1.3 Combining Transformations

1.3 COMBINING TRANSFORMATIONS

When is the order of transformations important?

( ( ))y k af b x h ( ( ))y af b x h k

( )y k af bx c ( )y af bx c k

Multiple transformations can be applied to a function using the general model:

( ( ))y k af b x h Perform in any order: • Reflections and Stretches• One Horizontal and one Vertical transformation• A Horizontal translation and vertical stretch or reflection• A Vertical Translation and Horizontal stretch or reflection

Specific Order in the function equation: • A Horizontal stretch or reflection with

a horizontal translation. • A Vertical stretch or reflection with a

horizontal translation.

Factor

ST

R

For a function equation:

When performing transformations from a graph of a function, the transformations can be done in any order, however, you may not get the same graph

Order is Important Any Order

( )y af x h

( )y af x k

( )y f b x h

( )y k af bx

2 (3 )y f x

2 (4 1) 5y f x

RST

F

Combinations of Transformations

y = | x |y = | x |

y = - | x |

Graph y = -| x - 3 | + 2 using transformations of the parent graph.

y = | x |

y = - | x |

y = - | x - 3| + 2RST

F

A point (a, b) lies on the graph of y = f(x). What are the coordinates of the image point after the transformations:

Combinations of Transformations

12 4 53

y f x

,a b 3 4,2 5a b

The graph of y = f(x) is reflected in the x-axis, stretched vertically about the x-axis by a factor of ⅓, and stretched horizontally about the y-axis by a factor of 4 to create the graph of y = g(x).

For the point (-3, 6) on the graph of y = f(x), the corresponding point on the graph of y = g(x) is

A. (9, 24) B. (-12, -18) C. (1, 24) D. (-12, -2)

Describe the combination of transformations that must be applied to the function y = f(x) to obtain the transformed function. Sketch the graph.

) 2 ( ) 3a y f x

1) 22

b y f x

Vertical stretch about the x-axis by a factor of 2 then a vertical translation 3 units down.

1 42

y f x

Horizontal stretch about the y-axis by a factor of 2 then a horizontal translation 4 units right.

Graphing Combinations of Transformations

y 3 (x 4) 3

Rewrite this function as

-3 (-( 4)) 3.y f x

Given f (x) x , list the domain and range of y 3 3 f ( x 4)

: ( , 4]Do

: ( ,3]Ra

Three transformations have been applied to the graph of f(x) to become the graph of g(x).The transformations include:• a horizontal stretch by a factor of 2• a horizontal reflection in the y-axis• a horizontal translation 2 units left

Where the transformations done in the order FRST?(function equation)

When given a graph and its image graph, the order do not necessarily follow this order!!

In what order where the transformations performed ?

The graph of the function y = g(x) represents a transformation of the graph of y = f(x). Determine the equation of g(x) in the form y = af(b(x - h)) + k. Locate key points on the graph of f(x) and their image points on the graph of g(x).

Compare the distances between key points. In the vertical direction, 4 units becomes 8 units.

(0, 0) is unaffected by any stretch so it can be used to determine the translations.(0, 0) → (-7, 2) h = -7 and k = 2

In the horizontal direction, 8 units becomes 2 units.

There is a vertical stretch by a factor of 2

There is a horizontal stretch by a factor of ¼.

g(x) = 2f(4(x + 7)) + 2.

Write the Equation of a Transformed Function Graph

A point on the graph of y = f(x) is (a, b). The point (2, 3) is on the graph of the transformed function y = 4f(-4x - 12) + 3. What are the original coordinates (a, b)?

ASSIGNMENT:Page 38:1b, 2, 3, 4, 5a, 6a,b,d, 7f, 8a, 9, 10, 12, 13

Describe a sequence of transformations required to transform the graph of to the graph of .y x 1 8 10

2y x