Chapter 3: The Nature of Graphs Section 3-1: Symmetry Point Symmetry: Two distinct points P and P’ are symmetric with respect to point M if M is the midpoint of Symmetry with respect to the Origin: The graph of a relation S is symmetric with respect to the origin if (-a, -b) are elements of S whenever (a, b) are in S. (they are called odd relations).

Section 3-1: Symmetry Example: Determine if is symmetric with respect to the origin. Algebraically, this can be verified if f(-x) = - f(x) for all x in the domain.

Section 3-1: Symmetry Line Symmetry: Two distinct points P and P’ are symmetric with respect to a line k if k is the perpendicular bisector of Symmetry with respect to the x and y axes: The graph of a relation S is symmetric with respect to the x-axis if (a, -b) are elements of S whenever (a, b) are in S. The graph of relation S is symmetric with respect to the y-axis if (-a, b) are elements of S whenever (a, b) are in S. (they are called even relations.)

Section 3-1: Symmetry Symmetry with respect to the lines y = x and y = -x: The graph of a relation S is symmetric with respect to the line y= x if (b, a) are elements of S whenever (a, b) are in S. The graph of relation S is symmetric with respect to the line y = -x if (-b,-a) are elements of S whenever (a, b) are in S.

Section 3-1: Symmetry All Polynomial Functions whose powers are all Even are Even Functions (symmetric to y-axis). All Polynomial Functions whose powers are all Odd are Odd Functions (symmetric to Origin).

Section 3-1: Symmetry Example: Determine if the graph of is symmetric with respect to the… x-axis y-axis y = x y = -x the origin

Section 3-1: Symmetry Example: Complete the graph below so that it is symmetric with respect to… a)x-axis b)y-axis c) y = x d) y = -x e) the origin

5-Minute Check Lesson 3-2A

5-Minute Check Lesson 3-2B

Section 3-2: Analyzing Families of Graphs A family of graphs is a group of graphs that displays one or more similar characteristics. A parent graph is a basic graph that can be transformed to create the other members in the family. Some of the functions that you should be aware of: constant, identity, polynomial, square root, absolute value, greatest integer, rational

Section 3-2: Analyzing Families of Graphs A line reflection flips a graph over a line called the axis of symmetry. The shifting of a graph either vertically or horizontally is called a translation. A dilation is a stretching or compressing of a graph.

Section 3-2: Analyzing Families of Graphs The graph of y = f(x) + a translates the graph of f up ‘a’ units In general, for a function y = f(x), the following are true when a > 0 … The graph of y = f(x) - a translates the graph of f down ‘a’ units The graph of y = f(x + a) translates the graph of f to the left ‘a’ units

Section 3-2: Analyzing Families of Graphs The graph of y = f(x- a) translates the graph of f right ‘a’ units In general, for a function y = f(x), the following are true when a > 0 … The graph of y = -f(x) reflects the graph of f over the x-axis The graph of y = f(-x) reflects the graph of f over the y-axis

Section 3-2: Analyzing Families of Graphs The graph of y = a*f(x) is stretched vertically if a > 1. In general, for a function y = f(x), the following are true when a > 0 … The graph of y = a*f(x) is compressed vertically if 0 < a < 1.

Section 3-2: Analyzing Families of Graphs The graph of y = f(ax) is compressed horizontally if a > 1. In general, for a function y = f(x), the following are true when a > 0 … The graph of y = f(ax) is stretched horizontally if 0 < a < 1.

Section 3-2: Analyzing Families of Graphs Example: if the graph of f(x) is shown, draw the graph of each function based on f(x)… a)f(x) + 1 b)f(x – 2) c)f(-x) d)2*f(x)

Section 3-2: Analyzing Families of Graphs Example: Describe the transformations that have taken place in each of the related graphs:

5-Minute Check Lesson 3-3A

5-Minute Check Lesson 3-3B

Section 1-7: Piecewise Functions A Piecewise Function is one that is defined by different rules for different intervals of the domain. When graphing piecewise functions, the various pieces do not necessarily connect.

Section 1-7: Piecewise Functions One example of a piecewise function is the Absolute value function.

Lesson Overview 1-7B

Lesson Overview 1-7A

Section 1-7: Piecewise Functions Graph the following piecewise function:

5-Minute Check Lesson 1-8A

Section 3-5: Continuity and End Behavior All Polynomial Functions are continuous. A function is continuous if there are no ‘breaks’ or ‘holes’ in the graph (you can graph the function without ‘lifting your pencil’).

Section 3-5: Continuity and End Behavior There are 3 types of discontinuity that can occur with a function: Infinite, Jump, and Point. Infinite Point Jump Discontinuity Discontinuity Discontinuity

Lesson Overview 3-5A

Lesson Overview 3-5B

Section 3-5: Continuity and End Behavior Here are three examples of discontinuous functions: Infinite Jump Point

Section 3-5: Continuity and End Behavior Example: Determine whether each function has infinite, point, or jump discontinuity. a) b) c)

Section 3-5: Continuity and End Behavior Continuity Test A function is continuous at x = c if it satisfies the following conditions: 1.The function is defined at c ( f(c) exists ) 2.The function approaches the same y-value on the left and right sides of x = c 3.The y-value that function approaches from each side is f(c).

