and Functions Equations Part VII Script 3-04 --Slide 1 Title This section defines and investigates power and polynomial functions. Specifically, we will look at how the graphs of these functions look at the left and right ends. We can think of this as wha t the y values are doing as x gets really, really big and as x gets really, really small. --Slide 2 Title We will also see the connection between the roots of a function and its factored form. We will cover rational functions at the end of this section but we'll start with polynomial and power functions now. --Slide 3 Power function As it says, a power function will be defined to be a one-term function where the exponent on the x is a positive integer. I will use the words degree and coefficient a lot so know what they mean. --Slide 4 These are some examples and counterexamples of power functions. Notice what makes a function a power function. The top three examples all have positive integers for the exponent on the x. Notice this exponent is called the degree. Make sure you can pick out the degree from a power function. It will play a big part in what we do this section. The bottom three counterexamples are not power functions. You should be able to see why each is not a power function. Power functions have exactly one term and the exponent on the x must be a positive integer. --Slide 5 Polynomial function This is rather complicated looking but we can get through it. It needs to be complicated since it is trying to encapsulate every possible polynomial. We have actually seen this formula before in The Review Part II. First, notice the first term is a power function. In fact, every single term could be thought of as a power function. Notice how the subscript and exponent for each term match. Think through the exponents this way. The first term has the highest exponent. It must be a nonnegative integer. So maybe it's 10 or 5 or 3. The subscript of "a sub n" merely indicates it goes with this term. It really is just a label. Notice the exponents for the successive terms are labeled "n minus 1", "n minus 2", etcetera. This is a way to ensure the exponents of the terms are all integers. Again, the subscripts of the coefficients are merely labels to associate them to each term. We write all the terms where their exponents range from n to "n minus 1" to "n minus 2" to "n minus 3", all the way to "n minus n" or zero. Notice the last term could be thought of as "a sub 0, times x to the zero".
--Slide 6 expls: Here are some examples. Notice they all follow the general form given. The first two are the quadratic and linear functions we have seen before. Notice each term has a positive integer exponent. The last three examples are more complicated but are still polynomials. Consider "y equals x to the fifth, minus 4 x to the fourth, plus 2 x cubed, minus 7". Notice its first term has the largest exponent. The degree of the function is this 5. Each successive term's exponent can be thought of as "5 minus 1", "5 minus 2", "5 minus 3", "5 minus 4", and zero. Actually, you'll notice there are no "x squared" or x terms written. Imagine the "zero times x squared, plus zero x " terms in there. Then it follows the general form exactly. Now look at f and g. First, notice the coefficients can be under a radical sign, since they are still real numbers. However, the x should never be under a radical sign. This sometimes tricks students. Also, notice how the terms of g are not written in decreasing-exponent order. Since we could use the Commutativity of Addition to rewrite the terms, this is still a polynomial function. The big thing to remember is that the exponents on the x's must be nonnegative integers. --Slide 7 counterexamples: Here are some counterexamples. The first has a negative exponent so it's not a polynomial. Notice the second function has "the square root of x" which can be thought of as "x to the ". This breaks the positive integer exponent rule and so is not a polynomial. The third counterexample does not follow the general form at all. Notice the general form is a sum of many terms. There is no division in the general form of a polynomial. The third counterexample breaks this form. It is actually an example of a rational function, which we will see later in this section. --Slide 8 Terminology Use this terminology as you think through the problems. I will use it a lot so you've got to get used to it. The top thought bubble shows an example of a polynomial function. Alongside each definition, I've shown the applicable parts from the example. For instance, "y equals -4 x cubed, plus 2 x squared, minus 7" is a polynomial function. You can think of this as "y equals -4 x cubed, plus 2 x squared, plus 0 x, minus 7". Its coefficients are "-4, 2, 0, and -7". The leading term and coefficient will be very important in this section. Notice the leading term is merely the term with the highest exponent on the x's, not necessarily the first term written. So the leading term is "-4 x cubed". The leading coefficient is -4. The degree is 3.
--Slide 9 Characteristics of graphs Graphs of polynomial functions all share some basic characteristics. Keeping them in mind will aid graphing. First, let's discuss the domain. Remember, this is the set of x values that work in the function. In figuring domain, there are two things to worry about, division by zero and square roots of negative numbers. If ever an x value would make us do either of these two things, we exclude it from the domain. Think about the many polynomial functions we have seen. Since the x's are never under square root symbols or in denominators, these two things never come up. So the domain of any polynomial function is "all real numbers". Second, the graphs are always continuous . This means that we could trace the graph from the very left end to the very right end without lifting our pencil. The two pictures illustrate this. The left one can be traced without lifting the pencil. To trace the right one, you'd need to lift the pencil and replace it to span the gap. This right graph could not be a polynomial. --Slide 10 The third characteristic is that the graph will look like a smooth curve. It has no sharp corners. You'll recall the absolute value function looks similar to the graph on the right with that sharp corner. It is not a polynomial. This characteristic might be a little confusing. Sometimes the calculators do not have enough pixels and so the graph will appear as though it has sharp corners when it does not. When this happens, it's a good idea to look at the formula to see if it's a polynomial. The fourth characteristic of polynomials is their end behavior. It is what it sounds like. We will look at how the graph behaves at the left and right ends of the graph. We will investigate end behavior by doing several worksheets. --Slide 11 Worksheets The worksheet "Power functions: End behavior" looks at what is happening at the left and right ends of power function graphs. The notation used is important. An example follows this slide. The worksheet "Polynomial functions: End behavior" explains the notatio n needed and provides practice for finding end behavior. We will see that the end behavior of a polynomial or power function depends on the leading term. The third worksheet, "Connection between polynomial and power functions", breaks a polynomial function down into its terms. We will see how the leading term dominates the function's y values. This should develop an understanding of why we use the leading term to find the end behavior of polynomial functions. --Slide 12 End behavior notation This is an example of the notation we will use for both power and polynomial functions. Again, remember what we are interested in is how the graph behaves near the ends. Notice on the left end, as x approaches negative infinity, the graph is soaring upward. And, on the right end, as x approaches positive infinity, the graph is soaring upward.
