Investigations on Relative Extrema

1.	We will get our start on this lesson by looking the parabola for the quadratic function.  Let us look at the graph of the parabola looking at the Graphing Calculator output.  Try to get it to match what is given below.

a.	What are the coordinates for the vertex of this parabola?

b.	How can you find the x-coordinate of the vertex?  How is related to the first two coefficients?

2.	Let’s try another parabola and quadratic function to double check our work.  The quadratic function is .
a.	Does your graph match the Graphing Calculator output that is below?

b.	What are the coordinates of the vertex of this parabola?

c.	How can relate the x-coordinate of the vertex to the first two coefficients of this parabola?

3.	Let’s move up to the third power.  The first cubic polynomial we will look at is the cubic polynomial .
a.	Can you get your graph to match the Graphing Calculator output below?

b.	In this graph, there are two places that are similar to vertices of the parabola.  Where does it appear that these points are located?

c.	What is the connection between the number of vertices and the degree of the parabola?

d.	What do you notice about the connection between your answers from and the solutions to the quadratic equation ?

e.	Do you see a connection between  along with ?

4.	Let’s try another cubic polynomial: .
a.	What is the degree of this polynomial?

b.	Graph the polynomial on your own.

c.	The points that resemble vertices of a parabola are called extrema.  According to your graph, where (in terms of coordinates) are the extrema of this cubic polynomial?

d.	Let’s get an idea of where these extrema are by generating tables of values for selected values of the domain.

x	f(x)	x	f(x)	
-1/3		4		
0		5		
1/3		6		
2/3		7		
1		8		

e.	What do you notice about your answers from the previous problem and the solutions to the quadratic equation ?

f.	If you have not already, do you see a connection between  along with ?

5.	One last cubic polynomial: 
a.	What is the degree of the polynomial?

b.	Graph the polynomial on your own.

c.	According to your graph, where (in terms of coordinates) are the extrema of this cubic polynomial.

d.	Let’s get an idea of the extrema by looking at a table of values.
x	f(x)	
-3		
-2		
-1		
0		
1		
2		
3		

e.	Do you think the table of values contain the extrema?  If so, where are they?

f.	What do you notice about the extrema and the solutions to the quadratic equation ?

g.	What do you notice about the number of extrema and the degree of the polynomial?

h.	If you have not already, do you see a connection between  along with ?

6.	Let’s look at a few fourth degree polynomials.  The first polynomial is .  The graph of the polynomial is given below.  We might need to look at the graph in two parts.
 and 
a.	Can you see the three extrema?  What are the coordinates of the extrema?

b.	What is the connection between the number of extrema and the degree of the polynomial?

7.	Try a fourth degree polynomial of your own.  Look at the graph of the polynomial 
a.	Graph the function on your own.

b.	The first two extrema are easier to see with a typical set of coordinate axes.  What are the coordinates of the extrema, according to the graph you have seen?

c.	Let us look at a table of values and see if there does indeed look like possible extrema exist.
x	f(x)	
-4		
-3		
-2		
-1		
0		
1		
2		
3		
4		

d.	Why do you think you might have extrema in the interval [-4, 4]?

e.	Do you still see a connection between the degree of the polynomial and the number of extrema?

8.	If we look at the fourth degree polynomial , we have fewer terms to look at.
a.	Graph the function on your own.

b.	How many extrema do you see?  Where are they?

c.	Confirm your ideas by filling in the following table of values.
x	f(x)	
-3		
-2		
-1		
0		
1		
2		
3		

d.	Why do you think you might have extrema?

e.	Which of the extrema look different than the others?  How do they look differ?  

f.	If we classify extrema as relative maximum or relative minimum, which extrema is which?

9.	Concluding exercise:
a.	What is the greatest number of extrema that a polynomial can have?

b.	How can you tell a number can be an extreme or not?

c.	Do you see a possible connection between the coefficients of the original polynomial and the coefficients of the equation that can be used to find the extrema (see exercise 3-5)?