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 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.

2.1 Graphs of Basic Functions and Relations 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 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. f increases on I if, whenever f decreases 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 The Square Root and Cube Root Functions

2.1 Absolute Value Function decreases on and increases on It is continuous on its entire domain, Definition of Absolute Value |x|

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