Polynomial Functions and their Graphs

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1 College Algebra - MAT 11 Page: 1 Copright 009 Killoran Polnomial Functions and their Graphs Polnomial Functions have the general form: p./ D a n n C a n 1 n 1 C C a C a 1 C a 0 And the all share a common propert of being Continuous for all Real Numbers.R/ and Smooth. Definition 1 Continuous Functions: Functions that are defined for all real numbers, graphs that are created "mentall" where the pencil never leaves the paper These are NOT Pol Function Graphs Definition Smooth: Functions whose graphs have no "peaks" and sudden turns Eamples of what ou will NOT see in a Polnomial Function Graph: 1 End Behavior Looking at the graphs of f./ D and g./ D we see that as we go Far Left and Far Right, that each function shoots off to positive infinit. We call this the "End Behavior" of the Graphs, what is happening to the functions at ver large values (positive and negative) of. Sharp Corners Discontinuous f./ D g./ D The End Behavior of a Graph is determined b onl its highest degree term. When writen in decending powers it will be the lead term. Were both functions of the form: f./ D or g./ D then the end behavior of both graphs would be DOWN.

2 College Algebra - MAT 11 Page: Copright 009 Killoran Looking at the graphs of f./ D 3 and g./ D 5 f./ D 3 g./ D 5 Here we have "one up", "one down". If we looked at f./ D 3 the graph would be flipped over the -ais, but still one end would go to 1 and the other will go to C1: In general if the Degree of the Polnomial (the highest power of ) is Even, then the end behavior is both up " " or both down # #. Odd, then the end behavior is one up, one down # " or " #. Graph the base or base 3 to find the end behavior of these functions. (be careful if the lead (degree) term is negative!) Zero of a Function Definition 3 A zero of a function are all the values D c where f.c/ D 0: Also known as the graph of the function s -intercepts. Theorem 1 n-zeros Theorem: If the degree of p./ ; a polnomial function, is n; then the graph of p./ has at most n -intercepts. Eample 1 How man Zeros does the function g./ D 5 15 C C 05 8 have? Solution: Since the degree of the polnomial is n D 5, then it has at most 5 zeros. {0,1,,3,, or 5 zeros}

3 College Algebra - MAT 11 Page: 3 Copright 009 Killoran Finding Zeros can be done algebraicall, if the polnomial will factor: Eample Find the Zeros of the polnomial: p./ D C 5 First we will let p./ D 0 0 D C 5 and attempt to factor. In this case it is ver nice. Lets make the term positive: 0. 1/ D C 5. 1/ 0 D 5 C factors of that add to 5 0 D 1 0 D. 1/. C 1/. /. C / D f1; 1; ; g the zeros of p./ The graph of: p./ D C 5 is in the margin. Notice that the intercepts of the graph are the same as the zeros of the function. Also notice that the Degree of this Polnomial is, and there are 3 "turns" (3 relative ma/mins) in the graph. As a general rule the maimum number of turns of a polnomial is n 1; where n is the degree of the polnomial. It could be less # of turns. p./ D C 5 3 Roots with Multiplicit When we have a function with a perfect square or cube (or higher degree) these zeros are considered to have Multiplicit, (happen more than once). Though a repeated answer for an equation is unnecessar, it effects a graph dramaticall. Eample 3 Find the Zeros of f./ D. / and describe the Behavior of Graph at these point(s). The zeros of this function are D f; g since:. /. / D 0 Instead of writing a repetitive root, we state that D f w/ mult.of g Now the Graph of this is familiar, it is the Base Parabola moved units to the Right: We will describe the behavior of the Zero here as a BOUNCE. f./ D. /

4 College Algebra - MAT 11 Page: Copright 009 Killoran Eample Find the Zeros of g./ D. C 1/ 3 these point(s). and describe the Behavior of Graph at The zeros of the function are D f 1 w/ mult.of 3g The Graph of g./ is a Cubic Graph translated 1 unit to the left: We will describe the behavior of the Graph at the zero of D 1 as a LEVEL. What would happen if I put these functions together. f g/? Eample 5 Graph h./ D. /. C 1/ 3 b finding its Zero s, end behavior, and using the appropriate graphing behavior at these Zeros. The Zeros of h./ are D f w/ mult.of, become the -intercepts.; 0/ and. 1; 0/. 1 w/ mult.of 3g : On a graph these will The function h./ has a Degree of 5. This obtained from seeing that in the first binomial the highest power of would be and in the other it would be 3. Thus there will be a term 3 D 5 : Thus the end behavior will be one up-one down (low to high) 8 g./ D. C 1/ 3 Since the Multiplicit of 1 is 3 there will be a Level at D 1 and since the Multiplicit of is two, there will be a Bounce at D : Using all this information we get End Behavior Level at D 1 Bounce at D Notice that I did not label the -ais. All we can do is get a good feel for what this function is doing.

5 College Algebra - MAT 11 Page: 5 Copright 009 Killoran Intermediate Value Theorem: Theorem Given p./ with real coefficients and p.a/ p.b/ < 0 then there eists at least one c.a; b/ where p.c/ D 0 (there could be more than one such value for c) This can be demonstrated logicall if we remember that p./ is a function and continuous. Take two points on a graph such that.a; p.a// is above the -ais and.b; p.b// is below, one positive and one negative. Then the onl wa to connect the two points is either "straight line" and the line crosses the -ais between a and b or it curves around the endpoint to connect. a b The second condition we find right awa that it can NOT be the graph of a Function. So it must pass on the interval.a; b/ at some point D c so we have p.c/ D 0 since it is an -intercept. Eample Show that p./ D 35 C 50 has a zero (root) on the interval.; 3/ p./ D./ 35./ C 50 p./ D and p.3/ D.3/ 35.3/ C 50 p.3/ D 1 Since p./ p.3/ D ; there must eist at least one c.; 3/ such that p.c/ D 0