transformations transforming graphs. 7/9/2013 transformations of graphs 2 basic transformations...

Transformations

Transforming Graphs

Transformations of Graphs 27/9/2013

Basic Transformations

Restructuring Graphs Vertical Translation

f(x) to f(x) + k

Horizontal Translation f(x) to f(x – h)

y

x

• •

•

x′

y′

x′ + h

y = f(x)

y = f(x) + k

y = f(x – h)y′ + k

Note: When x = x′ + h ,

y = f(x – h)= y′

= f(x′ + h – h)= f(x′)

Vertical move k units


Basic Transformations Horizontal Translation

f(x) to f(x – h) Example

y

xx′

y′ • ••

• •

y = 2x

y = 2(x – 1)

x′ + 1

y = f(x) = 2xWhen x = x′ , y = 2x′

When x = x′ + 1,y = 2(x – 1) = 2((x′ + 1) – 1)

= 2x′

y = f(x – 1) = 2(x – 1)

= y′ = f(x′)



y

x

• •

x

y

x′

y = f(x)y = f(x′)

y

x′

y = f(x′)

x′ = 0 when x = h So, x – h = 0 = x′

f(x′) = f(x – h)

h

x′

•

Horizontal Translation

Assume h > 0y = f(x – h)

Replacing x with

moves the graph h units to the right

x – h



Restructuring Graphs Vertical Stretching

f(x) to af(x) , a > 1 stretch by factor of a

Note that when y = f(x) = 0 then y = af(x) = 0

y

x

y = f(x)

y = af(x)

•Stretch is away from x-axis

Note: •



Restructuring Graphs Vertical Shrinking

f(x) to af(x) , 0 < a < 1 shrink by factor of a

Note that when y = f(x) = 0 then y = af(x) = 0

y

x

y = f(x)

y = af(x)

••Stretch is toward x-axis

Note:



Restructuring Graphs Horizontal Stretching

f(x) to f(cx) , 0 < c < 1 stretch by factor of c

Note that when y

x

y = f(x)y = f(cx)

•Stretch is awayfrom y-axis

Note:

x = 0 then cx = 0so

y = f(x) = f(cx)



Restructuring Graphs Horizontal Shrinking

f(x) to f(cx) , c > 1 shrink by factor of c Note that when y

x

y = f(x)y = f(cx)

•Shrink is toward y-axis

Note:

soy = f(x) = f(cx)

x = 0 then cx = 0



Restructuring Graphs: Reflections Vertical reflection

through horizontal axis f(x) to -f(x)

y

x

y = f(x)

y = -f(x)

•

•

Each point on the graph of f(x) projects to acorresponding point on the graph of -f(x) reflected through the x-axis



Restructuring Graphs: Reflections Horizontal reflection

through vertical axis f(x) to f(-x)

y

x

y = f(x)y = f(-x)

••Each point on the graph of f(x) projects to a corresponding point on the graph of f(-x) … reflected through the y-axis Note: When x = 0 ,

y = f(x) = f(-x)


Translations

Vertical Translation y = f(x) Alters the value of y

Horizontal Translation y = f(x) Alters the value of x

x

yy = f(x) + k

k

y = f(x – h)

h


Examples

Examples y = -x2

y = -x2

x

yy = -x2 + 4

y = -(x – 3)2

y = -x2 + 4

y = -(x – 3)2

Moves graph up 4 units

Moves graph right 3 units


Examples

Vertical Stretching : y = f(x) Example:

Vertical Shrinking : y = f(x) Example:

x

y

y = af(x)

x

y

y = af(x)

y = 2(x – 4)2 – 6

a > 1

0 < a < 1

y = (x – 4)2 – 3

= 2y

y = (x – 4)2 – 3

y 12

32

–(x – 4)2 = 12= y

• •

••

f(x) = 0 = af(x)

f(x) = 0 = af(x)


Consider y = f(x) = 2x + 1 y = f(x)

Horizontal Shrinking & Stretching

x

yg(x) = f(2x)

●

●●

●

●

f(x)

= 2

x +

1

f(2x

) = 4

x +

1

●

●●

f(x)f(2x)f(x/2)

x 0 1 2 1 3 51 5 9? ? ?

●

= 4x + 1

What changes ?

The graph shrinks toward the y-axis




x

yh(x) = f((½)x)

●

●● ●

f(x)

= 2

x +

1

f(2x

) = 4

x +

1

f((½

)x) =

x +

1

●

f(x)f(2x)f(x/2)

x 0 1 2 1 3 51 5 91 2 3

●●●

= x + 1

What changes ?

The graph stretches away from the y-axis




x

y

f(x)

= 2

x +

1

f(2x

) = 4

x +

1

f((½

)x) =

x +

1

f(x)f(2x)f(x/2)

x 0 4 8 16 1 17 331 33 651 9 17

For constant c, c ≠ 0, point (x, y) on the graph of y = f(x) corresponds to on the graph of f(cx)

xc( , y)

••

•

•

••

84

Let c = 2

xxc

16

(8,17)

•17

xc

9173

½


How does it work ? y = f (x)

Horizontal Shrinking

x

y

y = g(x) = f(2x)

b

(b, f(b))●●●●

y = f(x)

x1

●

Note: g(x/2) = f(2(x/2))

For some x1 want g(x1) = f(b)

g(x1) = f(2x1) = f(b)

