation of the image of 2y x= after a
horizontal stretch about the y-axis by a factor of 3
4.
33
Ex) Write the equation of the image of 3 7y x= +
after a vertical stretch about the x-axis by a
factor of 4 and a horizontal stretch by a factor
of 15
about the y-axis.
Ex) In each case below, describe how the graph of
the second function compares to the graph of
the first function.
a) 2xy =
32 xy =
b) 3y x=
33 (2 )y x=
34
Ex) Given the thin line graph of the original
function, determine the equation of the
transformed thick line graph.
a) 2
6
1y
x=
+ transformed graph
b) ( 4)( 2)( 6)y x x x= + − − transformed graph
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Stretches Assignment:
1) Given the graph of ( )y f x= below, sketch the graph of each given
transformation.
a) 2 ( )y f x= b) (2 )y f x=
c) 1
( )2
y f x= d) 1
2y f x
=
36
2) Determine the equation of ( )g x as a transformation of ( )f x for each case
below.
a) b)
c) d)
3) Use mapping notation to describe how the graph of ( )y f x= is transformed
into each of the following.
a) (8 )y f x= b) 6 3
7 2y f x
=
c)
15
4y f x
− = −
37
4) Describe how the graph of ( )y f x= can be transformed into the graph of
13
7y f x
=
.
5) Describe how the graph of 5log ( )y x= can be transformed into the graph of
56log (3 )y x= − .
6) Describe what happens to the graph of ( )y f x= after each of the following
changes are made to its equation.
a) Replace x with 4x .
b) Replace y with 5y .
c) Replace y with 3
y−.
38
7) Determine the zeros of the function ( ) ( 16)( 52)h x x x= − + after each
transformation given below.
a) 4 ( )y h x= b) (4 )y h x=
8) Determine the equation of the transformed function if the graph of
12 8y x= − is stretched vertically about the x-axis by a factor of 2 and then
stretched horizontally about the y-axis by a factor of 3.
9) Determine the equation of the transformed function if the graph of
( 5)( 25)( 45)y x x x= − + + is stretched vertically about the x-axis by a factor
of 3 and then stretched horizontally about the y-axis by a factor of 1
5.
39
Combining Transformations:
Rules:
Stretches Vertical
Stretch
y = stretch
factor
Horizontal
Stretch y = stretch
factor
Reflections Reflect in
x-axis y =
Reflect in
y-axis
y =
Reflect in
The line y x=
y =
or
x =
Translations Vertical
Translation y =
Horizontal
Translation
y =
Invariant Points:
40
Ex) Given the graph of ( )y f x= sketch its image
after the required transformations.
a) Stretch the graph of b) Translate the graph of
( )y f x= horizontally ( )y f x= 3 units up, and
by a factor of 2 about then stretch it
the y-axis, and then horizontally by a factor
translate it 3 units up. of 2 about the y-axis.
c) Translate the graph of d) Stretch the graph of
( )y f x= 2 units down, ( )y f x= vertically by
and then stretch it a factor of 3 about the
vertically by a factor of x-axis, and then translate
3 about the x-axis. 2 units down.
41
**Sometimes the order in which we apply the required
transformations matters.
To simplify the process, transformations should be
applied and described in the following order:
• Stretches
• Reflections
• Translations
Ex) Given the graph of ( )y f x= , sketch the graph
of the indicated transformed function. In each
case identify any invariant points.
a) 2 ( )y f x= − b) ( )1 12 2
y f x− −=
42
c) ( )12 ( 5)2
y f x= + d) 2 ( 3) 1y f x= − − +
e) ( )1 3 82
y f x= + −
43
Ex) Determine the equation of the graph of ( )y f x=
after it has been horizontally stretched by a
factor of 14
about the y-axis, then reflected
over the x-axis, and finally translated 5 units
down.
Ex) The function 3( )G x x= is transformed into a new
function ( )y P x= . To form the new function
( )y P x= , the graph of ( )y G x= is stretched
vertically about the x-axis by a factor of 0.2,
reflected about the y-axis, and then translated 3
units to the right. Determine the equation of the
new function ( )y P x= .
44
Ex) Describe the series of transformations required
to transform graph A into graph B.
Description 1:
Description 2:
45
Ex) Describe the transformations required to
transform:
a) graph A to graph B
b) graph B to graph C
46
Combining Transformations Assignment:
1) Given the graph of ( )y f x= below, sketch the graph of each transformation.
a) 3 ( ) 4y f x= − + b) ( )2( 3) 6y f x= + +
c) (3 15) 4y f x= − − − d) 1
( 3)2
y f x= − +
47
2) If the graph of 2y x= is stretched horizontally about the y-axis by a factor of
2 and then reflected about the x-axis, determine the equation of the
transformed image.
