TI-83
An Introduction to
Graphing
http://jwelker.lps.org/lessons/
Mathematics Staff Development
Lincoln Public Schools
August 25, 2005
© Jerel L. Welker
Graphing a Function
Graph the function y = x - 5
Press Y=
Enter X - 5 in Y1=
Press ZOOM – 6 or ZStandard to set
the window to set the window to
[-10,10] x [-10,10]
Use the TRACE key to move the
cursor along the line with the left
and right arrow keys.
Using the Table
Values from the function can also be
viewed using the TABLE.
Press 2nd – TABLE to view the values
in table format.
Use the up/down arrows to scroll
through the table of values.
Press 2nd – TBLSET to change the
starting value or increment (  Tbl).
Graph a 2nd function.
1
Graph y=x - 5 and y  x  9
3
on the same graph.
Press Y=
Enter (1/3)X + 9 in Y2
Graph the functions.
Changing the Window
Change the window to view the
intersection of the two functions.
Press WINDOW.
To see more of the graph change:
Xmin to change the left side.
Xmax to change the right side.
Xscl changes the increment of the
tick marks on the x-axis.
Ymin to change the bottom.
Ymax to change the top.
Yscl changes the increment of the
tick marks on the y-axis.
Graph the functions and adjust as
required.
Finding the Intersection
Method 1:
Use the TRACE button and
approximate the solution.
Method 2:
Use the TABLE (2nd - Table) to find the
x-value which has the same
y-value for both functions.
Finding the Intersection
Method 3:
Use the intersection feature of the
calculator.
Press 2nd – CALC followed by
intersect or 5.
Press ENTER on the first curve.
Use the up/down arrows to
changes curves if required.
Press ENTER on the second
curve.
Use the arrow keys to position the
cursor close to the intersection
and press ENTER. The answer
will be displayed.
On Your Own
Clear the previous equations
using the CLEAR key.
Find the intersection(s) of:
y = -(x + 3)2 – 13
y = 5x - 10
Notes:
Use ^ 2 or the x2 key for a square.
Note the difference between (-)
and minus – keys.
On Your Own Solutions
A Complete Graph
A complete graph should:
show the origin.
show all x- and y-intercepts if they exist.
show all turning points of the graph.
show the end behavior of the graph.
utilize the entire screen.
It may be possible that some graphs
require more than one screen for a
complete graph!
Find a Complete Graph
Find a complete graph of
y = x3 – 14x2 – 21x + 90
Hint: Only make changes to the
graph in one direction at a time.
Sketch the Graph
Sketch the graph and show the window
in [Xmin, Xmax] x [Ymin, Ymax] form.
[-10, 18] x [-600, 150]
Showing the window provides documentation
that a complete graph has been found.
Y-intercept
Find the y-intercept of the function.
The y-intercept occurs where the xvalue is 0. Substitute 0 for x or
Calculate the value of 0. 2nd – CALC
and choose VALUE or option 1.
Enter the value of 0.
The y-intercept is (0, 90).
X-intercept(s)
Find the x-intercept(s) of the function.
The x-intercepts occur on the x-axis
where the y-value is 0. They
are also called roots or zeros of
the function.
Finding X-intercept(s)
Press 2nd – CALC and choose zero or
option 2.
The calculator is asking for a left bound.
Use the arrow keys to position the
cursor on the LEFT side of the zero.
Press ENTER.
The calculator is asking for a right bound.
Use the arrow keys to position the
cursor on the RIGHT side of the zero.
Press ENTER.
Position the cursor over the zero and
press ENTER. The zero value will be
displayed.
Writing Factors
The x-intercepts or zeros are:
(-3, 0), (2, 0), and (15, 0)
Write the factors of the function.
If a is a zero of the function then:
x=a
x–a=0
(x – a) is a factor of the function.
x3 - 14x2 - 21x + 90 = (x + 3)(x - 2)(x - 15)
Turning Points
Turning points or extrema of the graph
are points where the slope of the line
tangent to the curve is zero.
Local maximum
Local minimum
Finding a Maximum
Press 2nd – CALC and choose maximum
or option 4.
The calculator is asking for a left bound.
Use the arrow keys to position the
cursor on the LEFT side of the
maximum. Press ENTER.
The calculator is asking for a right bound.
Use the arrow keys to position the
cursor on the RIGHT side of the
maximum. Press ENTER.
Position the cursor over the maximum
and press ENTER. The maximum
value will be displayed.
Finding a Maximum
The local maximum is 97.50
when x is -0.70.
Finding a Minimum
