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Saxon Calculus Homeschool: Online Textbook Help30 chapters | 247 lessons

Instructor:
*Michael Eckert*

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

The scientific calculator is a great tool in both increasing the accuracy and speed at which one can practice a plethora of mathematical operations. In this lesson, we seek to find the intersection points of two functions: y = x^2 and y = 2x using the Texas Instruments TI-84 Scientific / Graphing Calculator.

Learning how to use a **Texas Instruments Graphing Calculator (e.g. TI-83, 84 or 92)** is important in math because it serves as a means to quickly and accurately check work that might otherwise be done manually. Furthermore, using a graphic calculator increases the speed at which one can practice such operations. In this lesson, we are going to use the TI-84 to:

- graph two functions and
- to find if there exist any points (x, y) where the two functions y = x2 and y = 2x intersect within the Cartesian coordinate system.

Before we can even get started, let's look at the face of a TI-84 to make sure that we are familiar with some of the commands necessary in dealing with functions of lines e.g. graphing and/or finding their points of intersection. For the purposes of this lesson, we are going to be using the following commands: **MODE, WINDOW, Y =, GRAPH, TRACE, and CALC**. These commands can be seen on the face of the TI-84 as shown above.

Before we get started graphing y = x2 and y=2x, it is necessary to make sure that the calculator MODE is set correctly. In other words, whether we are able to deal with decimals or just whole numbers, radians or degrees, general functions or parametric equations, real numbers or imaginary, etc. Below is a picture for the MODE settings we desire for this exercise highlighted in bold.

Before we put our functions into the TI-84 to be graphed, we must also make sure that our Cartesian coordinate system is suitable for their projection; therefore, we set up the WINDOW as follows:

Xmin = -10, Xmax = 10, Xscl = 1, Ymin = -10, Ymax = 10, Yscl = 1 and Xres = 1.

Xscl and Yscl determine how many tic-marks that we will have along the x and y-axis . Along each axis, we will have 10 in the negative direction from the origin (0,0) and 10 in the positive direction from the origin along both the x and y-axis. Xres is a variable we can just take to be 1.

Now that the coordinate system is to scale, we can now plot the graphs of y = x2 and 2x. We plug y = x2 into Y1 = and 2x into Y2 = and simply press the GRAPH button.

This is the image projected onto our screen after pressing the GRAPH command Note that the image will take a few seconds to load.

Once we have our two functions' lines projected onto the screen, we now have everything we need to find the exact two points where these two functions intersect one another. At first glance, it is easy to see that they intersect at two points and perhaps we could estimate their location; however, in the event that these points were not whole numbers or if we were dealing with more complex functions, it is easy to just use the intersection command. To access this, we press 2ND and then press CALC. We then scroll down to command 5: intersect and press ENTER.

After a projection of the graph appears, we will then be prompted by the TI-84 to press ENTER. As seen below in the projection of the two graphs: it will read 'First curve?'.

We will then be prompted again to press ENTER by 'Second curve?'.

Last, we are given one more prompt 'Guess?' before pressing ENTER for the third and final time.

We are given our first point of intersection at (0, 0).

To find the second point we TRACE near to it -explained below:

Above: Press the TRACE command. By using the side to side arrow keys on the face of the TI-84, we will move a visible dot along either function in the direction of the second point of intersection. As before, we use the intersect command to find this second point.

We repeat: 2ND, CALC, 5: intersect and then we follow the three intersection prompts: 'First curve?' ENTER, followed by 'Second curve?' ENTER, followed by 'Guess?' ENTER. We are given the second point of intersection at (2,4).

In this lesson, we obtained the two points of intersection of the two functions y = x^2 and y = 2x by using the graphing feature on the TI-84 Scientific Calculator. By merely putting the equations of the two functions into our calculator, using the graph and intersection feature, we were able to obtain intersection points of (0,0) and (2,4). What once was done manually using algebraic manipulation can now be done with a scientific calculator (in this case with the TI-84 Texas Instrument). This is just one of the operations that can be quickly and accurately performed on the TI-84, either in practicing mathematical operations or in applied mathematics.

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Saxon Calculus Homeschool: Online Textbook Help30 chapters | 247 lessons

- Basic Features & Functions of a Graphing Calculator
- How to Evaluate & Graph Functions on a Graphing Calculator
- Working With Functions on a Graphing Calculator
- Evaluating Exponentials & Logarithms on a Graphing Calculator
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- Approximating Slopes of Curves on a Graphing Calculator
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