3) Describe how the graph of ( )y f x= could be transformed into the graph of
3 (4 16) 10y f x= − − − .
4) Given the graph of ( )y f x= below, sketch the transformed graph if ( )y f x=
is stretched vertically about the x-axis by a factor of 2, stretched horizontally
about the y-axis by a factor of 1
2, and then translated 3 units up.
48
5) If the point ( )12, 18− is on the graph of ( )y f x= , determine the coordinates
of its image under each of the following transformations.
a) 6 ( 4)y f x+ = − b) 4 (3 )y f x= c) 2 ( 6) 4y f x= − − +
d) 2
3 6 53
y f x−
= − −
e) ( )1
3 2( 6)3
y f x−
+ = +
6) Given the graph of ( )y f x= below, determine the equation of each
transformed graph.
a) b)
49
7) If the graph of ( )y f x= is stretched vertically about the x-axis by a factor of
3, reflected in the x-axis, and then translated 4 units to the left and 5 units
down, determine the equation of the transformed function.
8) If the graph of ( )y f x= is stretched horizontally about the y-axis by a factor
of 1
3, stretched vertically about the x-axis by a factor of
3
4, reflected in both
the x and y-axis, and then translated 6 units to the right and 2 units up,
determine the equation of the transformed function.
9) Describe using mapping notation the transformations on ( )y f x= for each of
the following.
a) 2 ( 3) 4y f x= − + b) (3 ) 2y f x= − −
c) ( )1
( 2)4
y f x−
= − + d) ( )3 4( 2)y f x− = − −
e) 2 3
3 4y f x
− − =
f) 3 6 ( 2 12)y f x− = − +
50
10) Describe how the graph of y x= could be transformed into the graph of
14 10
2y x
−= − +
11) Given the thin line graph of the original function, determine the equation of
the transformed thick line graph.
a)
b)
51
c)
12) If the domain and range of function ( )h x are such that
D : 12 18,x x x R− and R : 36,y y y R , determine the domain
and range of each transformation on ( )h x .
a) ( )3 2( 3) 5y h x= − + + b) 1
( 3 ) 82
y h x= − −
13) The graph of the function 22 1y x x= + + is stretched vertically about the
x-axis by a factor of 2, stretched horizontally about the y-axis by a factor of
1
3, and translated 2 units to the right and 4 units down. Determine the
equation of the transformed function.
52
14) If the x-intercept of the graph of ( )y f x= is located at ( ), 0a and the
y-intercept is located at ( )0, b , determine the x-intercept and y-intercept after
each of the following transformations on the graph of ( )y f x= .
a) ( )y f x= − − b) 1
22
y f x
=
c) 3 ( 4)y f x+ = − d) 1 1
6 ( 4)2 4
y f x
+ = −
e) 3 ( 7) 4y f x= − + + f) 2 (3 18) 34
yf x= − +
53
Answers
Function Notation Assignment:
1. a) 6− b) 34 20x − c) 364 5x − d) 3x e) 3( 5) 5x + −
f) 3 5x− −
2. a) ( ) 1f x − b) ( 3)f x + c) (2 1)f x − d) 3 ( )f x−
3. a) ( ) 3f x + b) ( 3)f x + c) 3 ( )f x d) (3 )f x e) 4 ( ) 7f x −
f) 4 ( ) 28f x − g) ( )22 1f x− − h) 3 ( 2)f x− − −
Transformations Assignment:
1. a) b)
c) d)
2. A
3. a) ( ) ( ), 10, x y x y→ − b) ( ) ( ), , 6x y x y→ −
c) ( ) ( ), 7, 4x y x y→ + + d) ( ) ( ), 1, 3x y x y→ + +
54
4. a) 1
45
yx
= ++
b) ( )2
8 6y x= − + c) 10 8y x= − −
d) ( 7) 12y f x= + −
5. a) The graph of ( )f x is translated b) The graph of ( )f x is translated
4 units to the left and 3 units 2 units to the right and 4 units
down. down.
c) The graph of ( )f x is translated d) The graph of ( )f x is translated
2 units to the right and 5 units up. 3 units to the left and 2 units up.
6. The graph of
3( )f x x= is translated 4 units to the left and 9 units up.
7. 3 units up
8. 24 units to the left
9. a) 2( 7) 2y x= + + b)
3 2( 12) 17( 12) 90( 12) 147y x x x= − + − + − +
10. If ( )g x x= , then ( ) ( ) 4 ( 4) 4f x g x g x x= + = + = +
11. a) 2( 4) 9y x= − − b) The graph of
2y x= is translated 4 units to
the right and 9 units down.
