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

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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

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

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

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

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

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

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

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

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

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

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 the square root and cube root functions

2.1 The Square Root and Cube Root Functions


2 1 absolute value function

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

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

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.


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