We start with the parent functions: f(x) = x2 to f(x) = x7. My students have already been exposed to the concept of parent functions for quadratic, linear, absolute value, and rational functions. I present each one and allow time for student to make quick sketches in their notes. I mark the vertex or center of each parent graph as well as the two points one unit way. Focusing on these points gives the students a pattern to follow as well as points to accurately show all transformations, including stretches/shrinks, in future lessons.
Once we have covered all six, the students talk with their partners about any patterns they see, followed by a group discussion. I make sure to ask WHY each observed pattern is true. Then the students record a statement describing the pattern(s) including the WHY in their notes.
This leads into a discussion on end behavior for even degree functions. First we define even functions. We do a think-pair-share on functions we have already seen that are even. Next we discover/discuss end behavior of even functions. I choose not to give a formal definition yet since I want my students to develop a solid conceptual understanding. I find that students often get stuck on formal definitions and never really grasp what is being said. A formal definition can always be introduced later once students master the concept.
I have students define with their partner what the graph ends are "doing" and then share out as a whole class. Definitions should include [increasing or decreasing] behaviors for BOTH extremes of the graphs. I encourage the students to evaluate their definitions for completeness and accuracy (Math Practice 3). To wrap up the conversation, students write down their definitions including diagrams that may be helpful.
I then repeat the conversation now for the opposite [negative] even functions and again for odd functions. Each discussion should move along faster since students are building on their initial concept of end behavior.
Graphing Polynomial Functions
To sketch any polynomial function, you can start by finding the real zeros of the function and end behavior of the function .
Steps involved in graphing polynomial functions:
. Predict the end behavior of the function.
. Find the real zeros of the function. Check whether it is possible to rewrite the function in factored form to find the zeros. Otherwise, use Descartes' rule of signs to identify the possible number of real zeros.
. Make a table of values to find several points.
. Plot the points and draw a smooth continuous curve to connect the points.
. Make sure that the graph follows the end behavior as found in the above step.
Graph the polynomial function .
Predict the end behavior of the function.
The degree of the polynomial function is odd and the leading coefficient is positive.
The degree of the polynomial is and there would be zeros for the functions.
The function can be factored as . So, the zeros of the functions are and .
Make a table of values to find several points.
Plot the points and draw a smooth continuous curve to connect the points