12. The graph of 1
yx
= is translated 5 units to the right and 3 units up.
13. 2 and 9
55
Reflections Assignment:
1. a) 3y x= − b) 2 1y x= − − c) 1
yx
−=
2. a) 3y x= − b) 2 1y x= + c)
1y
x
−=
3. a) ( ) ( ), , x y x y→ − b) ( ) ( ), , x y x y→ − − c) ( ) ( ), , x y y x→
4. a) 4− , 3 b) 3− , 4
5. a) 2 b) 4 c) 3 d) 1
6. a) ( )0, 4− b) ( )4, 4− − c) ( )4, 0
7. a) b)
8. a) 3
2
xy
−= b) 3 7 4y x= − +
56
9. a) 21 or 21x x − − b) 18 or 18x x
10. a) 32.2 C b) 9
325
y x= + y represents the temperature in Fahrenheit
and x represents the temperature in Celsius c) 89.6 F
d) The invariant point represents where the temperature in Celsius means
the same thing in Fahrenheit.
Stretches Assignment:
1. a) b)
c) d)
2. a) ( ) 4 ( )g x f x= b) ( ) (5 )g x f x= c) 1
( ) (4 )2
g x f x=
d) 1
( )3
g x f x
= −
3. a) ( )1
, , 8
x y x y
→
b) ( )2 6
, , 3 7
x y x y
→
c) ( ) ( ), 4 , 5x y x y→ − −
4. The graph of ( )y f x= is stretched vertically about the x-axis by a factor of 3
and then stretched horizontally about the y-axis by a factor of 7.
57
5. The graph of 5log ( )y x= is stretched vertically about the x-axis by a factor of
6, then it is reflected about the x-axis, and finally it is stretched vertically about
the y-axis by a factor of 1
3.
6. a) The graph of ( )y f x= is stretched horizontally about the y-axis by a factor
of 1
4.
b) The graph of ( )y f x= is stretched vertically about the x-axis by a factor
of 1
5.
c) The graph of ( )y f x= is stretched vertically about the x-axis by a factor
of 3 and then it is reflected about the x-axis.
7. a) 52− and 16 b) 13− and 4
8. 8 16y x= −
9. 375( 1)( 5)( 9)y x x x= − + +
Combining Transformations Assignment:
1. a) b)
58
c) d)
2.
21
2y x
= −
3. The graph of ( )y f x= is stretched vertically about the x-axis by a factor of 3,
then it is stretched horizontally about the y-axis by a factor of 1
4, next it is
reflected about the x-axis, and finally it is translated 4 units to the right and
10 units down.
4.
5. a) ( )8, 12− b) ( )4, 72− c) ( )6, 32− − d) ( )1, 49−
e) ( )12, 9− −
6. a) ( )( 2) 2y f x= − + − or ( 2) 2y f x= − − −
b) ( )2( 1) 4y f x= + − or (2 2) 4y f x= + −
7. 3 ( 4) 5y f x= − + −
8. ( )3
3( 6) 24
y f x−
= − − + or 3
( 3 18) 24
y f x−
= − + +
59
9. a) ( ) ( ), 3, 2 4x y x y→ + + b) ( )1
, , 23
x y x y
→ − −
c) ( )1
, 2, 4
x y x y−
→ − −
d) ( )1
, 2, 34
x y x y
→ + − +
e) ( )4 2
, , 3 3
x y x y− −
→
f) ( )1 1
, 6, 22 3
x y x y−
→ + +
10. The graph of y x= is stretched horizontally about the y-axis by a factor of
2, then it is reflected about the y-axis, and finally it is translated 8 units to the
left and 10 units up.
11. a) 4 2 2y x= − + − or 4 8 2y x= − + − b) 22( 4) 6y x= − + +
c) 2
25
( 3) 1y
x= −
− − + or
2
25
( 3) 1y
x= −
+ +
12. a) D : 9 6,x x x R− , R : 103,y y y R −
b) D : 6 4,x x x R− , R : 10,y y y R
13. ( )22 2(3 6) (3 6) 1 4y x x= − + − + − or 236 138 130y x x= − +
14. a) ( ), 0a− and ( )0, b− b) ( )2 , 0a and ( )0, 2b
c) ( )4, 0a + and ( )0, 3b − d) ( )4 4, 0a + and 1
0, 62
b
−
e) ( )7, 0a− + and ( )0, 3 4b + f) 1
6, 03
a
+
and ( )0, 8 12b +