Note that the trace values take up a
portion of the lower screen. If you cannot
see the minimum value, reset the window
to increase the space before continuing.
Finding a Minimum
Press 2nd – CALC and choose minimum
or option 3.
Repeat the process of finding the LEFT,
RIGHT and GUESS values for the
minimum value.
The local minimum is -520.02 when x is 10.03.
Domain
Find the domain of the function.
The graph is a continuous function with x
starting at  and continuing to  .
D:
(, )
Range
Find the range of the function.
The graph is a continuous function with y
starting at  and continuing to  .
R:
(, )
Check Your Skills
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
Find:
a complete graph and sketch.
the intercepts.
a linear factorization of the polynomial.
the extrema or turning points.
the domain and range.
Solutions - Graph
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
[-13, 3] x [-400, 500]
Now, find the intercepts.
Solutions – Y-intercept
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
The y-intercept is (0, -264).
Now, find the x-intercepts.
Solutions – Zeros
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
Zeros are (-11, 0), (-8, 0), (-3, 0), and (1,0).
Linear Factorization (x + 11)(x + 8)(x + 3)(x – 1)
Now, find the extrema or turning points.
Solutions – Extrema
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
Local min at -158.74 when x = -9.76.
Local max at 223.67 when x = -5.43.
Local min at -295.66 when x = -0.56.
Next, find the domain.
Solutions – Domain
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
The graph is a continuous function with x
starting at  and continuing to  .
D: (, )
Next, find the range.
Solutions – Range
For the function:
y = x4 + 21x3 + 123x2 + 119x - 264
The lowest point on the graph is -295.66 and
the graph continues upward to  .
R:
(295.66, )
Hidden Behavior
Some functions have behavior which is
“hidden” by the calculator in some
windows. This behavior may be extrema
or discontinuities which do not show in a
typical view.
For example, graph y = x3 – 0.1x
in standard view (Zoom – 6).
Hidden Behavior
“Flat spots” in a graph or areas hidden by
the axes can be difficult spots. Use Zoom
– Box or option 1 to take a closer look at
these places.
Zoom Box allows
the user to draw a
rectangle around a
portion of the graph
which should be
enlarged.
Hidden Behavior
Press Zoom – Box and move the cursor to
a corner of the rectangle to zoom.
Hidden Behavior
Press Enter and move the cursor to the
diagonal corner to form a rectangle.
Press Enter and the region within the
rectangle will be shown in a new window.
Hidden Behavior
Repeat the process until a satisfactory
graph is shown.
What first appeared to be a single zero
is actually three zeros.
Rational Functions
x3
Graph: f ( x) 
x2
The numerator and denominator of
rational functions must be enclosed
in parenthesis. Enter the function as:
Y1 = (x + 3)/(x – 2)
Graph in a standard window Zoom – 6.
Y-Intercept
Find the y-intercept.
Calculate the value of f(x) when x=0.
 3 
The y-intercept is  0, 
 2 
Rational Functions
x3
Graph: f ( x) 
x2
In connected mode, some may be
tempted to say the asymptote
shows on the graph.
Rational Functions
Change the mode to DOT by
pressing MODE and pressing
ENTER with the cursor on Dot.
Graph the function.
Rational Functions
In Dot mode, only points graphed by the
grapher are displayed. No connecting lines
are shown. Note the “line” is not shown in
this view and is not part of the graph.
If the line were part of the graph, there must
be an x-intercept where the line crosses the
x-axis. Solve for the x-intercepts
algebraically.
X-Intercept
f ( x) 
(x-2) 0 
x3
x2
x3
(x-2)
x2
0  x3
x  3
The only zero is x = -3 which is
shown on the graph. There is no
zero near the “line” on the
connected graph.
X-Intercept
Calculate the function value when x = -3.
The grapher confirms the function value
of -3 is zero.
Domain
The function is discontinuous. It cannot be
traced without lifting a pencil near x=2.
What happens as x = 2? What does a blank
Y= result indicate?
Domain
What happens as x approaches 2? Use
the calculate value function to investigate.
Calculate the values of f(x) as x
approaches 2.
x
f(x)
x
1.9
-49
2.1
51
1.99
-499
2.01
501
f(x)
1.999
-4,999
2.001
5,001
1.9999
-49,999
2.0001
50,001
1.99999
-499,999
2.00001
500,001
1.999999
-4,999,999
 x3
lim 
 

