Ch03+The+Nature+of+Graphs

Chapter Three - The Nature of Graphs
FLASH: Set of chapter review problems (and their answers) available at the bottom of this page. -JRH Oct30, 2013. Chapter objectives, topics, and assignments: Chapter Three highlights in Powerpoint: And here they are as a pdf file:

// 3-1A Graphing Calculators: Graphing Polynomial Functions //

 * [|Here] is a good tutorial on the basics of graphing functions on TI calculators (TI-83/84, TI-86, TI-89).

3-1 Symmetry

 * [|Here] is a 12-minute video that introduces the concepts of even and odd functions.
 * [|And here] is a 4-minute continuation that discusses the connections between even and odd numbers and functions.

3-2 Families of Graphs
Here is a worksheet for outlining the characteristics of a function:.

Here are eight basic "parent" functions you should understand thoroughly:

1. Constant function: f(x) = c 2. Linear function: f(x) = x 3. Quadratic function: f(x) = x 2 4. Cubic function: f(x) =x^3

5. Absolute value: f(x) = |x| 6. Greatest integer: f(x) = [x] 7. Square root: f(x) = (x)^0.5 8. Inverse: f(x) = 1/x

You will find it very helpful to characterize fully each of these eight basic using the "function summary" worksheet above. Here is a worksheet to help students review the concept of a piecewise function:.

Here are the Section 3.2 homework problems and their solutions:.

3-3 Inverse Functions and Relations

 * Here are three ways of creating an inverse from some function or relation:
 * Ordered-pair approach: For every point (//a,b//) in the original relation, plot the reversed pair (//b,a//) as part of the inverse relation.
 * Geometric approach: Sketch the original relation, then reflect that graph over the line //y = x//.
 * Algebraic approach: Rewrite the relation by swapping the letters for the independent (//x//) and dependent (//y//) variables, then solve the new equation for //y//. The resulting relation will define the inverse.

Here are the Section 3.3 homework problems and their solutions:.

We'll go back to 3-5, etc., after the quiz on Sections 3-4 and 3-8.)
>> > > (1) Point discontinuities - these can occur when a function has a "hole" in an otherwise continuous curve. > Example: f(x) = (x^2 - 1) / (x+1) – this rational function is undefined at x = –1, but is well behaved elsewhere. > (2) Jump discontinuities – these can occur with piecewise functions. > Example: f(x) = 2x where x < 5, and x+1 where x >= 5 : the function changes suddenly (jumps) at x = 5. > (3) Infinite discontinuities – these occur when a function has one or more vertical asymptotes where the function is undefined. > Example: f(x) = 1 / x^2 : an infinite discontinuity occurs at x = 0, where the function is undefined and has a vertical asymptote.
 * ==== ** Continuity and Discontinuity ** ====
 * For a function to be continuous at a point //x = c//, the function must satisfy three conditions:
 * The function must be defined for //x = c//.
 * As //x// approaches //c// from the left or from the right, //f(x)// must approach the same value.
 * As //x// approaches //c// from the left or from the right, //f(x)// must approach //f(c)//.
 * For a function to be continuous over an interval, the function must be continuous at every value of //c// within the interval.
 * Discontinuity Types – There are three "flavors" of discontinuities:

> A function //f//(//x//) is increasing if and only if //f//(//x// 1 ) < //f//(//x// 2 ) whenever //x// 1 < //x// 2. > A function //f//(//x//) is decreasing if and only if //f//(//x// 1 ) > //f//(//x// 2 ) whenever //x// 1 < //x// 2. > > > **End behavior** > The end behavior of the graph of a function refers to the behavior of f(x) as |x| becomes very large. > For instance, for the function g(x) = x 3, as x goes to infinity, g(x) also goes to infinity. Furthermore, as x goes to negative infinity, g(x) also goes to negative infinity. > > Here is another example: for the function h(x) = 1/x, as x goes to infinity, h(x) goes to zero. Also as x goes to negative infinity, h(x) again goes to zero (this time from the "underside" of the x-axis).
 * Increasing and decreasing functions.

3-4 Rational Functions and Asymptotes

 * A **//rational function//** is the quotient of two polynomials, having the form //f(x) = g(x) / h(x)//. where //h(x)// does not equal zero.
 * When we work with rational functions we must understand fully what the domain of //f//(//x)// is. We should be acutely aware of those values of //x//, if any, that would cause //h(x)// to equal zero, and understand that such values cannot lie within the domain of //f(x)//.
 * Here is a worksheet to practice determining the domain of the rational function //f(x): [[file:Rational Function Domain.pdf]] .//
 * //For extra practice with rational functions, here are some exercises from our textbook: [[file:Asymptotes.pdf]] .//
 * //Here is a summary of asymptote behavior:[[file:Asymptote summary.pdf]] .//

**Quiz on Sections 3-4 and 3-8**
Set of review questions for quiz on Sections 3-4 and 3-8:.

And here are the solutions to the review questions:. Do yourself a favor: Don't look at the solutions until after you at least have tried the problems.

**Quiz on Sections 3-5 through 3-7.**
Set of review questions for quiz on Sections 3.5, 3.6, and 3.7:. Answers at end of file.

Chapter 3 Test

 * [[file:PreCalc Ch03 Review.pdf]]is a set of fourteen review problems from various sources.
 * And here is a set of answers to the review problems: [[file:PreCalc Ch03 Review soln.pdf]].