Shifting of Graphs Transformation

Example 1

y = f( x ) + k Up k units

y = f ( x ) - k Down k units

Vertical Shifting

Below is the graph of a function y = f ( x ). Sketch the graphs of

a) y = f ( x ) + 1

b) y = f ( x ) - 2 y = f(x) + 1

y = f (x)

y = f (x)-2

Horizontal Shifting

y = f( x + h ) Left h units

y = f ( x - h ) Right h units

Continued…

Example 2.

Given the graph of a function

y = f ( x ). Sketch the graphs of

a) y = f ( x + 3 )

b) y = f ( x – 4 )

y = f ( x )

y = f ( x )

y = f ( x +

3)

y =

f ( x

- 4

)

Horizontal Shift 3 units to the left

Horizontal Shift 4 units to the right

Continued…Can you tell the effects on the graph of y = f ( x )

y = f( x + h ) + k

y = f( x + h ) - k

y = f( x - h ) + k

y = f( x - h ) - k

Example 3Below is the graph of a function y = f ( x ). Sketch the graph of y = f ( x + 2 ) - 1

y = f( x )

y = f( x + 2 ) - 1

Left h units and Up k units

Left h units and Down k units

Right h units and Up k units

Right h units and Down k units

Continued… Example 4

Below is the graph of a function . Sketch the graph of xxfy

3232 xxfy

y = f( x )

y =f(x-2)-3

The graph of the absolute value is shifted 2 units to the right and 3 units down

Solution:

Example 5

If the point P is on the graph of a function f. Find the corresponding point on the graph of the given function.

1) P ( 0, 5 ) y = f( x + 2 ) – 1

2) P ( 3, -1 ) y = 2f(x) +4

3) P( -2,4) y = (1/2) f( x-3) + 3

Solution: 1) P ( 0,5). y = f( x + 2 ) – 1 shifts x two units to the left and shifts y one unit down. The new x =0 – 2 = -2, and the new y = 5 – 1 = 4. The corresponding point is ( -2, 4 ).

2) P(3,-1). y = 2f(x) +4 has no effect on x. But it doubles the value of y and shifts it 4 units vertically up. Therefore the new x = 3(same as before ), and the new value of y = 2 (-1 ) + 4 = 2. Therefore, the corresponding point is ( 3,2 ).

3) P(-2, 4 ). y = (1/2) f( x-3) + 3 shifts x 3 units to the right and splits the value of y in half and then shifts it 3 units up. That is, the new value of

y = (1/2)(4) + 3 = 5. Therefore, the corresponding point is ( 1, 5 ).

Reflecting a graph through the x-axis

y = -f( x) Reflection through the x-axis

(x-axis acts as a plane mirror)

Example 11

Note: For any point P(x,y) on the graph of y = f(x), The graph of y = - f(x) does not effect the value of x, but changes the value of y into - y

Below is the graph of a function y = x2 . Sketch the graph of

1. y = - x2

x y = x2 y = -x2

2 4 -4

1 1 -1

0 0 0

-1 1 -1

-2 4 -4

Example 6

Vertical Stretching y = cf( x) ( c> 1 ) Vertical Stretch by a

factor c

y = (1/c)f ( x) ( c > 1 ) Vertical Compress by a factor 1/c

Note1 :When c > 1. Then 0 < 1/c < 1

Note 2 : c effects the value of y only.

Example 7Below is the graph of a function y = x2 . Sketch the graphs of

1. y = 5 x2

2. y = (1/5)x2

x y = x2 y=5x2 y=1/5x2

2 10 .4

1 1 5 .2

0 0 0 0

-1 1 5 .2

2 10 .42

2

Sketching a piece-wise function

Example 8

Definition: Piece-wise function is a function that can be described in more than one expression.

Sketch the graph of the function f if

12

1

152

)( 2

xif

xifx

xifx

xf

Solution:

Graph y = 2x + 5 and take only the portion to the left of the line x = -1. The point (-1, 3 ) is included.

1xIfGraph y = x2 and take only the portion where –1 < x < 1. Note: the points ( -1,1) and ( 1, 1 ) are not included

1xIfGraph y = 2 and take only the portion to the right of x = 1. Note: y = 2 represents a horizontal line. The point (1, 2 ) is included.

1If x

Sketching the graph of an equation containing an absolute value

Example 9

Sketch the graph of y = g ( x ) = 92 x

Note: To sketch an absolute value function . xfy

We have to remember that .0yAnd hence, the graph is always above the x-axis. The part of the graph that is below the x-axis will be reflected above the x-axis.

Strategy:

1. Graph y = f(x) = x2.

Solution:

2. Graph y = f( x ) - 9 = x2 – 9 by shifting the graph of f 9 units down

3. Graph g(x) by keeping the portion of the graph y = f( x ) - 9 = x2 – 9 which is above the x-axis the same, and reflecting the portion where y < 0 with respect to the x-axis.

92 xy

4. Delete the unwanted portion

Example 10

Below is the graph of y = f(x). Graph xfy

Let the animation talk about itself

Solution:

A picture can replace 1000 words

The End