Chapter 2: Analysis of Graphs of Functions

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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

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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.
2.1 Examples of Continuity

Determine intervals of continuity:

A. B. C.

Solution:

A.

B.

C.

Figure 2, pg 2-2

Figure 3, pg 2-2

2.1 Increasing and Decreasing Functions
• Increasing
• The range values increase from left to right
• The graph rises from left to right
• 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.
• fincreases on I if, whenever
• fdecreases on I if, whenever
• f is constant on I if, for every

Figure 7, pg. 2-4

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?”

2.1 The Identity and Squaring Functions
• 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 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.

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.
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.

2.1 Determine Symmetry Analytically
• Show analytically and support graphically that

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

Solution:

Figure 13 pg 2-10

2.1 Absolute Value Function

Definition of Absolute Value |x|

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

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.

2.1 Even and Odd Functions

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.)

Example

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