GRAPHING CALCULATOR



GRAPHING CALCULATOR

HANDBOOK

PRECALCULUS

NHHS Level 3

This handbook orients you to different functions of your graphing calculator. The first time you go through the handbook, work out everything on your calculator as it is explained so you experience firsthand the information and techniques. You are expected to be fluent with your graphing calculator in this course, so please see your teacher if you have any difficulty. You are expected to learn everything in this handbook, you may not use it as a reference on any in-class assessments.

HOME SCREEN

The home screen is what you get when you turn the calculator on and you press the keys [2nd][mode] and if you also press [clear] then the screen you see is blank with a flashing cursor in the upper left corner. This is the screen that you do mathematical calculations on.

The ANS feature

When carrying out calculations you can easily use the answer from one calculation to another. In the second line here by just typing “x” (multiply) the previous answer will be multiplied by what you type in. In the third line you can recall the answer by typing in [2nd][(-)] after you type the square root [2nd][x2].

Syntax error

There are a lot of different error messages, the best option to troubleshoot is to choose “go to” and it will bring you to the error. In this case a subtract symbol was used instead of the negative symbol.

EQUATION SCREEN

Press [y = ] and this is where you type in equations usually to graph or to examine in a table.

You type in a variable by using the variable key [X,T,θ,n] next to [ALPHA].

WINDOW SCREEN

Press [WINDOW] and this screen tells you the parameters of the graph you will see when you press [GRAPH]. Xmin tells how far to the left on the x-axis is shown, Xmax how far to the right, Ymin how far down and Ymax have far up on the y-axis. Xscl and Yscl tell you what intervals are marked on the axes. If you don’t want marks you can make them both zero. And Xres just indicates the resolution. The higher the number, the poorer the resolution but the quicker it graphs. I keep Xres = 1.

Choosing a Window

There are some “preset” window options found by going to [zoom]. You will get the standard window (see above) with [zoom][6]. When you graph trig you use [zoom][7], when you graph scatter plots after inputting data you use [zoom][9] ZoomStat. NEVER use [zoom][0] ZoomFit. This option distorts your graph and does NOT find a good window for you to analyze a function. Better that you manually create a window.

Evaluate a Function Value Suppose you have a function f(x) = -5x3 + 6x2 + 8x – 9 and you want to calculate f(x) if x = 4. There are three ways to do this.

(1) On the home screen retype the function with 4 inserted for x. This is great if you are going to evaluate just one value and not do anything else with the function.

(2) Type the function into the equation screen and graph it. Press [trace] then [4] enter. As long as your x-value is within in the window domain you’ll get a result. If it is outside the xmin-xmax domain just adjust your window.

(3) Use your table. After typing in your equation in [y=] go to table set up [2nd][window] and change the independent variable to ask. Then go to the table [2nd][graph] and type in 4 for x (if there are already values in the x column just delete them using [del]

There is a fourth funky option using the home screen and [vars] that your teacher can tell you about.

STORING STUFF

You can store values, up to 26 of them! You can also store equations (up to 100 of them) and pictures/graphs (up to 100 of them). Storing values is very important when you are doing calculations with no intermediate rounding(which should be all the time).

Example – suppose you have to solve lnx3 – 8 = 17 and you want to check your result by plugging it into the original equation. You go through all your solving steps and get 4160.262005…on your screen. You can store the full value (including all the digits you don’t see) by typing [sto >]{alpha][A] (you choose any letter you wish!) [enter]

You store equations and graphs through the DRAW MENU. Press [2nd][PRGM] to get to this menu. Then go over to STO.

If you want to store a graph (picture) select [1] and then type the number Pic you are going to store it in. (you can choose storage locations 0 through 9) If you want to store a set of equations (from [y=]) select [3] and type the number GDB you are going to store it in.

Then to recall either equations or graphs go through the DRAW MENU. Press [2nd][PRGM] to get to this menu. Then go over to STO. Press [2] and the Pic # to recall your picture. You can recall more than one picture at a time. Press [4] and the GDB # to recall your equations. Caution – when you recall your equations it wipes out any equations you have in the [y=] screen.

When you recall the PICS – they will recall on top of each other if you recall more than one at a time. If you recall a pic and then want to recall another without the first one being on the screen you can clear the first by going to [2nd][DRAW] and then press #1 ClrDraw – this clears your previous pic from the screen (not from the storage spot so don’t’ worry!)

**Whenever you store something (value, equation, picture) it will boot out anything already stored in the place you choose to store it in.