2x1 = b and x1 = b/2

g(b/2) = f(2(b/2)) = f(b)

Let

= f(x)f(b)g(b/2)

* *Note:We have assumed f is 1-1 on interval [x1, b]



Horizontal Shrinking - 2

x

y

y = g(x) = f(2x)

b

(b, f(b))●●●

y = f(x)

g(b/2) ●

Note: g(x/2) = f(2(x/2))

For some x2 want g(x2) = f(c)

g(x2) = f(2x2) = f(c)

2x2 = c and x2 = c/2

g(c/2) = f(2(c/2)) = f(c)

Let

= f(x)

●c

●● (c, f(c))

●x2

●

*

f(c)g(c/2)

*Note:We have assumed f is

1-1 on interval [x2, c]




x

y

y = g(x) = f(2x)

b

(b, f(b))●●●

y = f(x)

g(b/2) ●

Note: g(x/2) = f(2(x/2))

For some x3 want g(x3) = f(d)

g(x3) = f(2x3) = f(d)

2x3 = d and x3 = d/2

g(d/2) = f(2(d/2)) = f(d)

Let

= f(x)

●c

●● (c, f(c))●

*

f(c)g(c/2)

●

●●

d

(d, f(d))f(d)

x3 ●

●g(d/2)

*Note:We have assumed f is 1-1 on interval [b, d]




x

y

y = g(x) = f(2x)

b

(b, f(b))●

●

y = f(x)

g(b/2) ●

Note:g(x/2) = f(2(x/2))

Replacing x by 2x moves points (x,y) on the graph of f(x) to points (x/2, y) on the graph of f(2x)

Let

= f(x)●c

●● (c, f(c))●f(c)g(c/2)

●

●●

d

(d, f(d))f(d) ●g(d/2)

y = g(x) = f(2x)

●

This shrinks the graph toward the y-axis


Horizontal Shrinking y = f1(x)

Note:

Example

Examples

x

y

y = f2(cx)

y = | 2x |

, c > 1

x

(x, y)●●●

●●

= f1(x)

y

f2(c( )) = f1(x) = ycx

for x1 = cx

cxx1 =

f1(x) f2(cx)

f2(cx1) = f2(c( ))cx

y = | x |


x

y Horizontal Stretching

y = f1(x)

Examples

y = f2(cx)for 0 < c < 1

y = x 12

= f1(x)

for x1 = cx

f2(cx1) = f2(c( ))cx

Note: x

(x, y)●●●

●●

y

cxx1 =

f1(x)f2(cx) f2(c( )) cxf1(x) =

Example

y = | x |


Examples Vertical Reflection

y = f(x)

Example

y = |x|

Horizontal Reflection y = f(x)

Example

y = x + 1

x

y

y = –f(x)

y = f(-x)

y = –|x|

y = -x + 1

y

x

f(x)

–f(x)

f(x)

f(-x)


Exercise What Transformations Are These ?

x

yf1(x) – 2b = (ax + b) – 2b

f1(x) = ax + b

●(0, b)

●(-b/a, 0)

f2(x) = ax – b

●(0, -b)

●(b/a, 0)

f1(x) f2(x)?

= ax – b = f2(x)

f1(x – 2b/a) = a(x – 2b/a) + b = ax – 2b + b

= f2(x)

OR

Vertical Translation

Horizontal Translation

●



x

y

f1(x) = ax + b

●

(0, b)

●(-b/a, 0) ●(b/a, 0)

f2(x) = -ax + b

f1(x) f2(x)?

f1(-x) = a(-x) + b = -ax + b

= f2(x) Horizontal

Reflection



x

y

x

y

f1(x) = ax + b

● (0, b)●

(-b/a, 0)

● (0, -b)

f2(x) = -ax – b

f1(x) f2(x)?

-f1(x) = -(ax + b) = -ax – b

= f2(x) Vertical Reflection

f1(x) = ax + b

●(0, b)●

(-b/a, 0)●

f2(x) = 2ax + b

(-b/(2a), 0)

f1(x) f2(x)?

f1(2x) = a(2x) + b = 2ax + b

= f2(x) Horizontal Shrink



x

y

x

y

f1(x) = ax + b

●(0, b)

●(-b/a, 0)

●(0, -2b/a)

f1(x) f2(x)?

= f2(x) Horizontal Stretch

f2(x) = ax + b 12 f1(x) = ax + b

●●

(0, b)(-b/a, 0)

f2(x) = 2ax + 2b

●(0, 2b)

f1(x) f2(x)?

= f2(x) Vertical Stretch

2f1(x) = 2(ax + b) = 2ax + 2b ax + b 1

2=

a( x) + b

12f1( x) 1

2 =


Exercise What Transformation Is This ?

x

y

f1(x) = b●

(0, b)

f1(x) f2(x)?

= f2(x)

Horizontal Stretchf2( x) = b 1

2

... or Shrink

... or Reflection

f1(2x) = f2(x) = b

f1(-x) = f2(x) = b f1( x) = b

12


Exercise What Transformation Is This ?

x

y

f1(x) = b

●

(0, b)

f2(x) = -2b (0, -2b)

f1(x) f2(x)?

= f2(x)

Vertical Stretch and Reflection

-2f1(x) = -2b

●

... or Vertical Translation

f1(x) – 3b = b – 3b = –2b

= f2(x)


Think about it !

transformations transforming graphs. 7/9/2013 transformations of graphs 2 basic transformations...

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