f(x) + a
42
f(x) + a 0 2 4 6 8 x y = f(x) -2 -4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (1) f(x) = x 2 f(x) +2= x 2 + 2 f(x) - 5 = x 2 - 5 Vertical Translations In general f(x) + a gives a translation by the vector a 0
description
f(x) + a. 6. y = f(x). 4. In general f(x) + a gives a translation by the vector. 2. x. 4. -6. 2. 8. 6. -4. -2. -2. -4. -6. Graphs of Related Functions (1). Vertical Translations. f(x) +2= x 2 + 2. f(x) = x 2. 0. f(x) - 5 = x 2 - 5. 6. y = f(x). 4. - PowerPoint PPT Presentation
Transcript of f(x) + a
f(x) + a
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (1)
f(x) = x2
f(x) +2= x2 + 2
f(x) - 5 = x2 - 5
Vertical Translations
In general f(x) + a
gives a translation by the vector
a0
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (1)
Vertical Translations
In general f(x) + a
gives a translation by the vector
a
0
f(x)
f(x) + 3
f(x) - 2
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (2)
Horizontal Translations
f(x)
f(x - 5)
f(x + 2)In general f(x + a)
gives a translation by the vector
0
a
52
In other words, ‘+’ inside the brackets means move to the
LEFT
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (2)
Horizontal Translations
In general f(x + a) gives a translation by
the vector
0
a
f(x)
f(?)f(x - 5)
5
f(?)f(x + 3)
3
Worksheet 1
f(x)
x
y = f(x)
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)
x
y = f(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
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 1
Worksheet 1 Answersf(x)
x
y = f(x)
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)
x
y = f(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
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 1
-f(x)
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (4)
f(x) = x2
Reflections in the x axis
-f(x) = -x2
f(x) = x2 - 10x + 25
-f(x) = -x2 + 10x - 25
The graph of -f(x) is a reflection of f(x) in the x axis.
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (4)
f(x) = x2+ 1
Reflections in the x axis
f(x) = x2 - 10x + 23
-f(x) = -x2 + 10x - 23
The graph of -f(x) is a reflection of f(x) in the x axis.
-f(x) = -(x2 + 1)
= -x2 - 1
= -x2 - 1
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Reflections in the x axis
The graph of -f(x) is a reflection of f(x) in the x axis.
f(x) = x3 - 3x2 - 6x + 8
-f(x) = -x3 + 3x2 + 6x - 8
Graphs of Related Functions (4)
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Reflections in the x axis
The graph of -f(x) is a reflection of f(x) in the x axis.
f(x)
-f(x)
Graphs of Related Functions (4)
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Reflections in the x axis
The graph of -f(x) is a reflection of f(x) in the x axis.
f(x) = Sinx
-f(x) = -Sinx
Graphs of Related Functions (4)
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Reflections in the x axis
The graph of -f(x) is a reflection of f(x) in the x axis.
f(x) = 2Sinx
-f(x) = -2Sinx
Graphs of Related Functions (4)
Worksheet 2
f(x)
x
y = f(x)
Draw the graph of -f(x) for each case on the grids below.
y = f(x)
x
y = f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 2
Worksheet 2 Answers
f(x)
x
y = f(x)
Draw the graph of -f(x) for each case on the grids below.
y = f(x)
x
y = f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 2
f(-x)
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (5)
f(x) = x2 + 4x + 5
Reflections in the y axis
f(x) = x2 - 4x + 5
f(-x)
f(-x) = (- x)2 + 4(- x) + 5
= x2 - 4x + 5
The graph of f(-x) is a reflection of f(x) in the y axis.
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Reflections in the y axis
f(x) = x3 - 9x2 + 18x
f(-x) = (-x)3 - 9(-x)2 + 18(-x)
Graphs of Related Functions (5)
f(-x) = -x3 - 9x2 - 18x
The graph of f(-x) is a reflection of f(x) in the y axis.
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Reflections in the y axis
f(x)
Graphs of Related Functions (5)
The graph of f(-x) is a reflection of f(x) in the y axis.
f(-x)
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 & FridayPast paper practice #2
Calculator Paper(bring a calculator)
Worksheet 3
f(x)
x
y = f(x)
Draw the graph of f(-x) for each case on the grids below.
y = f(x)
x
y = f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 3
Worksheet 3 Answers
f(x)
x
y = f(x)
Draw the graph of f(-x) for each case on the grids below.
y = f(x)
x
y = f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x) f(x)
Worksheet 3
kf(x)
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Graphs of Related Functions (6)
The graph of kf(x) gives a
stretch of f(x) by scale factor k in the y direction.
f(x)
2f(x)
3f(x)
0Points located on the x axis remain fixed.
Stretches in the y direction
y co-ordinates doubled
y co-ordinates tripled
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
0
Graphs of Related Functions (6)
The graph of kf(x) gives a
stretch of f(x) by scale factor k in the y direction.
f(x)
½f(x)
1/3f(x)
0
y co-ordinates halved
y co-ordinates scaled by 1/3
2 4 6 8x
y = f(x)
-2-4-6
10
20
30
-10
-20
-30
Stretches in y
The graph of kf(x) gives a
stretch of f(x) by scale factor k in the y direction.
f(x)
2f(x)
Graphs of Related Functions (6)
3f(x)
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Sinx
2Sinx
Graphs of Related Functions (6)
3
-3
3Sinx
The graph of kf(x) gives a
stretch of f(x) by scale factor k in the y direction.
