Download CHAPTER 2: Functions and Their Graphs

Lauren Callahan and Cassie McClenaghan
Section 2.1: Functions
Relation- when the value of one variable is related to the value of a
second variable
 Function- the common link between 2 relations is each input
corresponds to exactly one output
•The set X is the domain. The
INPUT
OUTPUT corresponding Y is the range.
X
1
•The relation shown is a
function because the input
Y
2
corresponds to only one
output.
Z
3
•It is not a function if at least
one input has more than one
To find the value of a function ‘f’ defined by output.
f(x)= 2x2- 3x we take f(3) and solve the
•f(x) is another term for ‘y’
function by putting 3 where x is. For
example,
F(3)=2(3)2- 3(3)= 18-9=9

Section 2.2: Graph of a Function

Vertical Line Test- A set of points in the xy-plane is the graph of a
function, if and only if, every vertical line intersects the graph in at
most one point

A graph of a function is the collection of points (x,y) that satisfies
the equation y=f(x) .
Section 2.3: Properties of
Functions
EVEN AND ODD FUNCTIONS
A function is even, if and only if, whenever the point (x,y) is on a graph where
(-x,y) is also on the graph. f(-x)=x

A function is odd, if and only if, whenever the point (x,y) is on a graph where
(-x,-y) is also on the graph. f(-x)=-f(x)

To determine whether ‘f’ is even, odd, or neither, we replace x by –x in the
equation

INCREASING, DECREASING, AND CONSTANT GRAPHS



A function is increasing if for any x1 and x2, f(x1)<f(x2)
A function is decreasing if for any x1 and x2, f(x1)>f(x2)
A function is constant if for all choices of x, the values of f(x) are equal
LOCAL MAXIMA AND MINIMA

A function has a maxima(greatest value) and a minima(smallest value) in a set of
numbers.
AVERAGE RATE OF CHANGE

The average rate of change of from a to be is defined as: f(b)-f(a)/b-a
Section 2.4: Library of Functions;
Piecewise Defined Functions

1. Constant Function- f(x)=b

2. Identity Function- f(x)=x

3. Square Function- f(x)=x2

4. Cube Function- f(x)=x3

5. Square Root Function- f(x)=√x

6. Reciprocal Function- f(x)=1/x

7. Absolute Function- f(x)= |x|

8. Cube Root Function- f(x)= 3 √x

1.
3.
2.
8.
5.
4.
7.
Piecewise Function- when functions are defined by more than one equation
6.
Section 2.5: Graphing
Techniques-Transformations
•Vertical Shift Up- if a positive number k is added to the right side of the equation
y=f(x), the new equation would be y=f(x)+k
•Vertical Shift Down- if a positive number k is subtracted to the right side of the
equation, the new equation would be y=f(x)-k
•Horizontal Shift Right- if in the equation y=f(x), x is replaced with (x-h), the new
equation would be y=f(x-h)
•Horizontal Shift Left- if in the equation y=f(x), x is replaced with (x+h), the new
equation would be y=f(x+h)
•Vertical Stretch- if the right side of a function y=f(x) is multiplied by positive
number a, the graph of the new function is y=af(x). (multiply each y-coordinate by
a)
•Horizontal Stretch- if the function y=f(x) is multiplied by a positive real number
a, the graph of the new function is y=f(ax). (multiply each x-coordinate by a)
•X-axis Reflection- if the right side of the function y=f(x) is multiplied by -1, the
graph of the new function is y=-f(x)
•Y-axis Reflection- if in the function y=f(x) the x is replaced by –x, then the new
equation would be y=f(-x)
Section 2.6: Mathematical
Models-Building Functions
Using the concepts from 2.1-2.5 you can
build and analyze functions.
 Remember the distance formula
