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.