Investigations on Relative Extrema
 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.
 What
are the coordinates for the vertex of this parabola?
 How
can you find the xcoordinate of the vertex? How is related to the first two coefficients?
 LetŐs
try another parabola and quadratic function to double check our work. The quadratic function is .
 Does
your graph match the Graphing Calculator output that is below?
 What
are the coordinates of the vertex of this parabola?
 How
can relate the xcoordinate of the vertex to the first two coefficients
of this parabola?
 LetŐs
move up to the third power.
The first cubic polynomial we will look at is the cubic polynomial .
 Can
you get your graph to match the Graphing Calculator output below?
 In
this graph, there are two places that are similar to vertices of the
parabola. Where does it
appear that these points are located?
 What
is the connection between the number of vertices and the degree of the
parabola?
 What
do you notice about the connection between your answers from and the
solutions to the quadratic equation ?
 Do
you see a connection between along with
?
 LetŐs
try another cubic polynomial: .
 What
is the degree of this polynomial?
 Graph
the polynomial on your own.
 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?
 LetŐs
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


 What
do you notice about your answers from the previous problem and the
solutions to the quadratic equation ?
 If
you have not already, do you see a connection between along with
?
 One
last cubic polynomial:
 What
is the degree of the polynomial?
 Graph
the polynomial on your own.
 According
to your graph, where (in terms of coordinates) are the extrema of this
cubic polynomial.
 LetŐs
get an idea of the extrema by looking at a table of values.
 Do
you think the table of values contain the extrema? If so, where are they?
 What
do you notice about the extrema and the solutions to the quadratic
equation ?
 What
do you notice about the number of extrema and the degree of the polynomial?
 If
you have not already, do you see a connection between along with
?
 LetŐs
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
 Can
you see the three extrema?
What are the coordinates of the extrema?
 What
is the connection between the number of extrema and the degree of the
polynomial?
 Try a
fourth degree polynomial of your own. Look at the graph of the polynomial
 Graph
the function on your own.
 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?
 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


 Why
do you think you might have extrema in the interval [4, 4]?
 Do
you still see a connection between the degree of the polynomial and the
number of extrema?
 If we
look at the fourth degree polynomial , we have fewer terms to look at.
 Graph
the function on your own.
 How
many extrema do you see?
Where are they?
 Confirm
your ideas by filling in the following table of values.
 Why
do you think you might have extrema?
 Which
of the extrema look different than the others? How do they look differ?
 If
we classify extrema as relative maximum or relative minimum, which
extrema is which?
 Concluding
exercise:
 What
is the greatest number of extrema that a polynomial can have?
 How
can you tell a number can be an extreme or not?
 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 35)?