Graphs of Related Functions (1)
6 y = f(x)
Vertical
Translations
f(x) + a
f(x) +2= x2 + 2
4
f(x) = x2
-6
-4
2
0
-2
-2
f(x) - 5 =
x2
-5
-4
-6
2
4
6
In general
f(x) + a
gives a translation
by the vector
0
 
a
x
8
Graphs of Related Functions (1)
6 y = f(x)
Vertical
Translations
4
f(x) + 3
2
f(x)
-6
-4
0
-2
-2
f(x) - 2
-4
-6
2
4
6
In general
f(x) + a
gives a translation
by the vector
0 
 
a
x
8
Graphs of Related Functions (2)
6 y = f(x)
Horizontal
Translations
4
2
2
5
f(x - 5)
f(x)
-6
-4
0
-2
f(x + 2)
In other words, ‘+’
inside the brackets
means move to the
LEFT
-2
-4
-6
2
4
6
In general f(x + a)
gives a translation by
the vector
 a
 
0 
x
8
Graphs of Related Functions (2)
6 y = f(x)
Horizontal
Translations
4
3
5
f(xf(?)
- 5)
2
f(x)
-6
-4
f(xf(?)
+ 3)
0
-2
-2
-4
-6
2
4
6
In general f(x + a)
gives a translation by
the vector
 a
 
