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Transformation of Functions

Given y = f(x), we will investigate the function

y = af [k(x – p)] + q

for different values of a, k, p and q.

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1) Investigating y = f(x – p)

y x

4y x

3y x

(4, 2)

(0, 2)

(7, 2)

Horizontal shift (–4)

Horizontal shift (+3)

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2) Investigating y = f(x) + q

y x

3y x

5y x

(4, 2)

(4, 5)

(4, –3)

Vertical shift (+ 3)

Vertical shift (–3)

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3) Investigating y = af(x)

y x

2y x

1

2y x

(4, 2)

(4, 4)

(4, 1)

(vertical stretch by a factor of 2 or (x2))

vertical compression by a factor of ½ or x 12

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3) Investigating y = af(x)

y x

y x

2y x

(4, 2)

(4, –2)

(4, – 4)

(Reflection in the y-axis)

Reflection in the y-axis and a vertical stretch by a factor of 2

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4) Investigating y = af(x – p) + q

y x

2 5 3y x

(4, 2) (–1, 1)

Horizontal shift (-5)

vertical stretch by a factor of 2

vertical shift (–3)

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5) Investigating y = f(kx)

y x (4, 2)

2y x (2, 2)

horizontal compression by a factor of : 1

k

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y = af [k(x – p)] + q

Horizontal transformations – apply opposite operations!!

Add p units to the x-coordinate

Multiply the x-coordinate by 1

ky = af [k(x – p)] + q

vertical transformations

- Vertical stretch by a factor of a (if a > 1)

- Vertical shift of q units

- Vertical compression by a factor of a (if 0< a < 1)

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If the graph of f is given as y = x3, describe the transformations that you would apply to obtain the following:

a) y = (x + 1)3

b) y = x3 – 2

c) y = – 2x3

d) y = (– 3x)3

e) y = 2(x – 5)3

Horizontal shift (-1)

Vertical shift (-2)

vertical stretch ×2 and reflection in the x-axis

horizontal compression by a factor of 1/3 and a reflection in the y-axis,

right 5 and a vertical stretch by a factor of 2

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Example: Given the ordered pair

(3, 5) belongs to g.

List the ordered pairs that correspond to:

a) y = 2g(x)

b) y = g(x) + 4

c) y = g(x – 3)

d) y = – g(2x)

(3, 10)

(3, 9)

(6, 5)

(1.5, – 5)

Vertical stretch × 2

Vertical shift (+ 4)

Horizontal shift (+ 3)

Horizontal compression by a factor of ½ and a reflection in the x-axis

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If the graph of f is given as , describe the transformations that you would apply to obtain the following:

Horizontal shift (+ 3)

Vertical shift (+4)

horizontal compression (× 0.5))

1y

x

1a)

3y

x

1

b) 4yx

1c)

2y

x

1d) y

x reflection in the x-axis

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Describe the transformations that you would apply to f(x) to obtain the following:

a) f(2x + 6)

f [2(x + 3)]

b) f(– 3x + 12) + 5

f[–3(x – 4)] + 5

Horizontal shift (- 3)Horizontal compression by a factor of 0.5

Horizontal compression by a factor of 1/3.

Horizontal shift (- 3)

Reflection in y-axis

Vertical shift (+5)

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Describe the transformation on f (x):f(x)

x3 2x3

x 2x x4 (x – 3)4

( )f x (2 )f x

x2 (x + 1)2 + 4

( )f x 0.5 (0.5 )f x

Vertical stretch (x 2) Horizonal shift (-2), reflection in the x-axis

Horizonal shift (+3)

Horizontal compression (x 0.5) reflection in the x-axis Horizontal shift (-1)

Vertical shift (+4)

horizontal stretch (x2), vertical compression (x0.5).

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The graph of y = f(x) is given

Sketch the graph of

y = –f (2x+6) + 1

y = –f [2(x+3)] + 1

(x,y) (0.5x – 3,– y+1)

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y = –f [2(x+ 3)] + 1

× 2 × –1fadd 3x +1 y

left 3 ÷ 2 reflect in x-axis

up 1

P(–3, –2)

P

Q R

S

(–6, –2)

(–3, –2)

(–3, 2)

P´(–3, 3)

left 3

÷ 2

reflect in x-axis

up 1

P´

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y = –f [2(x+ 3)] + 1

× 2 × –1fadd 3x +1 y

left 3 ÷ 2 reflect in x-axis

up 1

Q(0, 2)

P

Q R

S

(–3, 2)

(–1.5, 2)

(–1.5, –2)

Q´(–1.5, –1)

left 3

÷ 2

reflect in x-axis

up 1

P´

Q´

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y = –f [2(x+ 3)] + 1

× 2 × –1fadd 3x +1 y

left 3 ÷ 2 reflect in x-axis

up 1

R(3, 2)

P

Q R

S

(0, 2)

(0, 2)

(0, –2)

R´(0, –1)

left 3

÷ 2

reflect in x-axis

up 1

P´

Q´ R´

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y = –f [2(x+ 3)] + 1

× 2 × –1fadd 3x +1 y

left 3 ÷ 2 reflect in x-axis

up 1

S(4, 0)

P

Q R

S

(1, 0)

(0.5, 0)

(0.5, 0)

S´(0.5, 1)

left 3

÷ 2

reflect in x-axis

up 1

P´

Q´ R´

S´