Exploring the End Behavior Of
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Honors Math 3 Name: Date:
Graphing Polynomials: End Behavior
Investigation
End behavior is the behavior of the graph of a function as x becomes infinitely large (+Ҙ or -Ҙ ).
You will need: a graphing calculator. Sketch a graph of each function and discuss with your groups any relationships that you notice among the functions degree, leading coefficient and end behavior.
2 3 4 5 f ( x ) x f ( x ) x f ( x ) x f ( x ) x f (x) x6
Does your conjecture change for the group of functions below? f (x) x2 f (x) x3 f (x) x4 f (x) x5 f (x) x6
Does your conjecture change for the group of functions below? f (x) 2x2 f (x) 2x3 f (x) 2x4 f (x) 2x5 f (x) 2x6 Summarize your conjecture:
2 Formalize your conjecture about the end behavior of a function of the form f (x) = ax n for each pair of conditions below.
a. when a > 0 and n is even b. when a < 0 and n is even
c. when a > 0 and n is odd d. when a < 0 and n is odd
n n - 1 If a polynomial function is written in standard form, f (x) = an x + an- 1x + ...+ a1x + a0, the leading coefficient is an . That is, the leading coefficient is the coefficient of the term of greatest degree in the polynomial.
The end behavior of a polynomial function depends on ______
Problem Set
1. Sketch the end behavior of the following functions. It’s okay if you’re not sure about the middle of the graph. a. f (x) = 3x 4 - 7x 3 - 13 b. f (x) = - 2x 3 + 3x 2 + 4x +1
2. Describe the end behavior of the graphs of the following functions. Decide if f (x) ® +/- Ҙ as x ® +/- Ҙ . You do NOT need to completely distribute the ones that are factored—all that matters is the term with the highest power of x.
as x ® Ҙ , f (x) ® ? as x ® -Ҙ , f (x) ® ? 3 a. f (x) = - 2x + 6x - 11 b. f (x) = x 4 - 5x 3 - x 2 + 2x +1
c. f (x) = 2(x - 1)(x + 3)(x - 5)
d. f (x) = - x 2(x + 7)(x - 3)
e. f (x) = 3x 2(x - 1)3 (x + 2)
3 3. The four functions below have each been graphed. Match the graphs with the functions without using your calculator. Use the x-intercepts and end behavior to guide you.
f (x) = (x + 3)(x - 1)(x - 5) h(x) = - x(x + 3)(x - 3) g(x) = 0.5(x + 5)(x +1)(x - 4) k(x) = - 1(x + 3)(x - 1)(x - 5)
30 30
A 20 B 20
10 10
0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10
-20 -20
-30 -30
30 30
C 20 D 20
10 10
0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10
-20 -20
-30 -30
4. Sketch a graph of each of the functions below. Plot the x- and y-intercepts and make sure that your end behavior is appropriate. You do not need to plot any additional points. The x-scale is one unit; the y-scale is ten units.
a. f (x) = 2(x - 5)(x +1)(x + 4) b. f (x) = - 0.5(x + 6)(x + 2)(x - 5)(x - 1)
50 50
30 30
10 10
-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8
-30 -30
-50 -50 4 c. f (x) = - 3x(x - 2)(x + 6) d. f (x) = 2x(x + 3)(x - 1)(2x - 7) 50 50
30 30
10 10
-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8
-30 -30
-50 -50
On the next two, you will need to factor the quadratic terms to find all the zeros. e. f (x) = (x - 5)(x +1)(x 2 - 3x + 2) f. f (x) = - 3(x 2 - 2x - 8)(2x 2 + 9x - 5)
50 50
30 30
10 10
-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8
-30 -30
-50 -50
You need to factor the next two fully before graphing. g. f (x) = x 3 - 2x 2 - 15x h. f (x) = - 3x 3 + 27x
50 50
30 30
10 10
-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8
-30 -30
-50 -50
6. Write a possible equation for each graph below. Use the form f (x) = a(x - __)(x - __)..... You do not need to find the value of a, but the end behavior should enable you to tell whether it is positive or negative. a. b.
30 30
20 20
10 10
0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10
-20 -20
-30 -30
c. d. 60 30
40 20
20 10
0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -20 -10
-40 -20
-60 -30
6