Calculator Help: The Calculate Menu – Part 1

I am using a TI-84 Plus Silver Edition for demonstration today; consult your manual for help locating similar menus and buttons on your device.

Before tackling this post, make sure you know how to input functions into your graphing calculator. If you need help with that or just need a reminder, review my blog post called Calculator Help: Input Functions. The very first section in my College Algebra textbook is called "Using Graphing Utilities." The applications portion of the exercise set requires students to find local extrema and intersections of functions. These two tasks are accomplished via the "Calculate" menu on most graphing calculators (TI-83 through TI-89 most specifically). This menu contains many other useful commands and I want to discuss all of them. Today, we'll begin by looking at the first four commands (value, zero, minimum, and maximum). Next time, we will cover commands 5 through 7 (intersect, derivative, and integral).

To find the Calculate menu, press 2ND then TRACE, which is located just below the graphing screen.

The Calculate menu contains seven commands: value, zero, minimum, maximum, intersect, derivative, and integral.

For practice, input the following function into your Y1 slot. Make sure all other function slots are empty and that all stat plots are turned off. (Refer to my post Calculator Help: Statistics for instructions on turning on/off stat plots.)

Use the following window values to obtain the graph seen below.

Now that we have a graph to work with, let's look at each of the Calculate commands individually.

HOW TO USE THE "VALUE" COMMAND

This command requires the input of an x-value and it will determine the y-value at that point. Let's try it out! In the Calculate menu, choose the first command 1: value. The calculator then prompts you for the x-value you want to evaluate. For this command to work, the x-value you enter must be within the viewing window.

Let's evaluate the function at x = 1 (which is in the viewing window of (-4, 8), so it is a valid input). Type in "1" and press ENTER.

The calculator will place an "x" mark on the point it located for us (circled in blue on the photo below) and identified the y-value that corresponds to the input (circled in red on the photo below).

Notice that the evaluated function is written at the top of this screen. If we input multiple functions, we can cycle through them after inputting an x-value by pressing the UP and DOWN arrow keys.

The "value" command is very helpful when you need to evaluate a function at a few points. For more than a few points, it is more helpful to consult the table, which we will look at in another post someday.

HOW TO USE THE "ZERO" COMMAND

This command confuses many students. The "zeroes" of a function are the x-values at which the function crosses the x-axis. This occurs whenever the y-value of the function is zero, hence the name. You may also see these values referred to as "x-intercepts." This command will NOT tell you the y-value at which x = 0 (which is the y-intercept). To find the y-intercept, use the "value" command we covered above and input "x = 0." To find the x-intercepts (a.k.a. zeroes) of a function, use the "zero" command. With the same function input into the calculator, return to the Calculate menu and choose the second command 2: zero.

The calculator places an "x" mark on the function (circled in blue on the photo above) and asks a question, "Left Bound?" (circled in red). The calculator is prompting you to move the cursor to a point on the graph that is "left bound" from the zero. Let's try to find the zero farthest to the left (circled in green above). To determine which side of the axis is left bound, place your finger on the zero and move it left, following the graph. From this point, left bound is below the x-axis. Use the LEFT arrow key to move the cursor to some point below the x-axis, left of the zero we want to identify. You can stop the cursor at any point left of the desired zero—there is no precise place to move it to. I usually stay fairly close to the axis while making sure I am definitely below it (below the axis, y-values are negative; above the axis, y-values are positive—use this to be sure you are below the axis in this case).

When you have positioned the cursor left of the desired zero, press ENTER. The calculator places a filled right-pointing arrow on the graph (circled in purple on the photo below) to mark where you have "saved" your cursor position and now prompts you to move the cursor to a point on the graph that is "right bound" from the zero (circled in red).

From the zero, right bound is above the x-axis, so move the cursor somewhere close, but above, the x-axis, then press ENTER.

Above, you see the next prompt that appears. The calculator is asking if you want to enter a guess for the answer. Entering a very close guess will save the calculator a bit of time finding the answer. You do not have to enter a guess. To continue from the guess screen without entering a guess, simply press ENTER. You can enter a guess by keying in an x-value or moving the cursor to the zero you are looking for. You can help the calculator find the zero faster also by choosing left and right bound points that are very close to the actual zero. The larger the interval you choose, the longer it will take the calculator to find your answer.