SEND & RECEIVE PROGRAMS – You will be loading the DEFAULTS program on your calculator among others this year. This program clears out data that you have input and puts the settings back to standard. It does not delete other programs and does not delete matrices or pictures in the drawing menu. But it is very useful especially if you play games on your calculator (which can change settings) or use your calculator in another class in which you input data (like statistics – just be sure you are done with the data before you run defaults). To send & receive programs you need two calculators (one with the program you want) and a connection cord.

1) attach cord to both calculators - be sure they are in tight!

2) press [2nd][X,T,θ,n] on both

3) calc which is receiving arrow right to receive and press [enter] and it will tell you it is

“waiting”

4) calc with program press [3]

5) arrow down to Defaults and press [enter]

6) arrow over to transmit and press [enter]

The program should transfer quickly and you are done. Press [2nd][mode] to get back to home screen on both calculators.

RUNNING DEFAULTS To run the program, just press [PRGM] and press the number in front of DEFAULTS and then press [ENTER]. You should see a coordinate plane graph flash on the screen and then you will see the word “Done” in the upper right corner. You can press [CLEAR] to get rid of that.

GRAPHING

We will use the graphing function of your calculator a lot in this class! You should be able to use the graph function to determine various characteristics of any function. It’s important that you first find the best window for your function (see previous).

You should be able to graph a function and find a window that shows the full function shape.

Most of your graph analysis is done using the CALCULATE MENU found at [2nd][trace].

Using the function f(x) = x2 + 3x – 12

FINDING THE VERTEX – this is either the maximum or minimum of the function. In this example it is the minimum. To find it graphically press [2nd][TRACE]. This screen gives you all kinds of calculating options for your graph. You choose [3] in this example. You will be asked a series of questions.

LEFT BOUND? Move the cursor to the left of the vertex with the left arrow key. Then press [ENTER].

RIGHT BOUND? Move the cursor to the right of the vertex with the right arrow key. Press [ENTER].

GUESS? Ha! We don’t have to guess, that’s silly. Just press [ENTER].

On the bottom of the screen you will have an x = # and y = #. See my snapshot below, The vertex is (-1.5, -14.25)

BE ALERT TO ARTIFACTS! Artifacts are the calculator making rounding errors. For example – the x-value on my screen for the previous example is -1.49999999 which is an artifact. It should be -1.5. You can even check that by pressing [TRACE][-1.5][ENTER] and you will get the same y-value. You might have gotten an artifact like -1.5000003 instead, again that is just -1.5 rounded funny by the calculator.

Another artifact example – type in [y=] –x2 – 4.125 press graph and you should see a downward opening parabola on the y-axis (guess what the x-value of the vertex is? In fact do you know the vertex easily without the calculator?). Use the steps above to find the vertex graphically (but use “maximum” this time). My calculator does NOT show the expected vertex of x = 0 and y = -4.125. Instead of zero for the x-value there is a weird scientific notation number – which is actually an artifact! In scientific notation it is a really really small number – and so is zero. Basically a weird scientific notation value with a negative E is the calculator having a hard time saying “zero”.

FINDING THE SOLUTIONS – Solutions to a function are also called “zeros” and zeros are where the function crosses the x-axis. Graph y = x2 + 3x – 12 again. (adjust that window if you have to). There are two zeros, you have to calculate each separately. Press [2nd][trace] to get to the CALCULATE screen again. Press [2] zeros. You get a series of questions here too. First decide which zero you are going to calculate first. LEFT BOUND move the cursor to the left of that zero. Then press [ENTER]. RIGHT BOUND move the cursor to the right of that zero. Press [ENTER] and then ignore the “Guess” directive and press [ENTER] again.

One solution is 2.275. then you repeat the process to find the other solution.

And the other solution is -5.275. Keep in mind, these same solutions can be found using the quadratic formula. And you will be learning other non-graphing methods for other polynomial functions this year.

FINDING X WHEN F(X) IS GIVEN (and how to find the intersection of two functions)

Let’s work with 3x2 + 5x – 9 and find all the values of x when f(x) = -2. Unless the values are whole numbers the table function won’t work that well. Instead press [y=] and in Y2 = type in -2 (the original equation is in Y1). Press graph and you should see a horizontal line go across the graph (you don’t really need to adjust the window because you don’t need the vertex now. You need to find where the horizontal line intersects with the original function. Press [2nd][TRACE] – CALCULATE screen again, this time choose [5] intersect. And you get prompted with questions again. This time you are asked FIRST CURVE – look in the upper left corner and it shows you which function your cursor is on (hopefully Y1). So press [ENTER] Then it asks you SECOND CURVE and the Y2 equation should show in the upper left corner – so press [ENTER] again. GUESS? Press [ENTER] and one of the intersections will appear. So one value for x when f(x) = -2 is 0.907.

