Math 110
In Class Exercises, Section 3.2
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Polynomial Functions : End Behavior
Goal: Understand that the first term of a polynomial (in standard form) will dominate the expression as X gets really
large/small
1) Graph y  0.04x 4  0.2x 3  x 2  2.6x  3.36 on the standard viewing window.
A. Now graph y  0.04 x 4 in the same window. Modify the window as stated below and
discuss.
a. X[-10, 10, 1], Y[-10, 100, 1]
b. X[-20, 20, 1], Y[-100, 1000, 1]
c. X[-30, 30, 1], Y[-1,000, 10,000, 1]
d. X[-50, 50, 1], Y[-10,000, 100,000, 1]
B. What conclusions can you make from the above observations?
3
2
2) Graph y  0.04 x  .2 x  3.36 on the standard viewing window.
A. Now graph y  0.04 x in the same window. Modify the window as stated below and
discuss.
a. X[-10, 10, 1], Y[-10, 100, 1]
b. X[-20, 20, 1], Y[-100, 1000, 1]
c. X[-30, 30, 1], Y[-1,000, 10,000, 1]
d. X[-50, 50, 1], Y[-10,000, 100,000, 1]
3
B. What conclusions can you make from the above observations?
3) What is the end-behavior of even and odd polynomials?
(This problem was originally written by David Whittaker)
Math 110
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Math 110
In Class Exercises, Section 3.2
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4) Given the following functions
f (x )  x 4  4x 3  16x  16
g (x )  x 4  4x 3  4x 2  16x
h (x )  x 4  x 3  8x 2  12x
a.
Now graph on the window X[-5, 5, 1], Y[-35, 15, 5] and discuss.
b.
Graph on the window X[-8, 8, 0], Y[0, 4000, 0] and discuss
(This problem was originally written by David Whittaker)
5) Given the following functions
f x   4 x 3  16 x  16
g x   4 x 3  4 x 2  16 x  16
hx   x 3  8x 2  12 x  16
a.
Now graph on the window X[-5, 5, 1], Y[-35, 15, 5] and discuss.
b.
Graph on the window X[-8, 8, 0], Y[0, 4000, 0] and discuss
(This problem was originally written by David Whittaker)
Math 110
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Math 110
In Class Exercises, Section 3.2
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Polynomial Functions : Zeros and multiplicity
Goal: Graph each zero, and explain how it’s multiplicity affects the graph.
6) Let’s say that you’re given the following, factored, polynomial:
3
2

4
f  x    x    x  3 2 x  2 
3

What are the roots of the polynomial?
What are the zeros of the polynomial?
For each zero, list the zero, and it’s multiplicity:
At what values of x does the function cross the x axis?
At each of those values (where the function crosses the x axis), does the function ‘reflect’ off the
axis, or cross the axis? How do you know?
Finally, you should determine what the end behavior of the function will be. You’ll need to figure
out what the degree of the polynomial is, and whether it’s leading term is positive or negative.
X
Lastly, graph the function:
10
9
8
7
6
5
4
3
2
1
0
-1 0 1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2 3 4 5 6 7 8 9 10
Y
Math 110
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Math 110
In Class Exercises, Section 3.2
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7) Sketch a graph of the function. Make sure your graph shows all x-intercepts and exhibits the
proper end behavior
f (x) = (x + 1)(x + 6)(x – 6)2
8) Sketch a graph of the function. Make sure your graph shows all x-intercepts and exhibits the
proper end behavior
f (x) = (2x + 5)(x – 5)3(x + 6)
9) Find a minimum-degree polynomial, given that is has the following zeros:
3, 4, -7
10) Find a minimum-degree polynomial, given that is has the following zeros:
1, -10, 10
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Math 110
In Class Exercises, Section 3.2
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11) Analyze the following function:
2
f  x    x  1
1. What is the degree of the polynomial? What is this polynomial's end behavior?
1. Find the x- and y- intercepts of the graph
2. Determine whether the function crosses, or touches/bounces at each intercept
3. Use a calculator to graph f(x)
4. Use the calculator to find any local minima / maxima that exist for the function
5. Use the graph to determine where f(x) is increasing, where it's decreasing, and if it's constant
anywhere. List those intervals here and clearly identify whether the function is increasing/decreasing/constant
on each interval you list.
6. Find the domain and range of f
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Math 110
In Class Exercises, Section 3.2
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12) Analyze the following function:
2
f  x    x 2  x  4   x  5
2. What is the degree of the polynomial? What is this polynomial's end behavior?
7. Find the x- and y- intercepts of the graph
8. Determine whether the function crosses, or touches/bounces at each intercept
9. Use a calculator to graph f(x)
10. Use the calculator to find any local minima / maxima that exist for the function
11. Use the graph to determine where f(x) is increasing, where it's decreasing, and if it's constant
anywhere. List those intervals here and clearly identify whether the function is increasing/decreasing/constant
on each interval you list.
12. Find the domain and range of f
Math 110
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