*Kelly Sjol*

# How to Solve Equations by Graphing on the CLEP Scientific Calculator

## Graphing on the CLEP Calculator

Let's take a quick look at graphing on the CLEP calculator.

You can enter the graphing mode on the CLEP calculator by choosing the mode that says 'Navigate: My Graphs', or, you can click on the little 'graph' button in the upper right corner. When you enter this mode, well, it looks pretty sparse. It just says My Graphs at the top, has a big blank space and then a bunch of buttons at the bottom: 'New', 'Window', 'Clear', 'Done', 'Table' and 'Graph'. You can try to graph right now, but all you'll get is this piece of, basically, graph paper.

In order to graph a function, you have to click 'New'. When you click 'New', you're first given a choice of 'Function', 'Polar' or 'Parametric'. Usually, you'll want to choose 'Function'; in fact, for the entirety of this calculus course, you'll want to choose only 'Function'. Not 'Polar', and not 'Parametric'. So let's pick 'Function' for now.

Now this window is exactly like the Solver window, *y(x):*, except that now you're not going to type *'y* equals something,' you're just going to type what's on the right side, so *'y(x):*' is just like *'y(x)* equals whatever you type.'

So let's graph the cosine of *x* times *pi*. Once we've typed it in here, cos(*x***pi*), we're going to click 'OK'. This takes us back to the My Graphs window, except now we have *y1*:cos(*x***pi*). This also gives us a couple of other options: First, there are the checkboxes on the far left. If this box is checked, then any time we go to our graphs screen, it will show this function. If it's not checked, this function won't show up on the graph. This column here shows you the type of line that will be graphed, and this shows the color of our graph. So in this case, we expect it to be a red solid line for *y*=cos(*x***pi*).

So let's click 'Graph' and see what this looks like. Sure enough, there's our graph of cos(*x***pi*) as a red, solid line. Let's zoom in. I'm going to click 'Zoom', and then 'Trig'. That gives me a graph from *x*=(-2)*pi* to *x*=2 *pi*, but let's zoom in a little bit further. I'm just going to click where I want to zoom. Looks like that's a little too far; let's zoom out some. Okay.

So now we're looking at this graph, and we've got our standard, default floating cursor for this graph. Now this cursor will give you the *x* and *y* coordinates of any point on this graph. But I don't really care that this point might be 3.2 or this point might be 2 *pi* in the *x* direction and 2 in the *y* direction. I want to know where the value of this graph is for certain values of *x*. So I'm going to choose 'Analysis: Trace/Evaluate'. And now I'm going to hover over various points along this graph.

So as soon as I choose 'Trace/Evaluate', I get this bottom window that first shows me what line I'm looking at. In this case, I'm looking at the red line, which is *y1*, which is equal to cos(*x***pi*). So any point is going to be along this line. Below that, I have the *x* and *y* coordinates of the point on that line that is denoted with this *x*. So I can trace the line, increasing *x*, and watch what happens to *y*.

## Finding the Zero Value

There are other options under the 'Analysis' menu, such as 'Zero', 'Minimum' and 'Maximum'. So let's click on 'Zero'. First, it says down in the bottom right-hand corner that I'm doing a zero analysis. What this does is it allows me to choose a range. So I click once to create the left side of the range, and I drag over and release to select the entire range. And what the calculator will do is it will find the first point where my function equals zero in that range. So here it's picked the point *x*=2.5. Obviously *y*=0, because I was looking for a point where *y* equaled zero in my range. Once you're in this analysis, you can select as many ranges as you want and find what *x* is for all of these zero, or root, values.

## Finding the Minimum Values

Let's try finding a minimum. It's the same process here. Now I click to select one side of my range, and I drag to select the entire range. So here, the range goes from this line to this line, and the calculator will find the minimum point within that range. You can do the same thing with maximum.

## Graphing Two Equations

Okay, so let's hit 'Done' for a second, and let's put in a second equation. So 'New', (yes, it's a function) *y(x) * will be *x* cubed minus 2 times *x*. Okay, this will be another red line. Wait a minute, well, which one is which? Let's go back and change it so that instead of looking at a red line, I'm looking at a blue or purple line for *x*^3-2**x*. That's much better.

So now, let's click 'Analysis: Trace' again. This time, as before, it gives me this lower window where it shows the function that I'm currently tracing: in this case, the red function, *y1*= cos(*x***pi*). If I want to instead trace my other function, I select this down arrow, and I click on *y2*. Now I can trace the *x*^3-2**x* line.

## Finding Intersections

Because I have multiple lines now on this graph, I have another analysis option: 'Intersection'. So let's select that one.

Notice now that it gives me, in this lower window, two places for functions. That's in case I have three, four or five functions - I can indicate which two functions I want to find the intersection of. So I want to find the intersection of *y1* and *y2*. I'm going to select a range, first by clicking to define one side of the range, and then, by letting go, to find the other. And the calculator will calculate, or find, an intersection within that range. So here, it says that these two functions intersect at the point (1,-1). Let's zoom out, though.

Okay, it turns out that these two functions intersect at at least three points. I see one here, here and here. So how might I calculate the other intersections? Let's go back and choose 'Intersection' and highlight the region on the left, here. Okay, the calculator found that the intersection is at an *x* value of -1.45 and a *y* value of -0.15 or so. I can find the middle intersection by highlighting the region with that, or I can find the right intersection by highlighting some region that includes that intersection. Okay, this is getting a little bit confusing.

So let's take a closer look at the *x*^3-2**x* graph, and let's not graph cos(*x***pi*). So back in the My Graphs window, I'm going to deselect *y1*. Because it's deselected, when I click on 'Graph', it won't be graphed.

Okay, so here I have my single function. Let's take a quick look at the minimum and maximum values. So now I'm going to find the minimum of this function. But what do you think is going to happen if I select this region? Within this region, where is the minimum? So let's find the minimum of this region. The minimum value of *x*^3-2**x* is actually on the endpoint, right here. The minimum of this larger region is right here. If I select the maximum, the same type of thing happens. First, I can find this local maximum, here, or in a larger region the maximum is up here. In fact, the maximum depends on how large my region is.

## Lesson Summary

So let's review. With the CLEP calculator, you can graph a whole bunch of explicit functions. When you graph these, you can find the intersections of multiple functions, or you can find the maximum or minimum values of a particular function within some region. And you can also find the zero values - that is, the roots - where this function crosses the *x* axis.

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