Week 2 – Graphing Polynomial Functions
This week’s fun thing (click to reveal)
A limerick is a type of poem, usually humorous, that has 5 lines, in which the first, second, and fifth lines rhyme together and are longer, and the third and fourth lines rhyme together and are shorter. Here’s an example:
A tall yet unassuming human being,
Wondered just what he was seeing,
His neighbor aghast,
Ran away really fast,
Shouting, “Don’t look around while you’re peeing!”
Can you come up with a Limerick that matches this mathematical equation??
((12 + 144 + 20 + 3*sqrt(4))/7)+(5*11) = 81 + 0
The solution is at the bottom of this page.
Up until now the primary skill we’ve developed with polynomials is how to factor them. We are building towards the ultimate goal, which you’ll have ideally mastered by the end of the week, of being able to graph most any polynomial function. Beyond just factoring polynomials, we also need to be able to extract important information from them just by looking at them. By examining the degree of the polynomial and the sign (positive/negative) of the leading coefficient (the coefficient of the term of the highest degree), we can immediately determine what is called the “end behavior” of the polynomial. The end behavior, simply put, is what is happening on either end of the polynomial. When we draw graphs, we typically draw arrows on either end of the curve to indicate that it keeps going on in the manner shown. End behavior describes where those arrows are going. The standard way we will describe end behavior is Limit Notation. Limit notation is something that is integral to calculus, and yet is a pretty simple idea. We can use limit notation to talk about the how a function behaves as it approaches a certain value. If we wanted to know what was happening to the function f(x) when x was 2, we would just evaluate the function by plugging in 2 for x. Alternatively, we can think about what is happening to the function as x approaches a value of 2 (from either side of the 2). By comparing those behaviors, we can make important conclusions about the function itself. For this class, however, we are focusing on using limit notation for end behavior, or x values of positive and negative infinity (what’s happening at the end…). Since we can’t actually plug in a value of infinity for x, limit notation is how we indicate what is happening at the ends of a function.
Odd degree polynomials have opposite behavior at their two ends, while even degree polynomials have the same behavior at both ends. All odd degree polynomials with positive leading coefficients start in the bottom left of the graph and progress towards the top right, just like y=x (an odd degree polynomial with a positive leading coefficient). Odd degree polynomials with negative leading coefficients go from the top left to the bottom right (like y=-x). Even degree polynomials with positive leading coefficients open upwards, meaning both ends go up, just like y=x^2. Even degree polynomials with negative leading coefficients open downwards.
For any polynomial given to us, we can systematically determine the end behavior (by analyzing the degree and the leading coefficient), the y-intercept, and x-intercept(s) (which usually requires some sort of factoring). The standard form of a polynomial is not particularly useful when it comes to graphing, so for that reason we want to factor it to identify the real zeros (x-intercepts). Once we know the end behavior of the polynomial and where it crosses the x and y axes, we can sketch in what the graph of the polynomial should look like. We can be more precise, but doing so typically requires a calculator (it is possible to do this by hand, though extremely tedious). We will use a calculator a bit later to provide more detail and more “calculus-oriented” information about the polynomial.
We can also work the other way, by looking at the graph of a polynomial, taking information from it, and generating the polynomial function it represents. The information we can take off of a graph are the real zeros (x-intercepts), the degree of the polynomial and the sign of the leading coefficient, and any number of other points (by looking at points on the grid which the curve passes through). We can take each real zero and write the corresponding factor (example, an x intercept of -1 corresponds to a factor of (x+1)), and then use any of the (x,y) coordinate points besides the x-intercepts to solve our equation for a, which is a “vertical modifier” of the function.
When we discover that there is an factor of a polynomial that is raised to a power greater than 1, we know that the corresponding zero has a “multiplicity”. For example, f(x)=(x-2)^2 has two solutions because it is a 2nd degree polynomial, but both solutions are 2. We can’t just say that the solution, or zero, is 2, because that would mean the graph crosses the x-axis at 2. We know that it doesn’t; it instead kind of “bounces” off the x-axis. The graph is a parabola with it’s vertex at the point (2,0). Even multiplicities “bounce” off the x-axis, while odd multiplicities cross the x-axis, but sort of curve along it for a while before crossing it. Check out this desmos graph for a demo.
Practice problems for basic polynomial graphing: THIS PDF, and PG 401 #5, 11, 18 (just find zeros and sketch graph)
In calculus, there is certain information about a polynomial that is useful when analyzing different things about it. I’ll save what those “things” are for you calculus teacher, but we can develop the ability to find that certain information in this class quite easily. We would like to know when a polynomial is greater than zero (positive, above the x-axis) and less than zero (negative, below the x-axis), as well as when the polynomial is increasing (the slope of the line drawn tangent to the curve is positive, or it is “going up the hill” if you consider the graph a roller coaster being ridden from left to right) and when the polynomial is decreasing (the slope of the line drawn tangent to the curve is negative, or going down the hill).
We can use the x-intercepts to determine when the polynomial is positive and when it is negative. In order to determine when the polynomial is increasing or decreasing, we need to be able to precisely identify the coordinates of what are known as “turning points”. Turning points of a polynomial are also often called relative minimums and maximums. These are the “humps” that you will often see on the graph of a polynomial. It is possible to find these coordinates by hand, but that is a very tedious process and our calculators enable us to do this very quickly. Once we find these coordinates, we can use them to write, in interval notation, during what parts of the domain of the polynomial function it is increasing, and during which parts it is decreasing.
For even-degree polynomials, it is also important to know the coordinates of these turning points, as they will be crucial in determining the polynomial’s range. While odd degree polynomials have a domain and range of negative infinity to positive infinity, even degree polynomials have a limited range, which will be governed by the y-coordinate of the turning point farthest from the x-axis.
Graphing Polynomial Functions
(video 2 of 3)
Graphing Polynomial Functions
(video 3 of 3)
Practice problems for graphing polynomial functions: PG401 #27, 30, 39 (find ALL information and sketch graph. This includes where f(x)>0, f(x)<0, f(x) is increasing, f(x) is decreasing. You may use desmos to find coordinate of turning points, but you should familiarize yourself with the process on a graphing calculator in the event you need to do this in the future and are not allowed to use desmos.)
When you’re ready, complete this google forms survey meant to check your understanding of the material presented this week. This is mandatory and should be done by Sunday, April 12th at 11:59PM (or before). This is not a quiz, will not be graded for correctness, but is worth points.
Solution to this week’s fun thing (click to reveal)
A dozen, a gross, and score,
Plus three times the square root of four,
Divided by seven,
Plus five times eleven,
Is nine squared and not any more.