In this video, I’d like to show you how to use a graphing utility to model polynomial equations as well as identify the locations of maximum and minimum points on the graph.

Graphing calculators are common in the Algebra 2 classroom. Texas Instruments makes popular models like the TI-84 and the TI-Nspire that you may have seen, or you may have one of your own. The graphing utility I’d like to use in this video comes from the website www.desmos.com. This graphing calculator is web-based and free to use. You can also download the Desmos app for your mobile device. I particularly like the point-and-click interface that allows you to easily find key points on a graph.

If we revisit an equation from a prior lesson, \(y = x^3 - 3x^2 - 4x + 12\), you might recall its end behavior, y-intercept and x-intercepts. We can confirm this by navigating to www.desmos.com and clicking the “start graphing” button. The polynomial can be entered in the input rows on the left. I’ll demonstrate by using the keyboard on the screen… This graph looks a lot like the sketch we made in the last lesson. We have the ability to zoom in and out by using the buttons on the right, or by pinching the screen if your device is touch-enabled. Also, we can identify the intercepts by pressing the screen where the intercept appears to occur… I encourage you to revisit polynomial equations from earlier lessons and graph them for practice using a graphing calculator or desmos.com

In addition to verifying intercepts and polynomial end behavior, graphing utilities often give you the ability to find additional information about the polynomial. At times we would like to know the points in a function that are represented by the high and low spots of the function’s graph. These spots are called relative maximum values if the point appears to be at a high point of the graph, and they are called relative minimum values if the point appears to be at a low point of the graph. There can be many relative maxima and minima within a polynomial since each point is only compared to those that are relatively close to it. Consider this real-world example. Picture the tallest building in your town or city. Relatively speaking, it’s the tallest building, but if you compare the height of this building to another group of buildings in a larger city, there are possibly other buildings that are taller. By focusing on a small range of values at a time, we find relative maximum or minimum points.

Can you identify the locations of the relative maxima and minima in this graph? Pause the video and try to mark the positions of these extreme points. Resume playback in a moment to check your work… Identifying precise locations of these points can be difficult since the scale of our graph isn’t very detailed.

Return to desmos.com and graph the following equation… \( [ y = x^4 - 2x^3 + sin (3x^2) ]\). Tap the screen in each spot that you identified an extreme value, and you’ll see a fairly precise coordinate is plotted. The mathematics required to calculate thes points requires a more complicated discussion than what we’ll get into here. For now, I’ll leave you with one last problem which shows the importance of identifying relative maxima and minima.

Suppose an electonics manufacturer has found that when the price of a clock is set at x dollars, the revenue, or amount of money recieved from purchases, is modeled by the equation, \( y = -28x^ 2 + 645x\). The variable x represents the price for each clock, and y represents the revenue from selling these clocks. This revenue function accounts for the number of clocks sold a particular price due to supply and demand and was likely determined using a mathematical method of its own. What price should the manufacturer set for the clock, and how much money would the company bring in at this price? This question is a matter of being able to find the maximum value of this revenue function. Pause the video now, and use the techniques from this lesson to answer the question. Resume playback shortly to check your work… The manufacturer should set the price to approximately $11.52 which will generate a revenue of approximately $3,714.51.