Exploring Polynomials Guided Discovery
Instructions: You and your group will be responsible for completing this packet by the end of the period or before.
There are two key questions you need to answer today
-What does “end behavior” mean
-What in an equation determines the end behavior, and explain why
You will use a graphing calculator to guide you through this exercise. After completing each step, check the checkbox marking
it complete. When your team has finished, get your TEACHER STAMP for the page. You may to get stamps at each section of
the packet or at the end if you are confident.
1. □ Using a highlighter, highlight all of the vocabulary words shown in the first diagram below. Then, in
the second diagram, identify the key points and write your own definition of the word.
Local Maximum:
Maximum y values
relative to points close to
them on the graph
Local Minimum: Minimum
y values relative to points
close to them on the graph
Roots: The zero’s of a
graph, where graph
crosses x axis
y- intercept: value when
x=0, graph crosses y
axis
End Behavior:
As x goes to negative infinity, the graph f(x) goes to negative infinity
x  -∞, f(x)  ____
As x goes to positive infinity, the graph f(x) goes to negative infinity
x  ∞, f(x)  ____
Local Minimum:
Local Max:
y- intercept:
Roots:
End Behavior:
As x goes to negative infinity, the graph f(x) goes to negative infinity
x  -∞, f(x)  ____
As x goes to positive infinity, the graph f(x) goes to negative infinity
x  ∞, f(x)  ____
2.
□ Write two complete sentences explaining what you think End Behavior might mean based on the diagrams
above.
Stamp #1
3. □ Highlight key information in the following passage
End behavior describes how the graph moves at the “extremes”. When we say “as x goes to negative infinity”, we mean
as we trace the graph to the left, so the x values get more and more negative. Then, the second part of the statement
describes the direction of the graph – does it go up to positive infinity, or down to negative infinity? Then we describe the
same thing as we move to the right – “as x goes to infinity”, meaning as the x values getting larger.
4. □ Consider the following examples, then complete the table.
End Behavior
Graph
End Behavior
10
8
6
x  -∞, f(x)  ____
4
x  ∞, f(x)  ____
2
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
0
2
4
6
0
2
4
6
8
10
0
2
4
6
8
10
0
2
0
2
-2
-4
-6
-8
-10
10
8
6
x  -∞, f(x)  ____
4
x  ∞, f(x)  ____
2
0
-10
-8
-6
-4
-2
8
10
-2
-4
-6
-8
-10
10
8
6
x  -∞, f(x)  ____
4
2
0
-10
-8
-6
-4
-2
-2
x  ∞, f(x)  ____
-4
-6
-8
-10
10
8
6
4
2
0
-10
-8
-6
-4
-2
-2
-4
-6
-8
-10
10
8
6
4
2
0
-10
-8
-6
-4
-2
4
6
8
10
-2
-4
-6
-8
-10
10
8
6
4
2
0
-10
-8
-6
-4
-2
4
6
8
10
-2
-4
-6
-8
-10
Stamp #2
5. □ Graph the following polynomials on your graphing calculator and sketch a small picture of the graph in
the space provided.
6. □ What patterns do you notice about the end behavior of the
even degree graphs?
y = x2
Y = x5
y = x4
7. □ What patterns do you notice about the end behavior of the
odd degree graphs?
Y = -x3
Y = x7
y = -2x6
8.
□ What happens to the graph when the degree increases?
y = x8
Y = x3
9. □ How does the degree of the function influence the end behavior of
the graph?
Y = x9
Y = -x9
Y = -x4
10. □ How does the sign of the leading coefficient influence the end
behavior of the graph?
Stamp #3
11. □ We can predict end behavior based on the degree of the polynomial and the leading coefficient. Write an
example equation and sketch the graph of the function.
Leading Coefficient Positive
Ex: f(x) =
Leading Coefficient Negative
Ex: f(x) =
Degree Even
Ex: f(x) =
Ex: f(x) =
Degree Odd
12. □ Consider the following list of equations. By looking at the equation, predict the end behavior of the graph by
completing the end behavior statement and sketching the endings of the graph. Then, graph the equation on your
calculator to see that you are correct.
Equation
y=
x2 +5x+7
y=
-2x3 -4x2+1
Prediction Sketch of End
Behavior
End Behavior Statement
Actual Sketch of End
Behavior (From
Calculator)
x  -∞, f(x)  ____
x  ∞, f(x)  ____
x  -∞, f(x)  ____
x  ∞, f(x)  ____
y = -3x6 -5x2+2x
y = 4x5 +2x2-5
y = 3x4 -5x3+7
Stamp #4
y=
-2x2 +5x+7