0 
x
8
Worksheet 1
Grid 1: Sketch or trace (a) f(x) - 4 (b) f(x + 4) (c) f(x - 3)
Grid 2: Sketch or trace (a) f(x + 4) - 2 (b) f(x - 3) + 1 (c) f(x - 3) - 5
y = f(x)
1
2
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace (a) f(x) + 2 (b) f(x - 3) - 4 (c) f(x + 3) + 3
Grid 4: Sketch or trace (a) f(x) + 3 (b) f(x + 7) + 2 (c) f(x - 3) - 2
y = f(x)
3
f(x)
x
4
y = f(x)
f(x)
x
Worksheet 1
Grid 1: Sketch or trace (a) f(x) - 4 (b) f(x + 4) (c) f(x - 3)
Grid 2: Sketch or trace (a) f(x + 4) - 2 (b) f(x - 3) + 1 (c) f(x - 3) - 5
Worksheet 1
Worksheet
1
Answers
2
1
y = f(x)
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace (a) f(x) + 2 (b) f(x - 3) - 4 (c) f(x + 3) + 3
Grid 4: Sketch or trace (a) f(x) + 3 (b) f(x + 7) + 2 (c) f(x - 3) - 2
y = f(x)
3
f(x)
x
4
y = f(x)
f(x)
x
Graphs of Related Functions (4)
6 y = f(x)
Reflections in the
x axis
f(x) = x2 - 10x + 25
f(x) = x2
4
The graph of -f(x)
is a reflection of
f(x) in the x axis.
-6
-4
-f(x)2
0
-2
2
4
6
-f(x) = -x2
-2
-4
-f(x) = -x2 + 10x - 25
-6
x
8
Graphs of Related Functions (4)
6 y = f(x)
Reflections in the
x axis
f(x) = x2+ 1
4
f(x) = x2 - 10x + 23
The graph of -f(x)
is a reflection of
f(x) in the x axis.
-6
-4
2
0
-2
2
4
6
-f(x) = -(x2 + 1)
= -x2 - 1
-2
= -x2 - 1
-4
-6
-f(x) = -x2 + 10x - 23
x
8
Graphs of Related Functions (4)
30 y = f(x)
Reflections in the
x axis
20
f(x) = x3 - 3x2 - 6x + 8
The graph of -f(x)
is a reflection of
f(x) in the x axis.
-6
-4
10
-2
0
2
4
-10
-f(x) = -x3 + 3x2 + 6x - 8
-20
-30
6
x
8
Graphs of Related Functions (4)
30 y = f(x)
Reflections in the
x axis
The graph of -f(x)
is a reflection of
f(x) in the x axis.
20
10
f(x)
-6
-4
-2
0
-10
-20
-30
4
2
-f(x)
6
x
8
Graphs of Related Functions (4)
y = f(x)
Reflections in the
x axis
The graph of -f(x)
is a reflection of
f(x) in the x axis.
2
f(x) = Sinx
1
x
-360
-270
-180
-90
0
90
-1
-f(x) = -Sinx
-2
180
270
360
Graphs of Related Functions (4)
y = f(x)
Reflections in the
x axis
The graph of -f(x)
is a reflection of
f(x) in the x axis.
f(x) = 2Sinx
2
1
x
-360
-270
-180
-90
0
-1
-2
-f(x) = -2Sinx
90
180
270
360
Draw the graph of -f(x) for each case on the grids below.
y = f(x)
1
Worksheet 2
y = f(x)
2
f(x)
x
x
f(x)
y = f(x)
3
f(x)
y = f(x)
4
f(x)
x
x
Worksheet 2
Worksheet 2
Draw the graph of -f(x) for each case on the grids below.
y = f(x)
1
y = f(x)
2
f(x)
x
x
f(x)
y = f(x)
3
f(x)
y = f(x)
4
f(x)
x
x
Worksheet 2
Answers
Graphs of Related Functions (5)
f(-x)
Reflections in the
6 y = f(x)
y axis
4
2
f(x) = x2 - 4x + 5
f(x) = x2 + 4x + 5
-6
-4
0
-2
2
4
6
f(-x)
The graph of f(-x)
is a reflection of
f(x) in the y axis.
-2
f(-x) = (- x)2 + 4(- x) + 5
= x2 - 4x + 5
-4
-6
x
8
Graphs of Related Functions (5)
30 y = f(x)
Reflections in the
y axis
20
f(x) = x3 - 9x2 + 18x
10
-6
-4
-2
0
2
4
6
-10
f(-x) = (-x)3 - 9(-x)2 + 18(-x)
The graph of f(-x)
is a reflection of
f(x) in the y axis.
-20
-30
f(-x) = -x3 - 9x2 - 18x
x
8
Graphs of Related Functions (5)
30 y = f(x)
Reflections in the
y axis
20
f(x)
-6
-4
f(-x)
10
-2
0
-10
The graph of f(-x)
is a reflection of
f(x) in the y axis.
-20
-30
2
4
6
x
8
Page 511 Ex E17.3
• A and A* questions
• Use a scale of 2 squares in your book = 1
square in the diagram
Next lesson (Monday)
Transformations of graphs
part 2 – stretches.
Transformations of sine,
cosine
Next Thursday & Friday
Past paper practice #2
Calculator Paper
(bring a calculator)
Worksheet 3
Draw the graph of f(-x) for each case on the grids below.
y = f(x)
1
y = f(x)
2
f(x)
x
x
f(x)
y = f(x)
3
4
y = f(x)
f(x)
f(x)
x
x
Worksheet 3
Worksheet 3
Draw the graph of f(-x) for each case on the grids below.
y = f(x)
1
y = f(x)
2
f(x)
x
x
f(x)
y = f(x)
3
4
y = f(x)
f(x)
f(x)
x
x
Worksheet 3
Graphs of Related Functions (6)
30 y = f(x)
Stretches in
the y direction
y co-ordinates
tripled
3f(x)
20
2f(x)
f(x)
10
-6
-4
-2
y co-ordinates
doubled
2
0
4
6
0
The graph of
kf(x) gives a
stretch of f(x) by
scale factor k in
the y direction.
-10
-20
-30
Points located on the
x axis remain fixed.
kf(x)
x
8
Graphs of Related Functions (6)
30 y = f(x)
y co-ordinates
halved
1/3f(x)
20
½f(x)
y co-ordinates
scaled by 1/3
f(x)
10
-6
-4
-2
2
0
0
The graph of
kf(x) gives a
stretch of f(x) by
scale factor k in
the y direction.
-10
-20
-30
4
6
x
8
Graphs of Related Functions (6)
30 y = f(x)
Stretches in y
3f(x)
The graph of
kf(x) gives a
stretch of f(x) by
scale factor k in
the y direction.
-6
-4
20
2f(x)
10
f(x)
2
-2
-10
-20
-30
4
6
x
8
Graphs of Related Functions (6)
3 y = f(x)
Stretches in y
3Sinx
2
2Sinx
Sinx
1
x
-360
-270
-180
-90
0
-1
The graph of
kf(x) gives a
stretch of f(x) by
