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6.01 Reflections

Worksheet
Reflection across the x-axis
1

Write the vertical reflection of the graph of y = f \left( x \right) across the x-axis using function notation.

2

Each of the following points A, lie on the curve y =f\left(x \right). State the coordinates of point A', the location of point A following the reflection y =-f\left(x \right).

a
A = \left( 3, -4 \right)
b
A = \left( 0, -7 \right)
c
A = \left( -5, -2 \right)
d
A = \left( -8, 0 \right)
3

For each of the following graphs of f \left( x \right), sketch the graph of -f \left( x \right):

a
-2
2
4
6
8
x
-12
-10
-8
-6
-4
-2
2
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
4

The graph of y = P \left(x\right) is shown:

Sketch the graph of y = - P \left(x\right).

-5
-4
-3
-2
-1
1
2
3
4
5
x
-40
-30
-20
-10
10
20
30
40
y
5

Consider the cubic function y = \left(x + 3\right) \left(x - 2\right) \left(x - 5\right).

a

Determine the x-intercepts.

b

A second cubic function has the same x-intercepts, but is a reflection of the original function across the x-axis. State the equation of the reflected function.

6

For each of the following functions f \left( x \right) and its corresponding graph:

i
Write an expression for the function -f \left( x \right).
ii
Sketch the graph of -f \left( x \right).
a

f \left( x \right) = x^{2} + 3

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b

f \left( x \right) = \left(x - 5\right)^{2}

1
2
3
4
5
6
7
8
x
-4
-3
-2
-1
1
2
3
4
y
c

f \left( x \right) = 2 x + 2

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d

f \left( x \right) = \left| 3 x - 12\right|

1
2
3
4
5
6
7
8
x
-4
-3
-2
-1
1
2
3
4
y
e

f \left( x \right) = x^{3} + 2

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
7

The graph of a function f \left( x \right) is shown:

a

State the domain of f \left( x \right).

b

State the range of f \left( x \right).

c

Sketch the graph of the reflection - f \left( x \right).

d

State the domain of the reflected function -f \left( x \right).

e

State the range of the reflected function - f \left( x \right).

-4
-3
-2
-1
1
2
3
4
5
x
-4
-3
-2
-1
1
2
3
4
5
y
8

Consider the function f \left( x \right) = 3 x^{2}.

a

State the domain of f \left( x \right).

b

State the range f \left( x \right).

c

Sketch the result of reflecting f \left( x \right) across the x-axis:

d

State the domain of the reflected function.

e

State the range of the reflected function.

f

Write down anything you notice about the domain and range of a function when it is reflected across the x-axis.

Reflection across the y-axis
9

Write the horizontal reflection of the graph of y = f \left( x \right) across the y-axis using function notation.

10

Each of the following points A, lie on the curve y =f\left(x \right). State the coordinates of point A', the location of point A following the reflection y =f\left(-x \right).

a
A = \left( 3, -4 \right)
b
A = \left( 0, 7 \right)
c
A = \left( 5, -12 \right)
d
A = \left( -3, 0 \right)
11

Consider the given graph of f \left( x \right):

a

Sketch the graph of f \left( - x \right).

b

What do you notice about the graphs of f \left( x \right) and f \left( -x \right). Explain why this is.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
12

The graph of y = P \left(x\right) is shown. Sketch the graph of y = P \left( - x \right).

-5
-4
-3
-2
-1
1
2
3
4
5
x
-40
-30
-20
-10
10
20
y
13

Consider the given graph of the function f \left( x \right) = \dfrac{1}{x + 6} - 3.

a

State the equations of the asymptotes of f \left( x \right).

b

For the related function f \left( - x \right), state the equations of the asymptotes.

c

Sketch the graph of the function f \left( - x \right).

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
14

Consider the function f \left( x \right) = \dfrac{3}{x + 1}.

a

State the domain of f \left( x \right).

b

State the range of f \left( x \right).

c

Sketch the result of reflecting f \left( x \right) across the y-axis.

d

State the domain of the reflected function.

e

State the range of the reflected function.

f

When a function is reflected across the y-axis, does it affect the domain or range?

15

For each of the following functions f \left( x \right) and its corresponding graph:

i
Write an expression for the function f \left( -x \right).
ii
Sketch the graph of f \left( -x \right).
a

f \left( x \right) = \left(x - 3\right)^{2}

-8
-6
-4
-2
2
4
6
8
x
3
6
9
12
y
b

f \left( x \right) = \dfrac{1}{3} x + 3

-4
-3
-2
-1
1
2
3
4
x
1
2
3
4
5
6
7
8
y
c

f \left( x \right) = \left| 2 x - 4\right|

-8
-6
-4
-2
2
4
6
8
x
1
2
3
4
5
6
7
8
y
d

f \left( x \right) = x^{3} - 2

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Reflection across both axes
16

For each of the following graphs of f \left( x \right) and g \left( x \right), write g \left( x \right) in terms of f \left( x \right) by considering the reflections that have occured on f \left( x \right) to get g \left( x \right):

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
17

Consider the function f \left( x \right) = x^{3} - 7 x^{2} + 10 x.

a

State the y-intercept of the function.

b

Determine the x-intercepts of the function.

c

Sketch the graph of:

i

f \left( x \right)

ii

f \left( - x \right)

iii

- f \left( x \right)

iv

- f \left( - x \right)

18

The graph of a particular function f \left( x \right) has x-intercepts at \left( - 9 , 0\right) and \left(4, 0\right). State the \\ x-intercepts of the following graphs:

a

y = f \left( - x \right)

b

y = - f \left( x \right)

c

y = - f \left( - x \right)

19

A particular function f \left( x \right) is increasing on the interval \left(-\infty, 4\right) and decreasing on the interval \left(4, \infty\right). Describe the behaviour of the following functions on the interval \left(-\infty, \infty \right) :

a
f \left( - x \right)
b
- f \left( x \right)
c
- f \left( - x \right)
20

Describe the overall effect on the graph of y = f \left( x \right) if is transformed to y = -f \left( -x \right).

21

Consider the graph of f \left( x \right) = x^{3} - 16 x :

a

Sketch the graph of - f \left( - x \right).

b

Comment on the graphs of f \left( x \right) and -f \left( -x \right).

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-25
-20
-15
-10
-5
5
10
15
20
25
y
22

Each of the following points A, lie on the curve y =f\left(x \right), State the coordinates of point A', the location of point A following the transformation y =-f\left(-x \right).

a
A = \left( 2, -4 \right)
b
A = \left( 0, 7 \right)
c
A = \left( -5, -9 \right)
d
A = \left( -3, 0 \right)
23

For each of the following functions f \left( x \right) and its corresponding graph:

i
Write an expression for the function -f \left( -x \right).
ii
Sketch a graph of -f \left( -x \right).
a

f \left( x \right) = x^{2} - 3

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b

f \left( x \right) = \left(x + 8\right)^{2}

-12
-8
-4
4
8
12
x
-12
-8
-4
4
8
12
y
c

f \left( x \right) = 2 x + 3

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d

f \left( x \right) = \left| 2 x - 10\right|.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
e

f \left( x \right) = x^{3} + 4

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
24

For each of the following graphs of f \left( x \right), sketch the graph of - f \left( - x \right):

a
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-7
-6
-5
-4
-3
-2
-1
1
y
b
-2
2
4
6
8
10
12
14
16
18
x
-2
2
4
6
8
10
12
14
16
18
y
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MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

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