Stretches in y
Graphs of Related Functions (6)
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
3
-3
Cosx
½Cosx
2Cosx
3CosxThe graph of kf(x) gives a
stretch of f(x) by scale factor k in the y direction.
Worksheet 4
f(x)x
y = f(x)
Grid 1: Sketch or trace the graph of 2f(x)Grid 2: Sketch or trace the graph of 3f(x)
y = f(x)
x
y = f(x)
Grid 3: Sketch or trace the graph of ½f(x)
Grid 4: Sketch or trace the graph of 2f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x)f(x)
Worksheet 4
Worksheet 4 Answers
f(x)x
y = f(x)
Grid 1: Sketch or trace the graph of 2f(x)Grid 2: Sketch or trace the graph of 3f(x)
y = f(x)
x
y = f(x)
Grid 3: Sketch or trace the graph of ½f(x)
Grid 4: Sketch or trace the graph of 2f(x)
xx
y = f(x)3 4
1 2
f(x)
f(x)f(x)
Worksheet 4
f(kx)
½ the x co-ordinate
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (7)
f(x)f(2x)f(3x)
The graph of f(kx) gives a stretch of
f(x) by scale factor 1/k in the x direction.
1/3 the x co-ordinate
Stretches in x
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (7)
f(x)f(1/2x)f(1/3x)
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 3
All x co-ordinates x 2
Stretches in x
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (7)
f(x)f(2x) f(1/2x)
All x co-ordinates x 2
All x co-ordinates x 1/2
The graph of f(kx) gives a stretch of
f(x) by scale factor 1/k in the x direction.
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Graphs of Related Functions (7)
f(x) = Sinxf(x) = Sin2x
All x co-ordinates x 1/2The graph of f(kx) gives a stretch of
f(x) by scale factor 1/k in the x direction.
Stretches in x
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Graphs of Related Functions (7)
f(x) = Sinxf(x) = Sin3x
All x co-ordinates x 1/3The graph of f(kx) gives a stretch of
f(x) by scale factor 1/k in the x direction.
Stretches in x
90 180x
y = f(x)
0 270 360-90-180-270-360
1
-1
2
-2
Graphs of Related Functions (7)
f(x) = Cosxf(x) = Cos2x
All x co-ordinates x 1/2The graph of f(kx) gives a stretch of
f(x) by scale factor 1/k in the x direction.
f(x) = Cos ½ x
All x co-ordinates x 2
Stretches in x
Worksheet 5
f(x)x
y = f(x)
Grid 1: Sketch or trace the graph of f(2x)Grid 2: Sketch or trace the graph of f(3x)
y = f(x)
x
y = f(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)
xx
y = f(x)3 4
1 2
f(x)
f(x)
f(x)
Worksheet 5
Worksheet 5 Answers
f(x)x
y = f(x)
Grid 1: Sketch or trace the graph of f(2x)Grid 2: Sketch or trace the graph of f(3x)
y = f(x)
x
y = f(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)
xx
y = f(x)3 4
1 2
f(x)
f(x)
f(x)
Worksheet 5
GCSE Q’s Mark scheme1. (a) Graph translated 2 units upwards through points(–4, 2), (–2, 4), (0, 2) and (3, 5)SketchM1 for a vertical translationA1 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 2M1 for reflection in x-axis or y-axisA1 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 1 mark
3. (a) (4, 3) 1 markB1 for (4, 3)(b) (2, 6) 1 markB1 for (2, 6)
4.(a)y = f(x – 4) 2 marksB2 cao
(B1 for f(x – 4) or y = f(x + a), a ≠ –4, a ≠ 0)
(b)
2
4
-2
-4
1 8 0 3 6 0 5 4 00
x
y
2
B2 cao(B1 cosine curve with either correct amplitude or correct period, but not both)
2
4
6
8
1 0
1 2
-2
-4
-6
-8
-1 0
-12
-14
-16
-18
2 4 6 8 1 0-2 -4 0 x
y
-6-8-1 0
2
B2 parabola max (0,0), through (–2, –4) and (2, –4)To accuracy +/- ½sq
(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)
Q5(a)
2
4
6
8
1 0
1 2
-2
-4
-6
-8
-1 0
2 4 6 8 1 0 -2-4-6-8-1 0 0 x
y
2-1
2
B2 parabola max (0, 4), through (–4, 0) and (4, 0)To ½sq
(B1 parabola with single maximum point (0, 4))To ½sq
Q5(b)
f(x) + a
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (1)
f(x) = x2
f(x) = x2 + 2
f(x) = x2 - 5
Vertical Translations
In general f(x) + a
gives a translation by the vector
a0
0 2 4 6 8x
y = f(x)
-2-4-6
2
4
6
-2
-4
-6
Graphs of Related Functions (2)
Horizontal Translations
f(x)
f(x + 2)In general f(x + a)
gives a translation by the vector
2
Inside the brackets, “+” means
move the curve _____
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