We were trying to find a point at which the y-value is zero, but here, it is giving us a point with a y-value that is not zero. The x-value for this zero is an estimate—it is a decimal. Therefore, the calculator evaluated the function at this estimated point. The y-value given is, however, very, very close to zero.

Try finding another zero; say the one in the middle. This time, if you begin at the zero and move left, you will be moving above the x-axis and if you move to the right, you will move below the x-axis. It is very important that, when entering left bound and right bound points, you pay attention to where the graph is located. Left bound will not always be below—as in this case, it is above! You should find this zero to be at the point (0.65, 0).

HOW TO USE THE "MINIMUM" COMMAND

The "minimum" command is used to find a local minimum point, which is the point that forms a "valley" in the graph. With the same function and window set in the calculator, navigate to the Calculate menu and choose command 3: minimum. Visually locate the minimum. This is the point we want to numerically identify.

You will notice that the prompts for this command are the same as the prompts for the previous "zero" command. The first prompt is "Left Bound?" Move the cursor using the LEFT or RIGHT arrow key to a point that is left of the minimum we want to identify. Anywhere left of the point is sufficient, but remember that the farther your cursor is set from the point, the longer the calculator will take to find the answer.

Press ENTER when you've chosen a point left of the minimum. The calculator will place a solid right-pointing arrow on the screen to mark your spot and bring up the next prompt: "Right Bound?" As before, move the cursor to the right of the minimum and press ENTER. The calculator will once again request a guess, which you can skip by pressing ENTER again.

My calculator estimated the minimum to be at the point (3.36, -24.75) as you can see below.

HOW TO USE THE "MAXIMUM" COMMAND

The "maximum" command is used to find a local maximum point, which is the point that forms a "hill" in the graph. It works exactly as the "minimum" command works. You will move the cursor to the left of the maximum, press ENTER, move the cursor to the right of the maximum, press ENTER again, and then press ENTER once more to move past the guess screen. Here is my answer of (-0.69, 8.60):

That covers the first four commands in the Calculate menu. These commands are very helpful when solving problems that involve complicated formulas without "pretty" (whole) numbers. They also help when you learn to find these values by hand, as you can check your answers! In the next post, we will cover commands 5 through 7, so stay tuned.

Trigonometry: Evaluating Base Angles - The Hand Method

Notice: On 12/6/2012, I uploaded a YouTube video that corresponds to this blog post. You can find it by following this link.

Most Trigonometry students are encouraged (and sometimes required) to memorize the "base angle" evaluations for the three fundamental trigonometric functions. These five core angles (in degrees) are 0, 30, 45, 60, and 90. The three fundamental trig functions are sine, cosine, and tangent. So students spend hours attempting to memorize 15 values... or they don't memorize them and do poorly on the test.

Of course, a calculator will give you these values quite easily. Memorizing them is useful though because (1) these angles show up frequently in introductory trigonometric and calculus courses and (2) having the answers memorized saves time on exams and homework. But memorization is a chore and is difficult for most students. That's why I really love this "hand method" of remembering (or rather, determining) these values. I did not create this--I learned it while taking Pre-Calculus my junior year of high school, and I've been using it ever since!

Each finger on the left hand with the palm facing towards you (thumb up, pinkie down) represents one of the five base or core angles. Visualize your hand as the first quadrant--the pinkie is in the 0 degree position and the thumb is in the 90 degree position. Then it is easy to see that the ringer finger is the 30 degree finger, the middle is the 45 degree finger, and the pointer represents the 60 degree finger. Here is a visual with angles in degrees and radians. Knowing the fingers in both measurements will save you the time it would take to convert all radian measures into degrees!

Once you have the fingers identified, we can begin evaluating the base angles, those found in the first quadrant. We will learn later in the post how to evaluate angles that are constant multiples of these base angles.