And you have to find the other intersection also. When you do the other one you have to move the left arrow to get your cursor closer to the other intersection when you are identifying the curves.

You can also find these values using the quadratic formula – but you must put the equation in standard form first.

HIDDEN BEHAVIOR

Sometimes there is something going on in your graph that is not especially clear. You have to examine these graphs more carefully using ZOOM. Graph y = 0.65x2 – 8.1x + 25.5. (I hope you ran DEFAULTS first). You get a nice parabola but it looks sort of flat on the bottom. Is it touching the x-axis or just skimming it? Press [ZOOM] [2] this is zoom in and then move the cursor over to the right – right near the vertex of the parabola. Then press [ENTER]. Ah ha – it does not intersect the x-axis! So there are no real zeros. Find the vertex….

Some Practice:

Now let’s examine some other examples (run DEFAULTS before each one)….ANSWERS BELOW….

1) Here’s another example of hidden behavior you should try. You’ll have to do some window adjusting and zooming in (sometimes you have to zoom in more than once….). Find both the vertex and the zeros (answers below). Y = 0.86x2 + 16.69x + 80.9

2) f(x) = -9x2 + 12x – 85 - find the vertex & zeros……you may need to adjust the window here, remember “c” is the y-intercept.

3) Find the vertex & zeros for [pic] - this is more challenging to type in. Move everything to the left side of the equation. And to get the absolute value you have to go to [MATH] right arrow over to NUM and [1] is abs( which is absolute value and you type what you want inside like: abs(3x – 2).

4) Find the vertex & zeros for 2x2 = |x| – 2 - Again, move everything to left side of equation. And it’s difficult to see exactly what is going on at the vertex, it’s sort of flat (this is called HIDDEN BEHAVIOR)– you can zoom in. Press [ZOOM] [2] [ENTER] and you’ll see TWO vertexes. Find both!

5) A cubic doesn’t really have a vertex – the vertices are really turning points – one is a local maximum and the other is a local minimum. Put the following into function form first (y = by moving everything to the left) 3x + 6 = x3, then graph & find the zeros & turning points

ANSWERS

1) Hidden behavior example vertex (-9.703, -0.076) zeros (-10, 0) (-9.406, 0)

2) Vertex (2/3, -81), and no real zeros

3) vertex – you may get artifacts – (-0.5, -4.243) zeros -2.179 and 2.179

4) two minimums (-0.25, 1.875) and (0.25, 1.875) and no real solutions.

5) Y = -x3 + 3x + 6 zero (2.355, 0) minimum (-1, 4) maximum (1, 8)

RESTRICTING YOUR DOMAIN - PIECEWISE FUNCTIONS & CIRCLES

Sometimes you will want to graph parts of functions – parts of lines.

You type the function in parentheses and divide the function by the restriction in the parentheses as shown below:

Y1 = (x2 – 5)/(x > 1)

Also results in

And results in

The inequality symbols are found in [2nd][math]. Also “and” and “or” are found in [2nd][math]>LOGIC.

Circles

Circles are NOT functions and we cannot draw non-functions in the [y=] screen. But we can split a circle into two half circles – these pass the vertical line test and can be separately graphed in the [y=] screen.

The equation of a circle with center (0, 0) is x2 + y2 = r2. If you want a circle with center on the origin, you need to choose a radius length and then solve the above equation for y. For example if the radius you choose is 5, you would input in Y1 = [pic] this will give you the top half of the circle. To get the bottom half, type in Y2 = –[pic].

You can accomplish this in one Y= spot by typing in Y = {-1. 1}[pic]. (But most students want to put it in two separate Y spots so it counts as two equations!) Notice the circle isn’t quite closed? This is because of pixel limitations on your calculator. You can connect the parts of the circle by using some draw menu functions such as Pen [2nd][prgm] or Pt-On [2nd][prgm]>points. This is something you might want to play around with when you work on your graphing calculator design project.

If you want a circle whose center is not (0, 0) you use the equation (x – h)2 + (y – k)2 = r2 and find your desired center and radius (to find the desired center – when you are on the graph screen of your design, just press one of the arrow keys (up, down, right or left) and a cursor will appear. At the bottom of the screen you will see an ordered pair that indicates where your cursor is. Move over to where you want the center is to estimate the center). Input them into the equation and solve for y. For example a circle with center (-4, 5) and radius 4 would be

Y1= [pic] and Y2 = –[pic]

There is lots more you can do with your calculator! This guidebook gives you an overview of actions commonly used in a Pre-Calculus course. But there are lots of other things we can do with the calculator. Later in the course we will do parametric graphing, radian/degree conversion and more!

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