Graph Behavior

As you have seen, there are more than just lines and parabolas when it comes to graphing. This lesson focuses on those types of graphs but also explores the behavior of more complex graphs, their domains and range, increasing and decreasing intervals, symmetry, and end behavior.

You have already learned about degree, roots, multiplicity, vertices, axis of symmetry, max/min, and the x- and y-intercepts. Recall that degree can tell you the maximum number of possible solutions, or roots, a graph can have. Multiplicity can occur when duplicates of factors occur. The axis of symmetry and max/min are related to the vertices where the graph will change direction. Each time a graph changes direction there is another opportunity for the graph to possibly cross the x-axis.

A graph can increase and decrease many times and it will always change direction at a vertex. There is an exception here though, most graphs will stop the increasing and decreasing intervals and eventually either increase or decrease to infinity. Let’s take a look!

As the graph grows from left to right we say the graph is increasing. The green path shows that this graph is increasing from the vertex, (0,0), to positive infinity.

As the graph falls from left to right we say the graph is decreasing. The green path shows that this graph is decreasing from negative infinity (x-axis) to the vertex, (0,0).

Below is a graph of the function Sin (x). Notice that the graph rises and falls, or increases and decreases over the intervals between vertices. This graph is a special graph and continues on forever in this manner. There are an infinite number of increasing and decreasing intervals in this graph.

I

I D

I II D D D

The graphs of constants, such as y = 1.5 or x = 1.5 also continue on without change. I II D D D y = 1.5 will not increase or decrease because it is the graph of a constant. x = 1.5 is not a function so we will ignore this or now.

Example: Indicate the increasing and decreasing intervals for the function y = x 3 – 5x 2 + 4x + 2. Hint: Graph the function and find the vertices/max/min first.

Example: Indicate the increasing and decreasing intervals for the function y = x 3 – 5x 2 + 4x + 2. Solution: The graph is increasing over the intervals of (- , 0.465) and (2.869,  ). The graph is decreasing over the interval of (0.465, 2.869) *It is always a good idea to zoom out a few times on a graph to make sure you are not missing an interval. I D I

Interval Notation: We found that the graph is increasing over the intervals of (- , 0.465) and (2.869,  ) and the graph is decreasing over the interval of (0.465, 2.869). But, we can also use interval notation to properly write the intervals. We use [ or ] when we have a stop and include the value, and we use ( or ) when the value is not included but all numbers up to that point are. Here we would write: increasing (- , 0.465] and [2.869,  ) decreasing [0.465, 2.869] I D I

You have already learned about domain and range. Domain of a function is the set of y-values (independent variable) that the graph uses. Range of a function is the set of x-values (dependent variable) that the graph uses. We will incorporate this in later, but review the domain and range now is needed.

Now that you know how to identify the increasing and decreasing intervals of a graph, and you remember how to identify the domain and range, let’s took a look at the final topic in this lesson, end behavior. End behavior is jut like it sounds. End behavior describes how the ends of the graph behave. They can increase or decrease thus approach positive or negative infinity.

It is easiest to remember the rules for analyzing graphs because sometimes when we graph on a calculator we forget to zoom out or in to make sure we have seen all of the graph. The rules for END BEHAVIOR are simple. a)If degree is even and the leading coefficient is positive then the graph approaches positive infinity as x approaches negative infinity, the graph approaches positive infinity as x approaches positive infinity. b)If degree is even and the leading coefficient is negative then the graph approaches negative infinity as x approaches negative infinity, the graph approaches negative infinity as x approaches positive infinity. c)If degree is odd and the leading coefficient is positive then the graph approaches negative infinity as x approaches negative infinity, the graph approaches positive infinity as x approaches positive infinity. d)If degree is even and the leading coefficient is negative then the graph approaches positive infinity as x approaches negative infinity, the graph approaches negative infinity as x approaches positive infinity. This is much easier to keep track of in an organizer, there is one made for you on the next slide

Degree is Even Leading Coefficient Positive Degree is Odd Leading Coefficient Negative END BEHAVIOR

Degree is Even Leading Coefficient Positive Degree is Odd Leading Coefficient Negative END BEHAVIOR Examples END BEHAVIOR Examples

Degree is Even Leading Coefficient Positive Degree is Odd Leading Coefficient Negative Domain & Range

Let’s wrap up this lesson by looking at symmetry. We explored the axis of symmetry already. That was found using the x-coordinate from the vertex. If we used that x-value and drew a vertical line through it we could see that the graph (or that region of the graph) was symmetric, a mirror image, on each side. We want to keep that concept separate from the concept here. Here, symmetry refers to whether a function is even or odd. A function that is symmetric about the y-axis is an even function A function that is symmetric about the origin is an odd function (Hint: Origin and Odd both start with “O” – they go together) We can take a look at symmetry with some examples! Let’s go!

Symmetry Examples Even Symmetric about the y-axis. Odd Symmetric about the origin. Neither Even nor Odd Not Symmetric about the y-axis nor the origin.

CAUTION! Symmetry Examples Odd Symmetric about the origin. Neither Even nor Odd Even though there is an odd exponent, this function is Not Symmetric about the y-axis nor the origin, so it is neither. In summary: 1.Even functions have even exponents 2.Odd functions have odd exponents 3.NOT all functions are even or odd… most are neither (even if they have an even or odd exponent!!)

We are providing you with a chart that will be very helpful to you. 1.Graph the function in the first column 2.Compare what you see to the information in the columns following. 3.If you do not see the info in the table on your graph…ZOOM in or out as needed! *Note that these are the basic function types, called parent functions. The information will change as terms are added to the equations.

Type of Parent Function Graph InterceptsMax/MinInc./Dec. Intervals End Behavior Symmetry (even, odd, neither) DomainRange Linear f(x) = x x-intercept (0,0) y-intercept (0,0) No Max No Min Increasing (- ,  ) Decreasing None Right Up Left Down Odd (- ,  ) Quadratic f(x) = x 2 x-intercept (0,0) y-intercept (0,0) No Max Min (0,0) Increasing (0,  ) Decreasing (- , 0) Right Up Left Up Even (- ,  )[0,  ) Cubic f(x) = x 3 x-intercept (0,0) y-intercept (0,0) No Max No Min Increasing (- ,  ) Decreasing None Right Up Left Down Odd (- ,  ) Square Root f(x) =. x-intercept (0,0) y-intercept (0,0) No Max Min (0,0) Increasing (0,  ) Decreasing None Right Up Left Ends at 0 Neither [0,  ) Cube Root f(x) =. x-intercept (0,0) y-intercept (0,0) No Max No Min Increasing (- ,  ) Decreasing (- ,  ) Right Up Left Down Odd (- ,  ) Absolute Value f(x) = |x| x-intercept (0,0) y-intercept (0,0) No Max Min (0,0) Increasing (0,  ) Decreasing (- , 0) Right Up Left Up Even (- ,  )[0,  ) Exponential f(x) = b x for b > 1 x-intercept (0,0) Asymptote y=0 No Max No Min Increasing (- ,  ) Decreasing As x approaches 0, -  ) Right Up Left Approaches, but does not pass, x-axis Neither (- ,  )[0,  )