David holds a Master of Arts in Education
What Does Finding the Zero Mean?
Imagine waking up one morning at 7am and the thermometer reads -3 degrees. Later that day at 5pm it reads 12 degrees. Can you use logic to assume that at some point during the day the thermometer read zero degrees? Well of course you can.
It's that same type of logic that allows a graphing calculator to find the zeros of a function/polynomial. Those zeros are the points in which the function moves from being negative to positive.
If you have a function with two variables then you can graph the solutions onto a x,y coordinate grid. The point(s) on the graph that are of interest in this lesson are the points that cross the x-axis. This point goes by several names: x-intercept, root, and zero. To find the zero of a function means to determine the x value that when inputted into the function will yield the y value of zero. Keep in mind that a function may have none, one, or more zeros.
When is the Calculator Necessary?
When the function is a lower order polynomial such as a linear or quadratic, a graphing calculator is not necessary. The zeros of these functions be easily found without one. For example, f(x) = 2x+1 is a linear function. You can find the zero of this function by substituting f(x) with 0 and then solving for x.
2x + 1 = 0 subtract the 1,
2x = -1, divide by 2,
x = -1/2.
-1/2 would be the zero of this function.
When the functions are not lower order polynomials then the process is greatly aided by the use of a graphing calculator.
How to Use a Graphing Calculator to Find the Zero
Note: There are many graphing calculators out there but the most common one is the TI-83, and TI-84 models, so we will refer to those. If you have another type, you may have to select different buttons.
The three buttons you will need to use are:
- the 'Y=',
- the 'GRAPH', and
- the 'CALC'.
Take a moment to find where these buttons are. The 'Y=' button is the top left button on the calculator. The 'GRAPH' button is the top right on the calculator. The 'CALC' button shares the button with the 'TRACE' button, which is the second one from the right in the top row.
Notice how the 'CALC' button is written above the button in a different color. This means that it is a second function button. To access 'CALC' you should select the 2ND button and then select 'TRACE/CALC' button.
Step One: Entering the Function.
This step requires the use of the 'Y=' button. Once that button is selected it will show on the screen several 'Y='s with different subscripts. If there is already something after the equal sign in any of the 'Y='s then clear them out. Be sure that your function/polynomial is solved for y and enter the function into the calculator.
Step Two: Graphing the Function.
This is done by selecting the 'GRAPH' button. You will need to make sure that the window that the calculator is showing is the correct size. If you cannot see the point(s) where the graph intersects the x-axis; adjust the window until you do see it. You can adjust the window by selecting 'ZOOM' or 'WINDOW' buttons on the top row.
Step Three: Calculating the Zero.
You are now telling your calculator to find the x at which the y value goes from negative to positive.
- You should select the 'CALC' button which requires the 2ND button first. It will give you a list of options.
- Choose zero (on some calculators this will say 'root'). The calculator will go back to the graph screen and the have the words 'Left Bound'.
- Move the cursor to a point that is left of the x intercept that you are interested in. Push 'ENTER'. It will then say 'Right Bound'.
- Move the cursor to the right of the same x-intercept and push 'ENTER'.
- It will then say 'Guess', hit 'ENTER' again.
You should get the word 'zero' on the bottom of the screen with a x and y value. The y should always say zero. The x will be your answer. Functions can have more than one x-intercepts. Repeat step three for all of the x intercepts.
Use a graphing calculator to find the zeros of y - 7 = 3 x^3 + 2x^2 - 5x
1. Enter the function into the calculator. It is not in 'Y=' form so we should add 7 to both sides.
y = 3x^3 + 2x^2 - 5x + 7
Hit the 'Y=' button on the calculator and enter the function.
2. Graph the function
Hit the 'GRAPH' button. You should see the graph with only one x-intercept, somewhere around -2. If you do not see this adjust your window or check your function to be sure that you entered the correct one.
3. Calculate the zero.
Hit the '2ND' button followed by the 'CALC' button. The screen should show a menu.
- Choose the 'zero' option
- Move the cursor left of the zero, hit 'ENTER'
- Move the cursor right of the zero, hit 'ENTER'
- The screen says 'Guess', hit 'ENTER'.
- x = -2.04 y = 0 (The y is not exactly zero, but it is such a tiny number that it rounds to zero)
- The zero is (-2.04,0)
Many functions are very difficult to solve for zero without the use of a graphing calculator. With the help of the graphing calculator the zeros are just a few buttons away.
- First you will enter the equation into the calculator using the 'Y=' button (top left).
- Second you should 'GRAPH' (top right) the function to see where the x-intercepts (zeros) are located.
- Thirdly use the calculate zero menu option found under the '2ND' and then 'CALC' button (second from the right in the top row) to have the computer find the x value in which the y value changes from negative to positive.
If your function has x-intercepts that are outside of the window your calculator is set on, you will need to adjust the size of the window using the WINDOW/ZOOM buttons.
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