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.
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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.
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.
TABLE SCREEN We don’t use this much, but it can be useful (we will use [GRAPH] more
often). Press [2nd][TABLE]. You won’t see much if you don’t have
an equation typed in the equation screen. You will just have an x
column with consecutive numbers listed. Your numbers might
start in a different place than mine do.
Let’s type an equation in and see what happens. Go to [y=] and
type in the equation y = x2 + 5x + 6. Then press [2nd][TABLE].
What you see in the table depends upon the table set up. Press
[2nd][WINDOW] to see what your table set up is. My table start is -6, so that’s where x starts.
The y column shows the corresponding y-values (or f(x)) for those
x-values. The “triangle” tbl means “change in x values” – Usually I
choose “1” but sometimes you might want to count by a higher
interval or lower.
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
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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 [2 nd][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 graph a function and find a window that shows the full function shape.
There are some basic functions that you learned in Algebra 2 that you should be very familiar
with. It is important that you know their general shapes so you can determine if the function
is “fitting on the window”.
The basic function shapes you should know are: y = x2, y = x3 , y = |x|, y = 1/x, y = ex
Graph & draw those shapes here (label each graph):
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Given an equation for any of the above functions you should be able to graph that function on
your calculator and determine what window shows the full function. Obviously you can’t see
the “full” function since it goes forever, but you can find a window that shows the full basic
shape.
For example in a quadratic you should see the full parabola without the vertex being cut off or
in a cubic you should see the full “s-curve”. To find the correct window you type the equation
into the [y=] and press graph (oops – I hope you ran the defaults program first, if you don’t
have defaults, press zoom 6 to get a standard graph). Then you examine what you see and
change the “mins” and “maxs” in the [WINDOW] screen, pressing graph after adjusting to see
if you have the full function.
For example – graph y = x2 + 3x – 12 and press [graph] – oops vertex cut off! Lower the ymin and you should see the vertex now. (keep in mind that the c-value of the function is the
y-intercept….). A y-min of -15 works nicely.
Now we can find any value and the vertex and the zeros of the function – GRAPHICALLY
(rather than algebraically like you did in Algebra 2).
FINDING A VALUE - Given f(x) = x2 + 3x – 12, find f(5) - yeah
you could plug in x = 5 but graphically (should have on parabola
on screen) press [TRACE] [5] and [ENTER]. On the bottom of
the screen you should see x = 5 and y = 28. So f(5) = 28.
You also see the function equation in the upper left corner, handy
if you have more than one function on the screen – it shows you
which one you are working with.
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)
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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][TRACE]
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.
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FINDING X WHEN F(X) IS GIVEN – 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….
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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
Now let’s examine some other examples (run DEFAULTS before each one)….ANSWERS
BELOW….
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 | 2 x  1 | 2 x  5 - 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) then graph & find the zeros & turning points
ANSWERS
1)
2)
3)
4)
5)
Hidden behavior example vertex (-9.703, -0.076) zeros (-10, 0) (-9.406, 0)
Vertex (2/3, -81), and no real zeros
vertex – you may get artifacts – (-0.5, -4.243) zeros -2.179 and 2.179
two minimums (-0.25, 1.875) and (0.25, 1.875) and no real solutions.
Y = -x3 + 3x + 6 zero (2.355, 0) minimum (-1, 4) maximum (1, 8)
BOOKWORK
Do the following problems to practice the above skills.
pp 83 – 84 #s 36 (graphically only), 39 – 47 odd, 51.
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