Continuity and End Behavior Before finishing this section

+ Continuity and End Behavior

+ Before finishing this section you should be able to: n Determine whether a function is continuous or discontinuous n Identify the end behavior of functions n Determine whether a function is increasing or decreasing on an interval

+ Most of the graphs that we have studied thus far have been smooth, continuous curves. However, some functions are known as discontinuous functions. You cannot trace the graph of the function without lifting your pencil. The chart below shows the different types of discontinuous functions. Function Name Infinite Discontinuity Graph Characteristics The graph becomes greater and greater as it approaches a given x-value.

Discontinuous Function Name Graph Characteristics Jump Discontinuity The graph stops at a given value of the domain and then begins again at a different range value for the same value of the domain. Point Discontinuity When there is a value in the domain for which the function is undefined, but the pieces of the graph match up. There is a hole in the graph.

What type of Discontinuity?

+ If a function is not discontinuous, it is said to be continuous. Linear and quadratic functions are continuous at all points. A function is continuous at x = c if it satisfies the following conditions: • The function is defined at c: in other words, f(c) exists the function approaches the same y-value on the left and right sides of x = c • The y-value that the function approaches from each side is f(c).

Example: Determine whether the function is continuous at the given x-value.

+ Find the End Behavior! 1. For the following function, graph to find and describe the end behavior. f(x) = x 4 – 4 x 3 + 3 x + 25 2. Write a function that demonstrated the end behavior of as x gets very large, y gets very small; as x gets very small, y gets very large.

+ Practice, Practice… 1. Is f(x) = 3 x 2 – 6 x + 8 continuous? If so describe the end behavior. If not, what type of discontinuity does it contain? 2. Is the following function continuous? If so describe the end behavior. If not, what type of discontinuity does it contain? 3. In 3 sentences describe all 3 types of discontinuous functions.

Another characteristic of functions used for analysis is the monotonicity of the function. This means that on an interval, the function is increasing or decreasing on that particular interval. Whether a graph is increasing or decreasing is always judged by viewing a graph from left to right.

The leading coefficient is positive and the highest power is odd. Therefore, as x -∞ y -∞ and as x ∞, y ∞ Use your graphing calculator to find the relative max and relative min to find your boundaries for the increasing and decreasing intervals.

The graph of the function is obtained by transforming the parent graph f(x) = x 2. The parent graph has been translated 1 unit to the left, and translated down 4 units. The function is decreasing for x < -1 and increasing for x > -1.

The graph of this function is obtained by transforming the parent graph p(x) = |x|. The parent graph has been reflected about the x-axis, translated 3 units right, and translated up two units. This function is increasing for x < 3 and decreasing for x > 3.

It changes direction at x = -1 and x = 1/3. The function is decreasing for x < -1. The function is also decreasing for x > 1/3. When -1 < x < 1/3 , the function is increasing.

Helpful Websites Discontinuity: http: //www. sparknotes. com/math/precalc/continuityandlimits/prob lems 3. rhtml http: //math. usask. ca/~maclean/101/Limits/Printables/BW/Continuit y. pdf End behavior: http: //www. purplemath. com/modules/polyends. htm 3 -5 Self Check Quiz: http: //www. glencoe. com/sec/math/studytools/cgibin/msg. Quiz. php 4? isbn=0 -07 -8608619&chapter=3&lesson=5&quiz. Type=1&header. File=4&state=