Equation: ______________________________________________
Degree: ___________
ODD or EVEN
Polynomial Name: _________________________
Leading Coefficient: ________
POS or NEG
Total number of roots: ___________
Zero:
Zero:
Zero:
Zero:
Zero:
End Behavior: As x  , f ( x)  ____
As x  , f ( x)  ____
x=________ Cross or Bounce
x=________ Cross or Bounce
x=________ Cross or Bounce
x=________ Cross or Bounce
x=________ Cross or Bounce
My Graphing Observations:
End Behavior:
Quadratics:
Roots/Zeros
Quadratics:
Turning Points (Max/Min)
Quadratics:
Cubics:
Cubics:
Cubics:
Quartics:
Quartics:
Quartics:
Quintics:
Quintics:
Quintics:
I notice the following patterns
with end behavior:
I notice the following patterns
with zeros/roots:
I notice the following patterns
with turning points:
Answer each of the following questions based upon our class observations:
1)
Which polynomials have the SAME end behavior as x   ? (Ends point in the same direction)
2)
Which polynomials have DIFFERENT end behavior as x   ? (Ends point in opposite directions)
3)
What are the maximum/minimum numbers of roots seen for each of the polynomial functions?
Function
Quadratic
Cubic
Quartic
Quintic
Max Roots
Min Roots
4)
What do you notice about the graph at a root when its factor is raised to an EVEN power? [Hint look at the
4
Quadratics and Quartics: i.e. y   x  1 ]
5)
What do you notice about the graph at a root when its factor is raised to an ODD power? [Hint look at the
5
Quintics: i.e. y  x  4 ]
6)
How many maximum/minimums (turning points) can each of the polynomial functions have?
Function
Quadratic
Cubic
Quartic
Quintic
# of max/min seen (list all noticed)
Graphing Activity
Focus Questions




How does the power/degree of a polynomial function affect its shape?
What is the maximum number of zeros/roots a polynomial function can have? What is the minimum number?
What determines if a graph “crosses” or “bounces” on the x-axis?
What are the similarities and differences in absolute/local maximums and minimums? How can they be seen in
a graph?
With your group, analyze each of the following functions. Be sure to pay
attention to critical information we identified and will help you answer our
focusing questions. Record your work on the sheets provided.
Group 1:
Group 2:
Group 3:
y  3x 2  4 x  5
y  x  3
y  2 x 2  3x  9
y  2 x3  9 x  5
y   x 3  3x 2  2 x
y  x3  5
y  ( x  1)4
y  x 4  18x 2  8
y   x 4  x3  5 x 2  1
y   x5  8 x 2  6 x
y   x5  8x 2
y  ( x  4)5
Group 4:
Group 5:
Group 6:
y  3x 2  5 x  2
y  ( x  3)( x  5)
y  x 2  5 x  10
y   x 3  3x  2
y  x3  4 x 2  9 x  2
y  x  2 ( x  3)( x  4)
y  ( x  4) 2 ( x  1)
y   x  2  ( x  6) 2
y   x 4  x 3  12 x 2  75
y  x5  7 x 4  10 x3  14 x 2  24 x  3
y   x5  3x 4  2 x3  7 x 2  5
y  x 5  x 4  15 x 3  9 x 2  54 x
2
2
2
Steps:
1. Copy the given equation
2. Determine the degree of the polynomial (highest power) and circle if this number
is odd or even
3. Determine the leading coefficient of the polynomial and circle if this number is
positive or negative
4. Use your graphing calculator to sketch each graph – you must show the roots of
each function and the general shape
5. Use your calculator to find the value of the roots, round to two decimal places if
necessary
6. Determine if the graph CROSSES through the root or BOUNCES at the root
7. Determine the end behavior of the graph
8. After you have finished ALL graphs complete the summary sheet by analyzing the
graphs you created