Transcript
2.6 FAMILIES OF FUNCTIONS
FAMILIES OF FUNCTIONS
There are sets of functions, called families that share certain characteristics. A parent function is the simplest form in a set
of functions that form a family. Each function in the family is a transformation
of the parent function
FAMILIES OF FUNCTIONS
TRANSLATIONS
One type of transformation is a translation. A translation shifts the graph of the parent
function horizontally, vertically, or both without changing shape or orientation.
EXAMPLEHow can you represent each translation of y = |x| graphically?
1.
2.
2g x x
1h x x
Shift the parent graph
down 2 units
Shift the parent graph
left 1 unit
EXAMPLE
How can you represent each translation of y = |x| graphically?
1.
2.
3 1g x x
2 3h x x
Shift the parent graph right 3 units and up 1 units
Shift the parent graph left 2 units and down 3 units
REFLECTIONS
A reflection flips the graph over a line (such as the x – or y – axis) Each point on the graph of the reflected function
is the same distance from the line of reflection as its corresponding point on the graph of the original function.
REFLECTIONSWhen you reflect a graph in the y-axis, the x values change signs and the y-values stay the same.
When you reflect a graph in the x-axis, the y-values change signs and the x-values stay the same.
REFLECTING A FUNCTION ALGEBRAICALLY
Let and be the reflection in the x-axis. What is a function rule for
?
3 3f x x g x g x
g x f x
REFLECTING A FUNCTION ALGEBRAICALLY
Let and be the reflection in the y-axis. What is a function rule for ?
3 3f x x g x g x
g x f x
2.6 CONTINUED
VERTICAL STRETCH AND VERTICAL COMPRESSION A vertical stretch multiplies all y-values of
a function by the same factor greater than 1. A vertical compression reduces all
y-values of a function by the same factor between 0 and 1.
Why do you think the value being multiplied is always positive?
EXAMPLEThe table represents the function f(x). Complete the table to find the vertical stretch and vertical compression. Then graph the functions.
EXAMPLE: COMBINING TRANSFORMATIONS
The graph of g(x) is the graph of f(x) = 4x compressed vertically by the factor ½ and then reflected in the y-axis. What is the function rule for g(x)?
EXAMPLE: COMBINING TRANSFORMATIONS
The graph of g(x) is the graph of f(x) = x stretched vertically by the factor 2 and then translated down 3 units. What is the function rule for g(x)?
EXAMPLE: COMBINING TRANSFORMATIONS
What transformations change the graph of f(x) to the graph of g(x)?
2 22 6 1f x x g x x
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