A previous post on solving equations with the TI-84 has proved very popular, so here are some tips and tricks when using the GDC for drawing graphs and associated functionality.
It’s easy enough to type in an equation and then press the graph button. However, the most important thing to do which helps make sense of any graph is to set up the correct window: the minimum and maximum value of the displayed coordinates, and the associated scale marks. The TI-84 will always display the last window which was used, which will probably be unsuitable for your current graph. So I suggest that instead of pressing the graph button immediately, press either Zoom:6 for standard axes (x and y both scaled from -10 to 10, with a tick mark every 1), or Zoom:7 for trigonometric axes (x scaled from -360º to 360º, tick marks every 90º, and y from -4 to 4, tick marks every 1).
For trig graphs, make sure you check first whether you should be in radians or degrees—the x scale will match the calculator setting. Incidentally, Zoom:8 (square scales) will show you the graph’s true shape—sometimes useful.
Setting the Window
But what if you don’t know what window to set? For example, you could easily miss turning points which are off the screen. Let’s see what happens if we draw the graph of y = 3x3 + x2 – 10x + 5.
Clearly some of the graph is missing, and we could use the x-axis more efficiently. We need to alter the window to focus on the x values around zero, and to bring the maximum point into view, but where is the maximum? We could use the Calc function to find it, but a more general method for setting up a window is to use a table of values. For the same function, here’s a table of values starting with x = -3.
(Use Tblset to choose the start value for x, and the step value, and then Table to view it).
We can see that y goes up to about 13 (it could be a bit more – the maximum value is not necessarily at x = -1). So, I think I’ll set the window as follows: x from -3 to 3 in steps of 1, y from -5 to 15 in steps of 1. Press the graph button, and we get …
…which is a much more useful display.
The zoom box
Another problem occurs when there’s graph activity close to the x-axis, and you need to zoom in to see what’s going on. For example, using the Standard Zoom, the graph of y = 4x3 – 12x2 + 11x – 3 looks like this …..
We need to enlarge the area around x = 1. We could reset the window, as before, but the zoom box is also a very useful, easy-to-use tool. Press Zoom:1, and then use the cursor keys to position the top left corner of the zoom box— press Enter. Then use the cursor keys again to drag the zoom box out to its bottom right corner (make the zoom rectangle roughly screen shaped) and press Enter again. The two images below show the display before and after pressing the second Enter.
Once you have drawn a graph, the Calc button offers 7 options connected to the graph:
1: value: enter an x value and the GDC returns the y value. For example, if you enter 2, you will get f(2).
2: zero: will return the x values where the graph cuts the axis, ie the solutions of f(x) = 0. Since only one value at a time is returned you are first asked for a value to the left of the zero (left bound), and then one to the right of the zero (right bound). You can either type these numbers are use the cursor to set them on the screen. Finally, you are asked for a guess – just press Enter.
3: minimum and 4: maximum. Use these to find turning points, going through the same procedure as for a zero.
5: intersect: Find the point(s) of intersection of two graphs. First you need to identify the graphs (you may have more than two on the screen), and then input a guess which is somewhere close to the point of intersection.
6: dy/dx: find the value of the gradient by inputting an x value.
7: : Enter a left bound and a right bound to evaluate the area under the curve.
Note that for all these options you will get an error if the x values you enter are off the screen.