How to Graph Functions on TI-83/84

Summary: It’s pretty easy to produce some kind of graph on the TI-83/84 for a given function. This page helps you with the tricks that might not be obvious. You’ll be able to find asymptotes, intercepts, intersections, roots, and so on.

See also: How to Evaluate Functions with TI-83/84
How to Graph Piecewise Functions on TI-83/84

The techniques in this note will work with any function, but for purposes of illustration, we’ll use

Graphing Your Function

Step 1: Clear unwanted plots.

You need to look for any previously set plots that might interfere with your new one.	Press [Y=] (the top left button). Look at the top of the screen. If any of Plot1 Plot2 Plot3 is highlighted, cursor to it and press [ENTER] to deactivate it. (No information is lost; you can always go back and reactivate any plot.) To verify that you have deactivated the plot, cursor away from it and check that it’s not highlighted.
(Sometimes you might want to graph more than one function on the same axes. In this case, make sure to deactivate all the functions you don’t want to graph.)	Now check the lines starting with Y1=, Y2=, and so on. If any = sign is highlighted, either delete the whole equation or deactivate it but leave it in memory. To delete an equation, cursor to it and press the [CLEAR] button. To deactivate it without deleting it, cursor to its = sign and press [ENTER].   My screen looked like this after I deactivated all old plots and functions.

Step 2: Enter the function.

If your function is not already in y= form, use algebra to transform it before proceeding.   Two cautions: For x, use the [x,T,θ,n] key, not the [×] (times) key. The TI-83/84 follows the standard order of operations. If there are operations on top or bottom of a fraction, you must use parentheses — for x + 2 divided by x − 3, you can’t just enter “x+2/x−3”.	Cursor to one of the Y= lines, press [CLEAR] if necessary, and enter the function.
Check your function and correct any mistakes.   For example, if you see a star * in place of an X, you accidentally used the times key instead of [x,T,θ,n].	Use the [◄] key and overtype any mistakes.   To delete any extra characters, press [DEL].   If you need to insert characters, locate yellow INS above the [DEL] key. Press [2nd DEL makes INS] and type the additional characters. As soon as you use a cursor key, the TI-83/84 goes back to overtype mode.

Step 3: Display the graph.

“Zoom Standard” is usually a good starting point. It selects standard parameters of −10 to +10 for x and y.

Press [ZOOM] [6].

Common Problems

If you don’t see your function graph anywhere, your window is probably restricted to a region of the xy plane the graph just doesn’t happen to go through. Depending on the function, one of these techniques will work:

• ZoomFit is a good first try. Press [ZOOM] [0]. (Thanks to Marilyn Webb for this suggestion.)

• You can try to zoom out (like going higher to see more of the xy plane) by pressing [ZOOM] [3] [ENTER].

• Finally, you can directly adjust the window to select a specific region.

For other problems, please see TI-83/84 Troubleshooting.

Tuning Your Graph

You can make lots of adjustments to improve your view of the function graph.

Zooming

The window is your field of view into the xy plane, and there are two main ways to adjust it. This section talks about zooming, which is easy and covers most situations. The next section talks about manually adjusting the window parameters for complete flexibility.

Here’s a summary of the zooming techniques you’re likely to use:

• You’ve already met standard zoom, which is [ZOOM] [6]. It’s a good starting point for most graphs.

• You’ve also met zoom fit, which is [ZOOM] [0]. It slides the view field up or down to bring the function graph into view, and it may also stretch or shrink the graph vertically.

• To zoom out, getting a larger field of view with less detail, press [ZOOM] [3] [ENTER]. You’ll see the graph again, with a blinking zoom cursor. You can press [ENTER] again to zoom out even further.

• To zoom in, focusing in on a part of the graph with more detail, press [ZOOM] [2] but don’t press [ENTER] yet. The graph redisplays with a blinking zoom cursor in the middle of the screen. Use the arrow keys to move the zoom cursor to the part of the graph you want to focus on, and then press [ENTER]. After the graph redisplays, you still have a blinking zoom cursor and you can move it again and press [ENTER] for even more detail.

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• Your viewing window is rectangular, not square. When your x and y axes have the same numerical settings the graph is actually stretched by 50% horizontally. If you want a plot where the x and y axes are to the same scale, press [ZOOM] [5] for square zoom.

There are still more variations on zooming. Some long winter evening, you can read about them in the manual.

Adjusting the Window

You may want to adjust the window parameters to see more of the graph, to focus in on just one part, or to get more or fewer tick marks. If so, press [WINDOW].

• Xmin and Xmax are the left and right edges of the window.
• Xscl controls the spacing of tick marks on the x axis. For instance, Xscl=2 puts tick marks every 2 units on the x axis. A bigger Xscl spaces the tick marks farther apart, and a smaller Xscl places them closer together.
• Ymin and Ymax are the bottom and top edges of the window.
• Yscl spaces the tick marks on the y axis.
• Xres is a number 1–8 inclusive. With 1, the default, the calculator will find the y value at x-values corresponding to every pixel along the x axis. With 2, the calculation occurs every 2 pixels, and so on. Higher values draw graphs faster, but fine details may be lost. My advice is, just leave this at 1.

Color TI-84s have two additional window parameters:

• Δx is the x distance between the centers of adjacent pixels. The calculator determines this automatically from Xmin and Xmax, so you don’t need to mess with it. However, if you do change it, the calculator will then determine Xmax from Xmin and Δx.
• TraceStep is the step size when you press ◄ or ► while tracing along a graph. By default it’s twice the value of Δ, but you can change it if you want to.

