<p> Honors Math 3 Name: Date:</p><p>Graphing Polynomials: End Behavior</p><p>Investigation</p><p>End behavior is the behavior of the graph of a function as x becomes infinitely large (+Ҙ or -Ҙ ).</p><p>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.</p><p>2 3 4 5 f ( x )  x f ( x )  x f ( x )  x f ( x )  x f (x)  x6</p><p>   </p><p>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</p><p>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:</p><p>2 Formalize your conjecture about the end behavior of a function of the form f (x) = ax n for each pair of conditions below.</p><p> a. when a > 0 and n is even b. when a < 0 and n is even</p><p> c. when a > 0 and n is odd d. when a < 0 and n is odd</p><p> 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.</p><p>The end behavior of a polynomial function depends on ______</p><p>Problem Set</p><p>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</p><p>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.</p><p> 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</p><p> c. f (x) = 2(x - 1)(x + 3)(x - 5) </p><p> d. f (x) = - x 2(x + 7)(x - 3)</p><p> e. f (x) = 3x 2(x - 1)3 (x + 2)</p><p>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. </p><p> 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)</p><p>30 30</p><p>A 20 B 20</p><p>10 10</p><p>0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10</p><p>-20 -20</p><p>-30 -30</p><p>30 30</p><p>C 20 D 20</p><p>10 10</p><p>0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10</p><p>-20 -20</p><p>-30 -30</p><p>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. </p><p> a. f (x) = 2(x - 5)(x +1)(x + 4) b. f (x) = - 0.5(x + 6)(x + 2)(x - 5)(x - 1)</p><p>50 50</p><p>30 30</p><p>10 10</p><p>-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8</p><p>-30 -30</p><p>-50 -50 4 c. f (x) = - 3x(x - 2)(x + 6) d. f (x) = 2x(x + 3)(x - 1)(2x - 7) 50 50</p><p>30 30</p><p>10 10</p><p>-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8</p><p>-30 -30</p><p>-50 -50</p><p>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)</p><p>50 50</p><p>30 30</p><p>10 10</p><p>-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8</p><p>-30 -30</p><p>-50 -50</p><p>You need to factor the next two fully before graphing. g. f (x) = x 3 - 2x 2 - 15x h. f (x) = - 3x 3 + 27x</p><p>50 50</p><p>30 30</p><p>10 10</p><p>-8 -6 -4 -2 -10 0 2 4 6 8 -8 -6 -4 -2 -10 0 2 4 6 8</p><p>-30 -30</p><p>-50 -50</p><p>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.</p><p>30 30</p><p>20 20</p><p>10 10</p><p>0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -10 -10</p><p>-20 -20</p><p>-30 -30</p><p> c. d. 60 30</p><p>40 20</p><p>20 10</p><p>0 0 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8 -20 -10</p><p>-40 -20</p><p>-60 -30</p><p>6</p>