 ## Getting the Most from Graphing Calculators

Your graphing calculator is a powerful tool. It can save you a lot of work if used properly but give you misleading information if misused. The FAQ list on this page will help you make the best use of your calculator. If there are other items which should be added, please let us know by email.

Note: This is NOT a "how-to" manual which tells you what to push in order to do calculations or generate graphs. Opportunities to learn the use of a graphing calculator are offered on a section-by-section and institution-by-insitution basis.

## The answer is 1/3. Is .333 correct?

No. The exact value of.333 is 333/1000 which is not the same as 1/3. In applying mathematics to physical reality, it suffices to work with good approximations. For many purposes .333 is just as good as 1/3. But they are not equal.

## Can I solve equations using my calculator?

Yes and no. Many hand-held calculators have built in "solvers". You can also use a graphing calculator to "see" where a function f(x) crosses the x-axis, thereby "solving" f(x)=0. But all calculations are limited by the precision of the calculator, and there are further limits to the accuracy of visual conclusions. Be aware that the solutions you obtain are approximations and that other methods must be used to get exact or symbolic answers.

## Can I use graphs to check answers?

Absolutely! Graphs can be effectively used to check for errors in algebraic calculations. For example, if you have erroneously computed that x2+ x + 1 has a zero at x = -1, a look at the graph on a scale between -2 and 1 will reveal that something is wrong.

We will use Calculus to draw conclusions about the behavior of functions. As an example, we will be interested in knowing when a functions is increasing or decreasing. This behavior is evident from a graph, so a graph can and should be used to check our conclusions.

## Does my calculator show asymptotes on graphs?

It shouldn't. But when graphing a function such as f(x) = 1/(x-1)2, which is undefined and has an asymptote at x = 1, the calculator will discover that the values of f near to 1 become very large. The calculator will then draw a spike at x = 1, and this often looks like an asymptote on the display. This is a bug, not a feature.

## Could the graph I see be completely wrong?

Sadly, yes. There are two reasons.

1. USER ERRORS: We all make mistakes. Common mistakes made when graphing functions are:

• Using the Wrong Window. If we set our range incorrectly, we might, for example, observe that a function has no root when the root we want is outside of our window.

• Using degrees instead of radians. The argument of all trigonometric functions in Calculus is measured in radians. Using degrees is equivalent to scaling the independent variable and will result in erroneous calculations and graphs.

• Using the Wrong Function: Sometimes the wrong function is graphed because of a typo or out of confusion among several stored functions. Other times, the mistake occurs in setting up the problem. For example, if a story problem really asks you to investigate some property of sin2(x) but you set it up wrong and think you need to look at tan(x), then your graphs will lead you to wrong conclusions.

TO AVOID USER ERRORS, WORK CAREFULLY AND TAKE TIME TO DOUBLE-CHECK WHAT YOU DO UNDERSTAND AND ASK OR READ ABOUT WHAT YOU DO NOT UNDERSTAND.

2. DESIGN CHARACTERISTICS: It is rare, but it can happen that a graphing calculator displays an extremely misleading graph. One cause is numerical precision. If the calculation needed to determine the y-coordinate of a point on a graph exceeds the precision of the calculator, the results will be unpredictable. Another cause is the calculator's display. The calculator (or computer acting as a calculator) displays graphs on a screen made up of tiny dots called pixels. Once a scale is set, each pixel actually represents a small range of values. The graph of a function cannot show large changes in the function value over this range.

## What are the dangers of using really small numbers?

You will get complete garbage if your calculations involve numbers smaller than the precision of your calculator. What makes this truly insidious is that sometimes a calculation involves really small numbers in a hidden way. For example, if you are attempting to compute the value of sin(x3)/x3, then a small number will be CUBED before sine is evaluated. This cubing makes the number even smaller. So to compute with numbers as small as 10-6, you will need a calculator which can handle numbers as small as 10-18.  