scale factor k in
the y direction.
-2
-3
90
180
270
360
Graphs of Related Functions (6)
3 y = f(x)
3Cosx
2
2Cosx
½Cosx
The graph of
kf(x) gives a
stretch of f(x) by
scale factor k in
the y direction.
1
Cosx
-360
-270
-180
-90
0
-1
-2
-3
90
180
270
x
360
Worksheet 4
Grid 1: Sketch or trace the graph of 2f(x)
Grid 2: Sketch or trace the graph of 3f(x)
y = f(x)
1
2
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace the graph of ½f(x)
Grid 4: Sketch or trace the graph of 2f(x)
y = f(x)
3
y = f(x)
4
f(x)
f(x)
x
x
Worksheet 4
Worksheet 4
Grid 1: Sketch or trace the graph of 2f(x)
Grid 2: Sketch or trace the graph of 3f(x)
y = f(x)
1
2
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace the graph of ½f(x)
Grid 4: Sketch or trace the graph of 2f(x)
y = f(x)
3
y = f(x)
4
f(x)
f(x)
x
x
Worksheet 4 Answers
Graphs of Related Functions (7)
6 y = f(x)
Stretches in x
f(3x)
f(2x)
f(x)
4
2
-6
-4
0
-2
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
-2
2
4
6
½ the x co-ordinate
1/3 the x co-ordinate
-4
-6
f(kx)
x
8
Graphs of Related Functions (7)
6 y = f(x)
Stretches in x
4
f(1/3x)
f(x)
f(1/2x)
2
-6
-4
0
-2
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
2
4
6
-2
All x co-ordinates x 2
All x co-ordinates x 3
-4
-6
x
8
Graphs of Related Functions (7)
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
6 y = f(x)
f(2x)
f(x)
f(1/2x)
4
2
-6
-4
0
-2
2
4
6
-2
All x co-ordinates x 1/2
All x co-ordinates x 2
-4
-6
x
8
Graphs of Related Functions (7)
y = f(x)
Stretches in x
2
f(x) = Sinx
f(x) = Sin2x
1
-360
-270
-180
-90
0
90
180
270
-1
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
All x co-ordinates x 1/2
-2
x
360
Graphs of Related Functions (7)
y = f(x)
Stretches in x
2
f(x) = Sinx
f(x) = Sin3x
1
x
-360
-270
-180
-90
0
90
180
270
-1
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
All x co-ordinates x 1/3
-2
360
Graphs of Related Functions (7)
y = f(x)
Stretches in x
2
f(x) = Cos2x
f(x) = Cos ½ x
f(x) = Cosx
1
x
-360
-270
-180
-90
0
90
180
270
-1
The graph of f(kx)
gives a stretch of
f(x) by scale factor
1/k in the x
direction.
All x co-ordinates x 1/2
-2
All x co-ordinates x 2
360
Worksheet 5
Grid 1: Sketch or trace the graph of f(2x)
Grid 2: Sketch or trace the graph of f(3x)
y = f(x)
1
2
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace the graph of (a) f(½x)
(b) f((1/3)x)
Grid 4: Sketch or trace the graph of f(½ x)
f(x)
y = f(x)
3
y = f(x)
4
f(x)
x
x
Worksheet 5
Worksheet 5
Grid 1: Sketch or trace the graph of f(2x)
Grid 2: Sketch or trace the graph of f(3x)
y = f(x)
1
2
y = f(x)
f(x)
f(x)
x
x
Grid 3: Sketch or trace the graph of (a) f(½x)
(b) f((1/3)x)
Grid 4: Sketch or trace the graph of f(½ x)
f(x)
y = f(x)
3
y = f(x)
4
f(x)
x
x
Worksheet 5 Answers
GCSE Q’s Mark scheme
1.
(a)
Graph translated 2 units upwards through points
(–4, 2), (–2, 4), (0, 2) and (3, 5)
Sketch
M1 for a vertical translation
A1 curve through points (–4, 2), (–2, 4), (0, 2) and (3, 5) ± ½ square
(b)
Graph reflected in x-axis through points
(–4, 0), (–2, –2), (0, 0) and (3, –3)
Sketch 2
M1 for reflection in x-axis or y-axis
A1 curve through points (–4, 0), (–2, –2), (0, 0) and (3, –3) ± ½ square
[4]
GCSE Q’s Mark scheme
2. (c)
Reflection in the y axis
3. (a) (4, 3)
B1 for (4, 3)
(b) (2, 6)
B1 for (2, 6)
1 mark
1 mark
1 mark
4.(a)y = f(x – 4)
B2 cao
2 marks
(B1 for f(x – 4) or y = f(x + a), a ≠ –4, a ≠ 0)
(b)
y
4
2
0
180
360
540 x
-2
-4
2
B2 cao
(B1 cosine curve with either correct amplitude or correct period, but
not both)
2
Q5(a)
B2 parabola max (0,0), through
(–2, –4) and (2, –4)
To accuracy +/- ½sq
y
12
10
8
6
4
2
-10
-8
-6
-4
-2
0
-2
-4
-6
-8
-10
-12
-14
-16
-18
2
4
6
8
10
x
(B1 parabola with single
maximum point (0, 0) or through
(–2, –4) and (2, –4),but not both
or the given parabola translated
along the y-axis by any other
value than -4 – the translation
must be such that the points (0,
4), (–2, 0), (2, 0) are translated
by the same amount.
To ½sq)
2
B2 parabola
max (0, 4),
through (–4, 0)
and (4, 0)
To ½sq
Q5(b)
y
12
10
8
6
4
2
-10
-8
-6
-4
-2
0
-2
-4
-6
-8
-10
-12
2
4
6
8
10
x
(B1 parabola
with single
maximum point
(0, 4))
To ½sq
Graphs of Related Functions (1)
6 y = f(x)
Vertical
Translations
f(x) + a
f(x) = x2 + 2
4
f(x) = x2
-6
-4
2
0
-2
-2
f(x) =
x2
-5
-4
-6
2
4
6
In general
f(x) + a
gives a translation
by the vector
0
 
a
x
8
Graphs of Related Functions (2)
6 y = f(x)
Horizontal
Translations
4
2
2
f(x)
-6
-4
0
-2
f(x + 2)
-2
-4
Inside the brackets, “+” means
move the curve _____ -6
2
4
6
In general f(x + a)
gives a translation by
the vector






x
8