To blow up a part of the graph for a more detailed view, increase Xmin or Ymin or both, or reduce Xmax or Ymax. Then press [GRAPH].

If you want to see more of the xy plane, compressed to a smaller scale, reduce Xmin and/or Ymin, or increase Xmax or Ymax. Then press [GRAPH].

The graph windows shown in your textbook may have small numbers printed at the four edges. To make your graphing window look like the one in the textbook, press [WINDOW] and use the numbers at left and right edges for Xmin and Xmax, the number at the bottom edge for Ymin, and the number at the top edge for Ymax.

Adjusting the Grid

The grid is the dots (dots or lines, in color TI-84s) over the whole window that line up to the tick marks on the axes, kind of like graph paper. The grid helps you see the coordinates of points on the graph.

If you have a black&white TI-83/84, and you see a lot of horizontal lines running across the graph, it means your Xscl is way too small, and the tick marks are running together in lines. Similarly, Yscl is the number of y units between tick marks. A bunch of vertical lines means your Yscl is too small. Press [WINDOW] and fix either of these problems.

To turn the grid on or off:

Locate yellow FORMAT above the [ZOOM] key. Press [2nd ZOOM makes FORMAT].   Cursor to the desired GridOn or GridOff setting, and press [ENTER] to lock it in.   Then press [GRAPH] to return to your graph.

Color TI-84s can present the grid as dots or lines. On the [2nd ZOOM makes FORMAT] screen, you can choose GridOff, GridDot, or GridLine, and you can also assign a color to the grid.

Exploring Your Graph

Domain and Asymptotes

First off, just look at the shape of the graph. A vertical asymptote should stick out like a sore thumb, such as x = 3 with this function. (Confirm vertical asymptotes by checking the function definition. Putting x = 3 in the function definition makes the denominator equal zero, which tells you that you have an asymptote.)

Color TI-84s have the ability to detect asymptotes: press [2nd ZOOM makes FORMAT] and change Detect Asymptotes to On. That often creates a more realistic picture of the graph, as in this case, but it can also make it harder to see an asymptote. Here are both versions:

The domain certainly excludes any x values where there are vertical asymptotes. But additional values may also be excluded, even if they’re not so obvious in the graph. For instance, the graph of f(x) = (x³+1)/(x+1) looks like a simple parabola, but the domain does not include x = −1.

Horizontal asymptotes are usually obvious. But sometimes an apparent asymptote really isn’t one, just looks like it because your field of view is too small or too large. Always do some algebra work to confirm the asymptotes. This function seems to have y = 1 as a horizontal asymptote as x gets very small or very large, and in fact from the function definition you can see that that’s true.

Function Values

While displaying your graph, press [TRACE] and then the x value you’re interested in. The TI-83/84 will move the cursor to that point on the graph, and will display the corresponding y value at the bottom.

The x value must be within the current viewing window. If you get the message ERR:INVALID, press [1] for Quit. Then adjust your viewing window and try again.

Intercepts

You can trace along the graph to find any intercept. The intercepts of a graph are where it crosses or touches an axis:

Most often it’s the x intercepts you’re interested in, because the x intercepts of the graph y = f(x) are the solutions to the equation f(x) = 0, also known as the zeroes of the function.

To find x intercepts:  You could naïvely press [TRACE] and cursor left and right, zooming in to make a closer approximation. But it’s much easier to make the TI-83/84 find the intercept for you.

Locate an x intercept by eye. For instance, this graph seems to have an x intercept somewhere between x = −3 and x = −1.	Locate yellow CALC above the [TRACE] key. Press [2nd TRACE makes CALC] [2]. (You select 2:zero because the x intercepts are zeroes of the function.)
Enter the left and right bounds.	[(-)] 3 [ENTER] [(-)] 1 [ENTER]   There’s no need to make a guess; just press [ENTER] again.

Two cautions with x intercepts:

• Since the TI-83/84 does approximations, you must always check the TI-83/84 answer in the function definition to make sure that y comes out exactly 0.
• When you find x intercepts, make sure to find all of them. This particular function has only one in its entire domain, but with other functions you may have to look for additional x intercepts outside the viewing area.

Finding the y intercept is even easier: press [TRACE] 0 and read off the y intercept.

This y intercept looks like it’s about −2/3, and by plugging x = 0 in the function definition you see that the intercept is exactly −2/3.

Multiple Functions

You can plot multiple functions on the same screen. Simply press [Y=] and enter the second function next to Y2=. Press [GRAPH] to see the two graphs together.

To select which function to trace along, press [▲] or [▼]. The upper left corner shows which function you’re tracing.

Intersection

When you graph multiple functions on the same set of axes, you can have the TI-83/84 tell you where the graphs intersect. This is equivalent to solving a system of equations graphically.

The naïve approach is to trace along one graph until it crosses the other, but again you can do better. We’ll illustrate by finding the intersections of y =(6/5)x − 8 with the function we’ve already graphed.

Graph both functions on the same set of axes. Zoom out if necessary to find all solutions.	Press [2nd TRACE makes CALC] [5].   You’ll be prompted First curve? If necessary, press [▲] or [▼] to select one of the curves you’re interested in. Press [ENTER].   You’ll be prompted Second curve? If necessary, press [▲] or [▼] to select the other curve you’re interested in. Press [ENTER].
Eyeball an approximate solution. For instance, in this graph there seems to be a solution around x = 2.	When prompted Guess?, enter your guess. In this case, since your guess is 2 you should press 2 [ENTER].
Repeat for any other solutions.

As always, you should confirm apparent solutions by substituting in both equations. The TI-83/84 uses a method of successive approximations, which may create an ugly decimal when in fact there’s an exact solution as a fraction or radical.