First, let's look at evaluating the cosine or sine of one of these core angles. In my visual below, I've chosen to evaluate at 60 degrees. Step 1 is to identify the finger that corresponds to the angle. Here, the pointer finger represents 60 degrees. Step 2 is to lower that finger, just fold it down, as in the image below. To find the cosine of this angle, simply count the number of fingers still standing above the lowered finger. Then, take the square root of this number and divide by 2. This is the cosine of the angle. To find the sine of this angle, count the number of fingers still standing below the lowered finger, which is 3 in this case. Take the square root of this number and divide by 2.

Easy enough, right? Evaluating tangent is simple too. You only have to do one extra thing. After lowering the required finger, flip your hand over so that the pinkie is on top and the thumb is on bottom. The fingers still represent the same angle values. Avoid confusion by lowering your finger first, then flipping your hand. After flipping your hand over, to find the tangent of the angle, count the number of fingers above (which is 3 here) and the number of fingers below (which is 1). Take the square root of the number above and divide by the square root of the number below.

See, I told you it was just as easy! Only one extra step, flipping the hand over. You can quickly evaluate any of these base angles at any of the three fundamental trig functions using this method. I've compiled a few for you to try. Evaluate these trig functions at the given angles using the hand method. Don't convert between radians and degrees--practice learning the fingers in both units. After you've evaluated all of them, use a calculator to check your answers (your calculator will give decimal approximations; enter your answer to see if the decimals match). Make sure your calculator is in the right mode for the type of angle you are evaluating!

You can also see the answers quickly by clicking here.

It is also possible to evaluate the trig functions at any constant multiple of the five core angles using this method. There is just an extra step involved, and that is to determine the sign (positive or negative) the answer will have. You know that all angles can be found on the unit circle. The unit circle contains 360 degrees (or 2pi radians). Angles between 0 and 90 degrees are in quadrant I of the standard x,y-plane; angles between 90 and 180 are in quadrant II; angles between 180 and 270 are in quadrant III; and angles between 270 and 360 are in quadrant IV.

But there are angles greater than 360 and angles less than 0. We discussed coterminal angles in another post. Review How to Find a Coterminal Angle in that post.

The first step to evaluating a trig function at a multiple of one of the five core angles is to determine which core angle is multiplied. To do this, find the angle that is coterminal with the given angle. For example, given the angle -405, the angle coterminal with this is 315 degrees. This angle is a multiple of 45 degrees: 315 = 7 * 45. Find this by trial and error--divide the angle by each of the five base angles. Whichever base angle gives a whole quotient, that is the base angle you are looking for.

The second step is to evaluate the trig function at the base angle you found.

The third step is to determine the quadrant of the original angle. Since -405 is coterminal with 315, and since 315 lies in the fourth quadrant, I know that -405 lies in quadrant IV.

Now, you can determine the sign of your final answer. Refer to the diagram below:

This little phrase will help you remember which trig functions are positive in which quadrants: All Students Take Calculus. "All" stands for, well, all. That is, in quadrant I, all of the trig functions are positive. "Students" stands for sine; so only the sine function is positive in the second quadrant. "Take" stands for tangent; so only the tangent function is positive in the third quadrant. And "Calculus" stands for cosine; so only cosine is positive in the fourth quadrant.

In my example, -405 degrees lies in the fourth quadrant where only cosine is positive. If I'm evaluating cosine at this particular angle, the answer is positive; if I'm evaluating sine or tangent at this angle, the answer is negative.

Here are a few examples:

In each instance, the steps are the same. First, identify the base angle by finding the angle that is coterminal to the given angle. Divide the given angle by each of the core angles to determine which angle to evaluate. Then determine which quadrant the angle is in and use it to determine the sign of the final answer.

Practice this method of angle evaluation enough, and it will be very helpful to you in future classes. After trigonometry, you will probably use it in calculus and maybe other subjects as well. You can impress your teachers or classmates with the speed at which you can answer evaluation questions. Of course, it only works for multiples of the five core angles, but you will see that these angles appear frequently in homework and test questions because trig functions evaluated at these angles yield nice numbers.

Edit: This post was edited on 8/16/2012 to correct a mistake in the "Examples" photo.

Edit #2: This post was edited on 12/6/2012 to add the notice at the top.