--Slide 13 For the left end, we will say "as x approaches negative infinity, y approaches positive infinity". Notice the short- hand notation with the arrows. We read the arrow as "approaches". For the right end, we will say "as x approaches positive infinity, y approaches positive infinity". Again, the short-hand notation of this is given. Remember to include both statements when you denote end behavior. The first statement talks about the left end and the second talks about the right end. --Slide 14 Summary of end behavior Here we summarize the four types of polynomial end behavior. Notice our last example falls into the category with even degree and positive coefficient. leading You would see this more clearly if you multiplied it all out. --Slide 15 Roots and x-intercepts This is a new concept. We'll use this function to discuss the connection between a function's formula and its roots. It really is quite interesting how the facts come together. First, remember that an x-intercept is where the graph crosses the x-axis. Since this point is on the x-axis, the y value is zero. Since the x value makes y equal to zero, we will call it a zero or root of the function. Notice also f is in factored form. It is written as something, times something, times something as opposed to something, plus something, plus something. The three factors of f are x, "x minus 3", and "x plus 1". --Slide 16 The graph shows us that the three zeros of the function are -1, 0, and 3. The point here is to see the connection between these three zeros and the factored form of f. Three is a zero of f, and notice "x minus 3" is a factor of f. Likewise, -1 is a zero, and "x minus -1" or "x plus 1" is a factor. And, 0 is a zero, and "x minus 0" or x is a factor. This would always be the case. Every root or zero corresponds to a factor. --Slide 17 Let's look at this in general. The first statement is the theorem in English. The Root/Factor Theorem starts off by introducing a function f and a number r such that f(r) is zero. We can call this number r a root or zero of f. It is also an x-intercept when we graph f. The third part refers to f's formula. Just like we saw with our example, if r is a root, then "x minus r" is a factor. The thought bubble uses the function "y equals x, times x minus 3, quantity squared, times the quantity x plus 1" as f and shows the different ways we can think of a particular root, like 3.
--Slide 18 expl: #13, pg 340 This is a common problem. It seems complicated but, once you've got the idea, is relatively easy. We are asked to form a third-degree polynomial function that has the roots -3, 0, and 4. Using the theorem we just discussed, we can form the factors of f. Since -3 is a root, then "x minus -3" or "x plus 3" must be a factor. Since 0 is a root, then "x minus 0" or x must be a factor. Since 4 is a root, then "x minus 4" must be a factor. Multiply it all out to get the finished answer. I would probably multiply the two factors "x plus 3, times x minus 4", and then distribute the x through. Notice what you end up with is a function of degree three. --Slide 19 Turning points The points we have been calling local maxs and mins are also sometimes called turning points because it's where the graph turns. It is sometimes helpful to know that a function of degree n has at most "n minus 1" turning points. Occasionally, we'll use this fact. --Slide 20 expl: To complete our notes on polynomial and power functions, we will complete this problem. We are interested in the x and y intercepts as always, the end behavior, and the turning points. We are skipping multiplicity so we will skip parts b and f. If we multiplied this all out, we'd see that f is a fourth-degree polynomial with a negative leading coefficient. So the end behavior should be both ends pointing down. This knowledge will help you get a complete graph on your calculator. --Slide 21 We'll answer parts d and e on this slide. Start by graphing on the Standard window. You'll notice two pieces that look unconnected. But all polynomial graphs are continuous. And we know the end behavior will be both ends pointing down. So we know it connects somewhere above the screen. Increase your ymax to 15 so we get a complete graph. Turning points occur where the graph goes from increasing to decreasing or vice versa. The important thing to see here is that there is not a turning point at zero, as the graph decreases the whole way down. The only turning point is at (-3, 13.5). This is also called a local maximum. --Slide 22 We'll answer parts a and c on this slide. We find the x and y intercepts as we always have. Find the x-intercepts by plugging in zero for y and solving, or use the Root or Zero function on your calculator. You can also find the x-intercepts by noticing that x and "x plus 4" are factors of f. Find the y-intercept by plugging zero in for x. The book asks for the power function that resembles f at large values of x. This is merely the leading term, "y equals negative one- half x to the fourth". You may need to multiply it all out to see this. Also, remember to use the notation described here to denote the end behavior. The book does not ask for this but I will. --Slide 23 Try the homework. Make sure you look over your notes before attempting the homework. Notice the change in directions for many of the problems.
--Slide 24 Title Mainly this semester, we have been studying polynomial and power functions. This section covers another kind of function called rational functions. Of course, as always, we will be interested in their domain. Domain will play a large part in what will be called vertical asymptotes. We are skipping horizontal and oblique asymptotes. --Slide 25 Rational function Remember, a rational number is just a fraction, like or . A rational function is a fraction also, but the top...