Graphing Functions on TI-83/84
Copyright © 2001–2017 by Stan Brown
Copyright © 2001–2017 by Stan Brown
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.
Evaluating Functions with TI-83/84
Graphing Piecewise Functions on TI-83/84
The techniques in this note will work with any function, but for purposes of illustration, we’ll use
Step 1: Clear unwanted plots.
|You need to look for any previously set plots that might interfere with your new one.||Press [|
|(Sometimes you might want to graph more than one function on the same axis. In this case, make sure to deactivate all the functions you don’t want to graph.)||Now check the lines starting with |
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.
|Cursor to one of the |
|Check your function and correct any mistakes.
For example, if you see a star
|Use the [|
To delete any extra characters, press [
If you need to insert characters, locate yellow
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 [|
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.
0]. (Thanks to Marilyn Webb for this
You can try to zoom out (like going
higher to see more of the xy plane) by pressing
Finally, you can directly adjust the window to select a specific region.
For other problems, please see TI-83/84 Troubleshooting.
You can make lots of adjustments to improve your view of the function graph.
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
6]. It’s a good starting point for
You’ve also met zoom fit, which is
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
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 [
2] but don’t
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.
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
5] for square zoom.
There are still more variations on zooming. Some long winter evening, you can read about them in the manual.
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 [
Xmaxare the left and right edges of the window.
Xsclcontrols the spacing of tick marks on the x axis. For instance,
Xscl=2puts tick marks every 2 units on the x axis. A bigger
Xsclspaces the tick marks farther apart, and a smaller
Xsclplaces them closer together.
Ymaxare the bottom and top edges of the window.
Ysclspaces the tick marks on the y axis.
If you want to blow up a part of the graph for a more detailed
Ymin or both,
Ymax. Then press
If you want to see more of the xy plane, compressed to a
smaller scale, reduce
Ymax. Then press
Many of the graph windows shown in your textbook will have small
numbers printed at the four edges. If you want to make your graphing window look
like the one in the textbook, press use the numbers at left and right
Xmax, the number at
the bottom edge for
Ymin, and the number at the top edge
The grid is the dots 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 see a lot of horizontal lines running
across the graph, your
Xscl is way too
small, and the tick marks are running together in lines.
Yscl is the number of y units between
tick marks. A bunch of vertical lines means your
is too small. Press [
WINDOW] and fix either of these
|To turn the grid on or off:||
Locate yellow |
Cursor to the desired
Then press [
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.)
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. 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.
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
adjust your viewing window and try again.
You can trace along the graph to find any intercept. The intercepts of a graph are where it crosses or touches an axis:
|x intercept||where graph crosses or touches x axis||because y = 0|
|y intercept||where graph crosses or touches y axis||because x = 0|
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 |
|Enter the left and right bounds.||
There’s no need to make a guess; just press [
Two cautions with x intercepts:
Finding the y intercept is even easier:
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.
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
To select which function to trace along, press
▲] or [
▼]. The upper left corner
shows which function you’re tracing.
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.
You’ll be prompted
You’ll be prompted
|Eyeball an approximate solution. For instance, in this graph there seems to be a solution around x = 2.||
When prompted |
|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.
Updates and new info: http://BrownMath.com/ti83/