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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 = {xx} and R = {yy}.

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, y approaches 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 27 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(xh) 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.

The graph of g(x) is the graph of the square root function reflected in the y-axis and then translated one unit down.

Exercises

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

  1. g(x) = 0.5 │x + 4│




  2. g(x) = 2x2 − 4



Chapter 1 28 Glencoe Precalculus