Chapter 2: Analysis of Graphs of Functions


Chapter 2: Analysis of Graphs of Functions

2.1 Graphs of Basic Functions and Relations; Symmetry

2.2 Vertical and Horizontal Shifts of Graphs

2.3 Stretching, Shrinking, and Reflecting Graphs

2.4 Absolute Value Functions: Graphs, Equations, Inequalities, and Applications

2.5 Piecewise-Defined Functions

2.6 Operations and Composition


2.1 Graphs of Basic Functions and Relations

• Continuity - Informal Definition– A function is continuous over an interval of its domain if its hand-drawn

graph over that interval can be sketched without lifting the pencil from the paper.

• Discontinuity– If a function is not continuous at a point, then it may have a point of

discontinuity, or it may have a vertical asymptote. Asymptotes will be discussed in Chapter 4.


Determine intervals of continuity:

A. B. C.

Solution:

A.

B.

C.

2.1 Examples of Continuity

Figure 2, pg 2-2 Figure 3, pg 2-2

),( ,on Continuous Point of discontinuity at 3, ( ,3) (3, )x Vertical asymptote at 2, ( , 2) ( 2, )x


2.1 Increasing and Decreasing Functions

1. Increasing – The range values increase from left to right – The graph rises from left to right

2. Decreasing – The range values decrease from left to right– The graph falls from left to right

• To decide whether a function is increasing, decreasing, or constant on an interval, ask yourself “What does the graph do as x goes from left to right?”


2.1 Increasing, Decreasing, and Constant Functions

Suppose that a function f is defined over an interval I.a. f increases on I if, whenever b. f decreases on I if, wheneverc. f is constant on I if, for every

Figure 7, pg. 2-4

)()(,2121

xfxfxx )()(,

2121xfxfxx

)()(, and 2121

xfxfxx


2.1 Example of Increasing and Decreasing Functions

• Determine the intervals over which the function is increasing, decreasing, or constant.

Solution: Ask “What is happening to the y-values as x is getting larger?”

)1,(on decreasing

]3,1[on increasing),3[on constant


• is increasing and continuous on its entire domain,

• is continuous on its entire domain, It is increasing on and decreasing on Its graph is called a parabola, and the point where it changes from decreasing to increasing, (0,0), is called the vertex of the graph.

2.1 The Identity and Squaring Functions

).,( xxf )(

2)( xxf ).,( ),,0[ ].0,(


2.1 Symmetry with Respect to the y-Axis

If we were to “fold” the graph of f (x) = x2 along the y-axis, the two halves would coincide exactly. We refer to this property as symmetry.

Symmetry with Respect to the y-Axis

If a function f is defined so that

for all x in its domain, then the graph of f is symmetric with respect to the y-axis.

( ) ( )f x f x

).()()( ,number realany For

.9)3()3( 16)4()4(

then,)( if example,For

22

2

xfxxxfx

ffff

xxf


2.1 The Cubing Function

•

• The point at which the graph changes from “opening downward” to “opening upward” (the point (0,0)) is called an inflection point.

3( ) increases and is continuous on its entire domain,

(- , ).

f x x


2.1 Symmetry with Respect to the Origin

• If we were to “fold” the graph of f (x) = x3 along the x and y-axes, forming a corner at the origin, the two parts would coincide. We say that the graph is symmetric with respect to the origin.

• e.g.

Symmetry with Respect to the Origin

If a function f is defined so that

for all x in its domain, then the graph of f is symmetric with respect to the origin.

)()( xfxf

).()()( ,number realany for or

,1)1()1( 8)2()2(

have we,)(Given

33

3

xfxxxfx

ffff

xxf


2.1 Determine Symmetry Analytically

• Show analytically and support graphically that

has a graph that is symmetric with respect to the origin.

Solution:

xxxf 4)( 3

.any for )()( that Show xxfxf

)(4)()( 3 xxxf xx )1(4)1( 33

xx 43 Figure 13 pg 2-10

)()4( 3

xfxx


2.1 The Square Root and Cube Root Functions

real. be tofor 0 :Note ).,0[ domain, entire itson continuous is and increases )( 2

1

fxxxxf

).,( domain, entire itson continuous is and increases )( 3

13

xxxf


2.1 Absolute Value Function

• decreases on and increases on It is continuous on its entire domain,

Definition of Absolute Value |x|

0 if

0 if )(

xx

xxxxf

xxf )( ]0,( ).,0[ ).,(


2.1 Symmetry with Respect to the x-Axis

• If we “fold” the graph of along the x-axis, the two halves of the parabola coincide. This graph exhibits symmetry with respect to the x-axis. (Note, this relation is not a function. Use the vertical line test on its graph below.)

e.g.

2yx

Symmetry with Respect to the x-Axis

If replacing y with –y in an equation results in the same equation, then the graph is symmetric with respect to the x-axis.

2222

2

)1()(

yxyxyx

yx


2.1 Even and Odd Functions

Example

Decide if the functions are even, odd, or neither.1.

2.

A function f is called an even function if for all x in the domain of f. (Its graph is symmetric with respect to the y-axis.)

A function f is called an odd function if for all x in the domain of f. (Its graph is symmetric with respect to the origin.)

)()( xfxf

)()( xfxf

xxxf 93

6)(

xxxf 52

3)(

odd. isfunction The )(96

)(9)(6)(

3

3

xfxx

xxxf

xx

xxxf

53

)(5)(3)(

2

2

odd.nor even neither is

,)()( and )()( Since

f

xfxfxfxf

Chapter 2: Analysis of Graphs of Functions

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