Transforming FunctionsTopic Index | Algebra2/Trig Index | Regents Exam Prep Center

There are numerous ways to apply transformations to functions to create new functions.
Let's look at some of the possibilities. Remember to utilize your graphing calculator to compare the graphs of your functions and their transformations.

Reflections and Functions: Examining -f (x) and f (-x)

Reflection over the x-axis -f (x) reflects f (x) over the x-axis.
A reflection is a mirror image. Placing the edge of a mirror on the x-axis will form a reflection in the x-axis. This can also be thought of as "folding" over the x-axis. If the original (parent) function is y = f (x), thereflection over the x-axis is function -f (x).

Reflection over the y-axis f (-x) reflects f (x) over the y-axis.
Placing the edge of a mirror on the y-axis will form a reflection in the y-axis. This can also be thought of as "folding" over the y-axis. If the original (parent) function is y = f (x), thereflection over the y-axis is function f (-x).

Translations and Functions: Examining f (x + a) and f (x) + a

Slide to the right or left f (x + a) translates f (x) horizontally

If the original (parent) function is y = f (x), the translation (sliding) of the function horizontally to the left or right is given by the function f (x - a).

if a > 0, the graph translates (slides) to theright.
if a < 0, the graph translates (slides) to theleft.

	Remember that you are "subtracting" the value of a from x. Thus f (x + 2) is
really f (x - (-2)) and the graph moves to the left.

Slide upward or downward f (x)+ a translates f (x) vertically
If the original (parent) function is y = f (x), the translation (sliding) of the function vertically upward or downward is the function f (x) + a. if a > 0, the graph translates (slides) upward. if a < 0, the graph translates (slides)downward. Remember that you are adding the value of a to the y-values of the function.

Stretch or Compress Functions: Examining f (ax) and a f (x)

Horizontal Stretch or Compress f (ax) stretches/compresses f (x) horizontally
A horizontal stretching is the stretching of the graph away from the y-axis. A horizontal compression is the squeezing of the graph towards the y-axis.If the original (parent) function is y = f (x), the horizontal stretching or compressing of the function is the function f (ax). if 0 < a < 1 (a fraction), the graph isstretched horizontally by a factor of a units.
if a > 1, the graph is compressed horizontally by a factor of a units. if a should be negative, the horizontal compression or horizontal stretching of the graph is followed by a reflection of the graph across the y-axis.

Vertical Stretch or Compress a f (x) stretches/compresses f (x) vertically
A vertical stretching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis.If the original (parent) function is y = f (x), the vertical stretching or compressing of the function is the function a f(x). if 0 < a < 1 (a fraction), the graph iscompressed vertically by a factor of a units. if a > 1, the graph is stretched vertically by a factor of a units.
If a should be negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis.

Parent Functions and Transformations

Parent Functions

A parent function is the simplest of the functions in a family.

Parent Function	Form	Notes
constant function	f(x) = c	graph is a horizontal line
identity function	f(x) = x	points on graph have coordinates (a, a)
quadratic function	f(x) = x²	graph is U-shaped
cubic function	f(x) = x³	graph is symmetric about the origin
square root function		graph is in first quadrant
reciprocal function		graph has two branches
absolute value function	f(x) = │x│	graph is V-shaped
greatest integer function	f(x) = [[x]]	defined as the greatest integer less than or equal to x; type of step function

Example Describe the following characteristics of the graph of the parent function f(x) = x³: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

The graph confirms that D = {x│x ∈

} and R = {y│y ∈

}.

The graph intersects the origin, so the x-intercept is 0 and the y-intercept is 0.

It is symmetric about the origin and it is an odd function:
f(-x) = -f(x).

The graph is continuous because it can be traced without lifting the pencil off the paper.

As x decreases, y approaches negative infinity, and as x increases, yapproaches positive infinity.

and

The graph is always increasing, so it is increasing for (-∞, ∞).

Exercise

Describe the following characteristics of the graph of the parent function f(x) = x²: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.

Chapter 1

Glencoe Precalculus

Transformations of Parent Functions

Parent functions can be transformed to create other members in a family of graphs.

Translations	g(x) = f(x) + k is the graph of f(x) translated...	...k units up when k > 0.
		...k units down when k < 0.
	g(x) = f(x − h) is the graph of f(x) translated...	...h units right when h > 0.
		...h units left when h < 0.
Reflections	g(x) = -f(x) is the graph of f(x)...	...reflected in the x-axis.
	g(x) = f(-x) is the graph of f(x)...	...reflected in the y-axis.
Dilations	g(x) = a · f(x) is the graph of f(x)...	...expanded vertically if a > 1.
		...compressed vertically if 0 < a < 1.
	g(x) = f(ax) is the graph of f(x)...	...compressed horizontally if a > 1.
		...expanded horizontally if 0 < a < 1.

Example Identify the parent function f(x) of , and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.