Math 30
Geogebra Activity
Polynomial Functions
Part I
Create a GeoGebra file with the polynomial function f  x   a( x  l )( x  m) . Create sliders for a,
l, and m.Set the slider for a to accept values between -5 and 5, with increments of 0.1. Set the
sliders for l and m to accept values between -10 and 10, and accept integral values only.
1.
2.
3.
4.
5.
State the degree of the polynomial.
What shape is the graph?
What is the effect of changing the a value?
Describe what l and m represent.
What happens when l and m are the same?
Part II
Create a GeoGebra file with the polynomial function g  x   a( x  l )( x  m)  x  n  . Create
sliders for a, l, m and n. Set the slider for a to accept values between -5 and 5, with increments of
0.1. Set the sliders for l and m and n to accept values between -10 and 10, and accept integral
values only.
6. State the degree of the polynomial.
7. What shape is the graph?
8. What is the effect of changing the a value?
9. Describe what l, m and n represent.
10. What happens when l and m are the same, but n is different?
11. What happens when l, m, and n are all the same?
12. Find values for the parameters that make the function have three distinct x-intercepts.
13. Find values for the parameters that make the function have two distinct x-intercepts.
14. Find values for the parameters that make the function have only one x-intercept.
Part III
Create a GeoGebra file with the polynomial function h  x   a( x  l )( x  m)  x  n  x  o  .
Create sliders for a, l, m, n and o. Set the slider for a to accept values between -5 and 5, with
increments of 0.1. Set the sliders for l, m, n and o to accept values between -10 and 10, and
accept integral values only.
15. State the degree of the polynomial.
16. What shape is the graph?
17. What is the effect of changing the a value?
18. Describe what l, m, n and o represent.
19. What happens when l, m, n and o are all the same?
20. Find values for the parameters that make the function have four distinct x-intercepts.
21. Find values for the parameters that make the function have three distinct x-intercepts.
22. Find values for the parameters that make the function have two distinct x-intercepts.
23. Find values for the parameters that make the function have only one x-intercept.
Part IV
24. Go back to your function, f  x   a( x  l )( x  m) , and change it to
f  x   a( x  l )( x  m)  d . Build a slider for d.
a. Determine values for a and d so that the function has no x-intercepts, and finishes
going up in the first quadrant.
b. Determine values for a and d so that the function has no x-intercepts, and finishes
going down to the right.
25. Go back to your function, g  x   a( x  l )( x  m)  x  n  , and change it to
g  x   a( x  l )( x  m)  x  n   d . Build a slider for d.
a. Determine values for a and d so that the function has no x-intercepts, and finishes
going up in the first quadrant.
b. Determine values for a and d so that the function has exactly one x-intercept, and
has a bump, and finishes going up to the right.
26. Go back to your function, h  x   a( x  l )( x  m)  x  n  x  o  , and change it to
h  x   a( x  l )( x  m)  x  n  x  o   d . Build a slider for d.
a. Determine values for a and d so that the function has no x-intercepts, and finishes
going up in the first quadrant.
b. Determine values for a and d so that the function has exactly one x-intercept, and
has at least one bump, and finishes going up to the right.
Part IV
On your own, without using GeoGebra, sketch an example of each of the following types of
functions, or state that it is impossible to draw.
1. A third degree polynomial, with exactly two x-intercepts.
2. A fourth degree polynomial with exactly one x-intercept.
3. A second degree polynomial with no x-intercepts.
4. A fifth degree polynomial with no x-intercepts.
5. A fourth degree polynomial, with two x-intercepts, each with multiplicity of two.
6. A third degree polynomial, with one x-intercept, with a multiplicity of three.
7. A fourth degree polynomial, with exactly three x-intercepts.