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f(x) + a

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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= x2 + 2

f(x) = x2

0

f(x) - 5 = x2 - 5

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) + 3

f(x)

0

f(x) - 2

6

y = f(x)

4

2

5

2

f(x - 5)

x

4

-6

2

8

6

-4

-2

f(x + 2)

-2

In general f(x + a) gives a translation by the vector

-4

-6

Graphs of Related Functions (2)

Horizontal Translations

f(x)

0

In other words, ‘+’ inside the brackets means move to the LEFT

6

y = f(x)

4

5

3

2

x

4

-6

2

8

6

-4

-2

-2

In general f(x + a) gives a translation by the vector

-4

-6

Graphs of Related Functions (2)

Horizontal Translations

f(x - 5)

f(?)

f(x)

0

f(x + 3)

f(?)

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)

2

1

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)

4

y = f(x)

3

f(x)

f(x)

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

y = f(x)

2

1

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)

4

y = f(x)

3

f(x)

f(x)

x

x

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Graphs of Related Functions (4)

Reflections in the x axis

f(x) = x2 - 10x + 25

f(x) = x2

The graph of -f(x) is a reflection of f(x) in the x axis.

0

-f(x) = -x2

-f(x) = -x2 + 10x - 25

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Graphs of Related Functions (4)

Reflections in the x axis

f(x) = x2+ 1

f(x) = x2 - 10x + 23

The graph of -f(x) is a reflection of f(x) in the x axis.

0

-f(x) = -(x2 + 1)

= -x2 - 1

-f(x) = -x2 + 10x - 23

= -x2 - 1

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.

10

x

4

-6

2

8

0

6

-4

-2

-10

-f(x) = -x3 + 3x2 + 6x - 8

-20

-30

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)

x

4

-6

2

8

0

6

-4

-2

-10

-f(x)

-20

-30

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

90

-90

-180

0

270

180

-270

360

-1

-f(x) = -Sinx

-2

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

90

-90

-180

0

-360

270

180

-270

360

-1

-2

-f(x) = -2Sinx

Draw the graph of -f(x) for each case on the grids below.

y = f(x)

y = f(x)

1

2

f(x)

x

x

f(x)

y = f(x)

y = f(x)

3

4

f(x)

f(x)

x

x

Worksheet 2

Worksheet 2

Draw the graph of -f(x) for each case on the grids below.

y = f(x)

y = f(x)

1

2

f(x)

x

x

f(x)

y = f(x)

y = f(x)

3

4

f(x)

f(x)

x

x

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Graphs of Related Functions (5)

Reflections in the y axis

f(x) = x2 - 4x + 5

f(x) = x2 + 4x + 5

0

f(-x)

The graph of f(-x) is a reflection of f(x) in the y axis.

f(-x) = (- x)2 + 4(- x) + 5

= x2 - 4x + 5

Graphs of Related Functions (5)

30

y = f(x)

Reflections in the y axis

20

f(x) = x3 - 9x2 + 18x

10

x

4

-6

2

8

0

6

-4

-2

-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

f(-x) = -x3 - 9x2 - 18x

-30

Graphs of Related Functions (5)

30

y = f(x)

Reflections in the y axis

20

f(-x)

f(x)

10

x

4

-6

2

8

0

6

-4

-2

-10

The graph of f(-x) is a reflection of f(x) in the y axis.

-20

-30

- 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)

Draw the graph of f(-x) for each case on the grids below.

y = f(x)

y = f(x)

1

2

f(x)

x

x

f(x)

y = f(x)

y = f(x)

3

4

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)

y = f(x)

1

2

f(x)

x

x

f(x)

y = f(x)

y = f(x)

3

4

f(x)

f(x)

x

x

y co-ordinates tripled

y co-ordinates doubled

Points located on the x axis remain fixed.

Graphs of Related Functions (6)

30

y = f(x)

Stretches in the y direction

3f(x)

20

2f(x)

f(x)

10

x

4

-6

2

8

0

6

-4

-2

0

-10

The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction.

-20

-30

y co-ordinates halved

y co-ordinates scaled by 1/3

Graphs of Related Functions (6)

30

y = f(x)

1/3f(x)

20

½f(x)

f(x)

10

x

4

-6

2

8

0

6

-4

-2

0

-10

The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction.

-20

-30

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.

20

2f(x)

10

f(x)

x

4

-6

2

8

6

-4

-2

-10

-20

-30

Graphs of Related Functions (6)

y = f(x)

3

Stretches in y

3Sinx

2

2Sinx

Sinx

1

x

-360

90

-90

-180

0

270

180

-270

360

-1

The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction.

-2

-3

Graphs of Related Functions (6)

3

y = f(x)

The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction.

3Cosx

2

2Cosx

½Cosx

1

Cosx

x

360

-360

90

-90

-180

0

270

180

-270

-1

-2

-3

Grid 1: Sketch or trace the graph of 2f(x)

Grid 2: Sketch or trace the graph of 3f(x)

y = f(x)

2

1

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)

4

y = f(x)

3

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)

2

1

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)

4

y = f(x)

3

f(x)

f(x)

x

x

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

½ the x co-ordinate

1/3 the x co-ordinate

-4

-6

Graphs of Related Functions (7)

Stretches in x

f(2x)

f(3x)

f(x)

0

The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction.

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

Graphs of Related Functions (7)

Stretches in x

f(1/3x)

f(x)

f(1/2x)

0

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 2

All x co-ordinates x 3

6

y = f(x)

4

2

x

4

-6

2

8

6

-4

-2

-2

-4

-6

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.

f(1/2x)

f(2x)

f(x)

0

All x co-ordinates x 1/2

All x co-ordinates x 2

Graphs of Related Functions (7)

y = f(x)

Stretches in x

2

f(x) = Sinx

f(x) = Sin2x

1

x

-360

90

-90

360

-180

0

270

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

Graphs of Related Functions (7)

y = f(x)

Stretches in x

2

f(x) = Sinx

f(x) = Sin3x

1

x

-360

90

-90

-180

0

270

180

-270

360

-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

Graphs of Related Functions (7)

y = f(x)

Stretches in x

2

f(x) = Cos ½ x

f(x)=Cos2x

f(x) = Cosx

1

x

-360

90

-90

360

-180

0

270

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

Grid 1: Sketch or trace the graph of f(2x)

Grid 2: Sketch or trace the graph of f(3x)

y = f(x)

2

1

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)

4

y = f(x)

3

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)

2

1

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)

4

y = f(x)

3

f(x)

x

x

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)Sketch2

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]

- (c)Reflection in the y axis1 mark
3.(a)(4, 3) 1 mark

B1 for (4, 3)

(b)(2, 6) 1 mark

B1 for (2, 6)

4.(a)y = f(x – 4)2 marks

B2 cao

(B1 for f(x – 4) or y = f(x + a), a ≠ –4, a ≠ 0)

(b)

2

B2 cao(B1 cosine curve with either correct amplitude or correct period, but not both)

Q5(a)

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)

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)

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) = x2 + 2

f(x) = x2

0

f(x) = x2 - 5

6

y = f(x)

4

2

2

x

4

-6

2

8

6

-4

-2

f(x + 2)

-2

In general f(x + a) gives a translation by the vector

-4

-6

Graphs of Related Functions (2)

Horizontal Translations

f(x)

0

Inside the brackets, “+” means move the curve _____