Section 3-5: Continuity and End Behavior Continuity on an Interval A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval. Example: is not continuous everywhere, but is continuous for x 0

Section 3-5: Continuity and End Behavior To determine the end behavior for a polynomial function, look at the term with the highest power of x. The end behavior of a function refers to the behavior of f(x) as |x| gets large (as x tends towards positive infinity or negative infinity).

Section 3-5: Continuity and End Behavior If the highest power term is positive with an even value of n: If the highest power term is positive with an odd value of n: In general, for a polynomial function P such that…

Section 3-5: Continuity and End Behavior If the highest power term is negative with an even value of n: If the highest power term is negative with an odd value of n: In general, for a polynomial function P such that…

Section 3-5: Continuity and End Behavior Example: Without graphing, determine the end behavior of the following polynomial function…

Section 3-5: Continuity and End Behavior Increasing and Decreasing Functions. Given that f is the function and are elements of f A function f(x) is increasing if whenever A function f(x) is decreasing if whenever

Section 3-5: Continuity and End Behavior Increasing and Decreasing Functions. Given that f is the function and are elements of f A function f(x) is constant if whenever

Section 3-5: Continuity and End Behavior Increasing and Decreasing Functions. Example: Determine if the following function is increasing or decreasing…

Section 3-5: Continuity and End Behavior Increasing and Decreasing Functions. Example: Determine if the following function is increasing or decreasing…

5-Minute Check Lesson 3-6A

5-Minute Check Lesson 3-6B

Section 3-6: Graphs and Critical Points of Polynomial Functions Graphs of Polynomial Functions have smooth curves with no breaks or holes. For a function f(x), f’(x) (known as the derivative) defines the slope of that function. When f’(x) = 0, the tangent line to the curve at that point is horizontal. Points when f’(x) = 0 are called critical points.

Section 3-6: Graphs and Critical Points of Polynomial Functions Critical points can occur on a graph in three different forms: relative maximum, relative minimum, or a point of inflection. Rel. Min. Rel. Max. Pt. of Inflection

Section 3-6: Graphs and Critical Points of Polynomial Functions By determining the critical points and the x and y intercepts of a polynomial function, you can create a much more accurate and complete graph of the function.

Section 3-6: Graphs and Critical Points of Polynomial Functions Example: Determine the critical points for the following function. Then, determine whether each point represents a minimum, maximum, or point of inflection.

Section 3-6: Graphs and Critical Points of Polynomial Functions Example: Determine the critical points for the following function. Then, determine whether each point represents a minimum, maximum, or point of inflection.

5-Minute Check Lesson 3-7B

Section 3-7: Rational Functions and Asymptotes The graph of a rational function usually includes vertical and horizontal asymptotes- lines towards which the graph tends as x approaches a specific value or as x approaches positive or negative infinity. A Rational Function is an equation in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions, and q(x) does not equal zero.

Section 3-7: Rational Functions and Asymptotes A Hole occurs at x = a whenever there is a common factor (x – a) in the numerator and denominator of the function. Example of a function with a hole:

Section 3-7: Rational Functions and Asymptotes A Vertical Asymptote is line x = a for a function f(x) if from either the left or the right. A Horizontal Asymptote is line y = b for a function f(x) if

Section 3-7: Rational Functions and Asymptotes To find Vertical Asymptotes, determine when the denominator of the function will be equal to zero (not including ‘holes’).

Section 3-7: Rational Functions and Asymptotes To find Horizontal Asymptotes, look at the limit of the function as x approaches infinity (3 scenarios): y = 0 is horizontal asymptote since degree of denominator is larger y = 9/5 is horizontal asymptote since degrees are equal No horizontal asymptote since degree of numerator is larger

Section 3-7: Rational Functions and Asymptotes Determine the asymptotes for the following function:

Section 3-7: Rational Functions and Asymptotes Example: Determine any holes, horizontal asymptotes, or vertical asymptotes for the following functions:

Section 3-7: Rational Functions and Asymptotes Example: Create a function of the form y = f(x) that satisfies the given set of conditions: a)Vertical asymptote at x = 2, hole at x = -3 b)Vertical asymptotes at x = 5 and x = -9, resembles

Section 3-7: Rational Functions and Asymptotes A Slant Asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator. Example of a function with a slant asymptote: The equation of the slant asymptote for this example is y = x.

Section 3-7: Rational Functions and Asymptotes To determine the equation of the slant asymptote, divide the numerator by the denominator, and then see what happens as Example: Determine the slant asymptote for the following function:

5-Minute Check Lesson 3-8A

5-Minute Check Lesson 3-8B

Test xy  for symmetry.

Test xy  for symmetry.

5-Minute Check Lesson 4-1A