Polynomial Functions I

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1 Name Student ID Number Group Name Group Members Polnomial Functions I 1. Sketch mm() =, nn() = 3, ss() =, and tt() = 5 on the set of aes below. Label each function on the graph Defn: A function ff that can be written in the standard form ff() = aa nn nn + aa nn 1 nn aa 1 + aa 0, with real number coefficients aa 0, aa 1,, aa nn where aa nn 0 and n is a nonnegative integer, is called a polnomial function. The degree of a polnomial function is n, the leading coefficient is aa nn, and the leading term is aa nn nn. The domain of a polnomial function is the set of all real numbers. The End Behavior of a Polnomial Graph: Given a polnomial ff() with leading term aa nn nn and n 1, When the degree n is even: 1. If the leading coefficient is positive (a > 0), as, ff() and as, ff(). If the leading coefficient is negative (a < 0), as, ff() and as, ff() When the degree n is odd: 1. If the leading coefficient is positive (a > 0), as, ff() and as, ff(). If the leading coefficient is negative (a < 0), as, ff() and as, ff()

2 Defn: The real roots of a function f are the real number solutions to the equation ff() = 0. The comple roots of a function f are all of the solutions to the equation ff() = 0. Ever nth-degree polnomial has eactl n comple roots counting multiplicities. Properties of the Multiplicit of a Real Root The multiplicit of a root is the number of times ( rr) appears in the root form of a polnomial function. The graph of a polnomial function passes through the -ais at a real root if the multiplicit of the root is odd and touches but does not pass through the -ais if the root s multiplicit is even. 1. Given the polnomial function jj() = , complete the following: a. End behavior: as, jj() as, jj() Eplain how ou know: b. Is the function odd, even, or neither? Show our work. c. Find the real roots of jj() b factoring, then sketch the graph d. Consider the function kk() = What characteristics of kk() are similar to jj()? What characteristics of kk() are different from jj()? Use a dashed line to sketch a graph of kk() on the same aes as jj().

3 Turning Points of Polnomial Graphs If ff() is a polnomial function of degree nn, then the graph of ff has at most nn 1 turning points. If the graph of a function ff has nn 1 turning points, then the degree of ff() is at least nn.. Given the polnomial function pp() = 3 5, complete the following: a. End behavior: as, pp() as, pp() Eplain how ou know: b. What is the maimum number of turning points possible? c. Is the function odd, even, or neither? Show our work. d. Find the real roots of pp() b factoring. Then sketch the graph on the aes below e. Consider the function qq() = 3 5. What characteristics of qq() are similar to pp()? What characteristics of qq() are different from pp()? Use a dashed line to sketch a graph of qq() on the same aes as pp(). 3

4 3. Given the polnomial function gg() = + 9, complete the following: a. End behavior: as, gg() as, gg() Eplain how ou know: b. What is the maimum number of turning points possible? c. Is the function odd, even, or neither? Show our work. d. Find the real roots of gg() b factoring. Then sketch the graph on the aes below e. Consider the function h() = What characteristics of h() are similar to gg()? What characteristics of h() are different from gg()? Use a dashed line to sketch a graph of h() on the same aes as gg().

5 . What is the significance of finding the roots of functions with regard to the graph? 5. What is the significance of noting whether the function is odd, even, or neither with regard to the graph? Be specific.. A function is considered to be positive when the graph is above the -ais and negative when the graph is below the -ais. Remember that function values are values. Consider the functions jj() from #1 and gg() from #3. Determine the intervals that make the following inequalities true. Write our answers in interval notation when appropriate. jj() 0: jj() > 0 jj() < 0 gg() > 0: gg() 0 gg() = 0 Intermediate Value Theorem Let ff be a polnomial function. For aa < bb, if ff(aa) and ff(bb) have opposite signs, then ff has at least one zero on the interval [aa, bb]. The IVT states that there must be a zero between a and b on the graphs below. On the second graph, draw in some polnomial that goes through both points. Is there an wa ou can draw this polnomial without going through the -ais? f(b) (b, f(b)) (b, f(b)) f(a) (a, f(a)) (a, f(a)) 5

6 Here is one eample of a polnomial that goes through the given points (on page 5). Note that because polnomials are continuous (we do not lift our pencil when drawing it), we MUST go through the -ais to connect the two points. On the graph below, circle the -intercept that we knew had to eist because of the given points and the IVT. f(b) (b, f(b)) f(a) (a, f(a)) 7. Determine whether or not the IVT guarantees that the function ff() = has a zero on the given interval. SHOW our work and eplain what it means. a. [ 3, ] b. [0,1] c. [1,]