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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 2.5 Piecewise-Defined Functions 2.6 Operations and Composition.
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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 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 • 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.
