Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
Vocabulary you need to know:
The general (standard) form of a polynomial is______________________________.
The leading coefficient of a polynomial is __________________________________ .
The degree of a polynomial is ____________________________________________.
The constant term of a polynomial is_______________________________________.
Right behavior is what y is doing as x is increasing.
Left behavior is what y is doing as x is decreasing.
Relative Minimum (minimum turning point) is the minimum value of the function relative to the
values around it.
Relative Maximum (maximum turning point) is the maximum value of the function relative to the
values around it.
Relative Extrema refers to the relative minima and the relative maxima.
Go to my website and click on “Geometer’s Sketchpad Activities” on the left.
Click on the link “Leading Coefficient Discovery Activity”
You are going to click on each example of a polynomial function and make some
conclusions. For each degree, “play” with the sliders to see how changing the
coefficients changes the graph. Using the sliders answer the following questions:
1. What is the graph of a function of degree 0? (Be specific)
2. What is the standard equation for each of the polynomial functions? Fill in the
following terms after each degree: quadratic, cubic, quintic, quartic, linear,
constant.
3.
Degree
Zero
One
Two
Three
Name
Standard Equation
Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
Four
Five
4. The next few questions only deal with polynomial functions of EVEN degree.
a. If the leading coefficient is positive, what is the left behavior of the graph?
In other words, as x decreases what is y doing?
b. If the leading coefficient is positive, what is the right behavior of the
graph? In other words, as x increases what is y doing?
c. If the leading coefficient is negative, what is the left behavior of the graph?
In other words, as x decreases what is y doing?
d. If the leading coefficient is negative, what is the right behavior of the
graph? In other words, as x increases what is y doing?
Graph below an example of a
Graph below an example of a
polynomial function of even degree
polynomial function of even degree
with a positive leading coefficient?
with a negative leading coefficient?
Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
5. The next few questions only deal with polynomial functions of ODD degree.
a. If the leading coefficient is positive, what is the left behavior of the graph?
In other words, as x decreases what is y doing?
b. If the leading coefficient is positive, what is the right behavior of the
graph? In other words, as x increases what is y doing?
c. If the leading coefficient is negative, what is the left behavior of the graph?
In other words, as x decreases what is y doing?
d. If the leading coefficient is negative, what is the right behavior of the
graph? In other words, as x increases what is y doing?
Graph below an example of a
Graph below an example of a
polynomial function of odd degree
polynomial function of odd degree
with a positive leading coefficient?
with a negative leading coefficient?
Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
Fill in the following chart:
Degree
Leading Coefficient
Even
+
Even
-
Odd
+
Odd
-
Left Behavior
Right Behavior
6. For each of the polynomial functions change the constant term.
a. Can you make a general statement for how changing the constant of a
polynomial function effects the graph?
b. How does the constant term relate to the y-intercept of the graph?
c. How does changing the constant affect the x-intercepts of the graph?
7. The following questions only deal with a third degree polynomial. Change the
values for b and c in the “cubic” polynomial function leaving a and d alone. How
do the values affect the shape of the graph?
8. Based on what you have learned and using the sliders write an equation of the
following:
a. A polynomial function that goes down to the left and up to the right.
b. A polynomial function that goes up to the left and up to the right.
c. A polynomial function that goes up to the left and down to the right.
Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
d. A polynomial function that goes down to the left and down to the right.
e. A cubic function that has a y-intercept of 5.
f. A polynomial function of degree 4 that has 2 x-intercepts.
g. A polynomial function of degree 2 that has no x-intercepts.
h. A polynomial function of degree 3 that has only 1 x-intercept.
i.
A polynomial function of degree 5 that has a y-intercept of –4.
j.
The equation f ( x)  x 3  2 x 2  4 x  1 crosses the y-axis at 1. What would
the equation of f (x) be if it was to shift up on the y-axis 5 units?
k. How does the shift of the equation in j. effect the x-intercepts? How does
the shift effect the y-intercept?
8. Write a paragraph below that answers the essential question at the top of the page.
Investigation of Polynomial Functions
EQ: How do the coefficients of a polynomial function affect the graph? How do I
determine the left and right behavior?
Fill in the following table:
Degree
Max # of Turns (Extrema)
Max # of Zeros
0
1
2
3
4
5
N
Graph the following functions in your calculator or using Geometer’s Sketchpad.
Fill in the table for each of the functions.
Function
f ( x)  x 3  3 x  2
f ( x)  x 3  7 x 2  7 x  15
f ( x)  x 3  3 x 2  6 x  8
f ( x)  x 3  x 2  80 x  300
f ( x)  x 3  x 2  34 x  56
For what values of x is
For what values of x is
f(x)>0?
f(x)  0?