Practice Page
Directions: Solve the following questions pertaining to polynomial graphs, zeros, and end behavior.
Please NO GRAPHING CALCULATORS for this practice page.

1.
Which of the following characteristics does not pertain to the graph shown at the right?   

Choose:
as x → -∞, f(x)→ -∞
as x →∞, f(x)→ ∞
three real zeros
increasing across the positive x-axis
pgraph1

 

 

2.
Which of the following characteristics does not pertain to the graph shown at the right?
Assume all roots are real.

Choose:
repeated root at x = 0
multiplicity of x is 3
as x → -∞, f(x)→ ∞
degree 4
pgraph2

 

 

3.
Find the zeros (and each multiplicity) for the polynomial P(x) = x2(x - 3)2(x + 1)(x + 4)3 .
The number in parentheses is the multiplicity of the preceding root (zero).
Choose:
x = 0 (2), 3 (2), -1 (1), -4 (3)
x = 0 (1), 3 (2), -1 (1), -4 (3)
x = 0 (2), -3 (2), 1 (1), 4 (3)
x = ± 1 (2), -3 (2), 1 (1), 4 (3)
searcher

 

 

4.
a) Assuming only real roots, what is the degree of the polynomial function sketched below?

sketch4

Choose:
2
4
6
8

b) Which statement describes the end-behavior for this function?
Choose:
as x → ∞, f(x)→ ∞;    as x → -∞, f(x)→ ∞
as x → ∞, f(x)→ ∞;    as x → -∞, f(x)→ -∞
as x → ∞, f(x)→ -∞;    as x → -∞, f(x)→ -∞
as x → ∞, f(x)→ -∞;     as x → -∞, f(x)→ ∞

c) Is the leading coefficient of this function positive or negative?
Choose:
positive
 negative 

 

 

5.
• State the roots of the polynomial
P(x) = x (x + 2)2 (x - 3)3.
• Indicate whether the graph crosses the x-axis at each root, or just touches the x-axis.
• Draw a sketch of the graph.
drawhand

 

 

6.
Which of the equations at the right, when graphed, will intersect with the x-axis only once? Explain how you arrived at your answer.

A:    y = 2x3 - 16x2 + 32x

B:    y = x3 - 2x2 + 4x - 8

C:    y = x3 - 3x2 - x + 3

 

 

7.
Given the conditions stated at the right, determine:
a) the degree of the polynomial function
b) the end behavior
c) a rough sketch
Explain your answers.
Conditions:
• the leading coefficient is positive
• Root x = -3 multiplicity 2
• Root x = 2 multiplicity 1
• Root x = -1 multiplicity 1
Assume only real roots.

 

 

8.
The polynomial function P(x) = x3 - 25x.
a) Factor and determine the roots.
b) Determine the end behavior of the graph.
c) Make a sketch of the graph.

Explain how you arrive at your answers.
polyend

 

 

9.
Given the polynomial function
f (x) = (2x + 1)(x - 3)(x + 1)
a) What is the y-intercept of the graph of the function?
b) Describe the end-behavior of this function.
c) For what values of x is f (x) > 0?
d) How many relative maximums does this function have?

 

 

10.
P(x) = 2x3 + 4x2 - 14x + 8
has (x + 4) as a factor.
a) Factor the polynomial into three linear terms.
b) Describe the end behavior.
c) Identify all intercepts.
d) Describe how you would go about sketching the graph of a function defined by this polynomial, without using a graphing calculator.

 

divider

NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
 and is not considered "fair use" for educators. Please read the "Terms of Use".
m