By using the 2^nd function feature of the key you can have your calculator find or calculate many values of interest.

When you press you see the CALCULATE menu shown below.



Option 1 allows you to type in an x–value and have the calculator give you the corresponding y-value. However, there is a quicker way to accomplish this with fewer keystrokes.

Suppose you have y = 2x² + 6x - 1 and the following graph of the function.



Now, what if you want to find out the y–value when x = 4? Press and then press .



When you press you will see the y–value at the bottom of the screen.



When using this option, the x–value must be in your viewing window, otherwise the calculator will display an error message. If you need help setting your window, click here.

So, when using the CALCULATE menu, you really do not need option 1. It just requires more keystrokes than the method shown here. However, the other options from this menu are extremely useful.

Option 2 allows you to find the zeros of a function. You may have also heard these referred to as roots or x – intercepts. No matter what you call it, this is where the graph touches the x – axis.

Let y = x² + x - 6 . By using factoring methods (this should link to Donna’s lesson on factoring trinomials) you can discover that the zeros are a x = 2 and x = -3. How can we use the calculator to come up with those same values?

To see the same screens shown below, graph y1 in a standard viewing window. You can get help with graphing or windows if you need it.

Press and you should see



By using this option, you have asked the calculator to find a zero for you. The calculator is now asking you to narrow its search. It begins by asking you for a Left Bound. Simply move your cursor with the arrow keys somewhere to the left of the zero. Since you’re pretty sure one zero occurs at x = -3, move the cursor to the left (or less than) that value. You will see ugly values at the bottom of your screen as you move the cursor. Don’t worry about this. Remember this can be avoided by using a friendly window, but it is not a concern for us during this lesson. The only important thing is to find a place to the left of the zero and press .

The calculator now wants a Right Bound. Use the to position the cursor to the right of the zero and press . You are now being asked for a Guess. You just have to press and the calculator will give you an answer.


This only gave you one of the zeros. You have to repeat the process to find the other zero of x = 2. Make sure you can do that here.

Of course, not all functions are as nice as the one we just worked with. Let’s look at y = 2x² - 4x - 3.

Use the zero feature to find the zeros at x = -0.58 and x = 2.58.




Depending on how you set your left and right bounds, you may see something strange for the y–value. Remember that when we are finding a zero (or x– intercept) that is where the y–value is zero. So when you see an x–value, the y–value should be zero. But look at the screen below. The y–value is not zero.



When you see y=-1E-12, this is the calculator notation for -1´10^-12. Changing this scientific notation into a decimal gives -0.000000000001. While mathematically speaking, -0.000000000001 is not equal to zero, it is REALLY close and acceptable for our needs. The calculator has limitations and sometimes this is as close to zero as it can get. And it will be close enough for us.

Use the zero feature to find the zeros of the following functions and make sure you get the answers given.

1. y = 6x² + 7x - 3

• x = -1.5 and x = 0.33
• Notice that by using factoring methods, you get exact answers of x = –3/2 and x = 1/3.

2. y = 2x² - 5x - 6

• x = -0.89 and x = 3.4
• Be careful. If you only look at the graph, you might think the zeros are at –1 and 3.5. Be sure to use the CALCULATE feature to find the precise values, not an eyeball estimate.

3. y = 4x² - 12x + 5

• x = 0.5 and x = 2.5
• It may be the case that you got 0.49999998 or 0.50000001 or some other value that was not exactly 0.5 (or 2.5). Don’t be fooled by these values. Just as there are sometimes calculator limitations on the y-value as we saw earlier, the calculator sometimes has limitations in finding the x–value exactly.

4. y = x³ - 2x² - 5x + 6

• x = -2, x = 1 and x = 3
• Notice that we have three zeros for this cubic equation. This means we have to use the zero feature three times instead of two.

5. y = 2x³ - 4x + 1

• x = -1.53, x = 0.26 and x = 1.27

6. y = 3x² + 4x² +2

• Notice the graph does not touch the x–axis. This means there are no real zeros for this function. You can find the non-real zeros by using the quadratic formula.

Finding zeros is just one feature of the CALCULATE menu. For finding a maximum or a minimum, go to Using the CALCULATE menu-Part II and for finding points of intersection, go to Using the CALCULATE menu-Part III.