# A Library of Parent Functions

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```					A Library of Parent
Functions
Section 2.4
College Algebra, MATH 171
Mr. Keltner
This Section
• The goal of this section is to make you able
to recognize the basic shapes of the graphs
of different types of functions.
 For instance, you know the graph of a linear
function in the form f(x) = ax + b is a line with
slope m = a and a y-intercept at (0, b).
 There are additional general shapes of graphs
that we will investigate further.
Linear Functions
• The graph of a linear function has the following
characteristics:
 The domain is the set of all real numbers, of (-∞, ∞).
 The range of the function is the set of all real numbers,
or (-∞, ∞).
 The graph has an x-intercept at (-b/m, 0) and a y-
intercept at (0, b).
 The graph is increasing if m>0 (positive) and
decreasing if m<0 (negative).
Special Linear Functions
• There are two special types of linear
functions:
 Constant functions, f(x) = c, which is a
horizontal line
 Identity function, f(x) = x, which is a 45°
angle, where the x- and y-coordinates are equal)
• Example 1: Write the linear function f for
which f(0) = -1 and f(5) = 8.
Squaring Functions
• A squaring function is one of the form f(x) = x2.
 The domain is the set of all real numbers of (-∞, ∞).
 The range is the set of all nonnegative real numbers [0,
∞).
 Is an even function, where f(-x) = f(x).
 Has an intercept at (0, 0)
 Decreasing for (-∞, 0), increasing for (0, ∞).
 Symmetric to the y-axis (characteristic of an even
function)
 Relative minimum at (0, 0)
Cubic Functions
• A cubic function is of the form f(x) = x3 and has
the following characteristics:
 The domain is the set of all real numbers, of (-∞, ∞).
 The range of the function is the set of all real numbers,
or (-∞, ∞).
 Is an odd function, where f(-x) = -f(x).
 Has an intercept at (0, 0)
 Increasing for (-∞, ∞).
 Symmetric with respect to the origin (characteristic of
an even function)
Square Root Functions
• A square root function is of the form
f(x) = √x.
 The domain of the function is the set of all
nonnegative real numbers, [0, ∞).
 The range of the function is the set of all
nonnegative real numbers, [0, ∞).
 The graph has an intercept at (0, 0).
 The graph is increasing on the interval (0,
∞).
Assessment
Pgs. 220-221:
#’s 5-60, multiples of 5

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