x 2  x  2 
2.000001 5,000,001
 x3
lim 


x2  x  2 
Domain
What is the domain of the function?
The function is continuous at all
points except for x = 2.
D:
(, 2)  (2, )
The function has a vertical asymptote
at x = 2.
Range
What is the range of the function?
We have already discovered that the
graph goes to  and  as x
approaches 2. What happens as the
graph extends horizontally?
Extend the window to show more “xvalues”. Trace toward the ends to
find what happens to the function
values as x approaches  .
Range
Tracing to both ends of the graph, we
find that the function value approaches
what value?
 x3
lim 
 1
x  x  2


 x3
lim 
 1
x  x  2


Range
What is the range of the function?
The function begins at  and
increases until approaching the
horizontal asymptote at 1. From 1,
the graph continues to  .
R:
(, 1)  (1, )
The function has a horizontal
asymptote at y = 1.
Check for Understanding
4  2x
g ( x) 
x3
Given g(x), find:
• the intercepts.
• the domain and range.
• the asymptotes.
Check for Understanding
4  2x
g ( x) 
x3
Given g(x), find:
• the intercepts.
 4
 0,  and
 3
 2, 0 
• the domain and range.
D: (, 3) U (3, ) R: (, 2) U (2, )
• the asymptotes.
y = -2 and x = -3
Hiding a Graph
At times, one may wish to keep a graph for later
use but not be visible in the grapher at all times.
It is possible to “hide” the graph from view without
clearing the equation from the grapher.
Leave the current equation in Y1.
Enter
3x  6
h( x ) 
x2
in Y2.
Hiding a Graph
Hide the graph in Y1.
Position the cursor over the = sign in
the equation line and press ENTER.
Move the cursor away from the =
sign.
When the = box is highlighted, the
function will graph.
When the = box is not highlighted,
the function will not graph.
Repeating the process of pressing
ENTER on the = sign will highlight
the box and graph the function.
Hiding a Graph
Both Y1 and Y2 will graph.
Only Y2 will be visible.
Hidden Behavior
View the graph of
3x  6
h( x ) 
x2
Is the graph what you expected?
Should there be an asymptote at x = 2?
What is the value of f(2)?
Hidden Behavior
Start with a window with Xmin of -10
and Xmax of 10. Change the
window so that x=2 is in the center
of the window, a “hole” appears in
the graph.
This “hole” in the graph is another
example of hidden behavior.
Domain and Range
The domain is all real numbers except
x = 2. This could be written as:
D:
 , 2 U 2, 
D:
x | x  2
The range is
 y | y  3
or
Removable Discontinuity
This particular “hole” in the graph is
called a removable discontinuity. By
redefining one point, the graph is a
continuous function.
Simplifying the function h(x), we get:
3x  6 3( x  2)
h( x ) 

3
x2
( x  2)
Since the “hole” appears where x=2 and
h(2) = 3, the graph has a removable
discontinuity at (2, 3). Graphing h(x) and
graphing the point (2, 3) would result in a
continuous function.
Removable Discontinuity
A removable discontinuity occurs at
the zero of a common factor of the
numerator and denominator of a
rational function. If x = c is a zero of
the common factor, the point (x, f(c)) is
a removable discontinuity.
Find the removable discontinuity of:
x 4
m( x)  2
x  3x  2
2
Removable Discontinuity
Find the removable discontinuity of:
x2  4
m( x)  2
x  3x  2
x2  4
( x  2)( x  2) x  2
m( x)  2


x  3x  2 ( x  2)( x  1) x  1
Since (x + 2) is a common factor of the
numerator and denominator, a
removable discontinuity occurs at x = -2
If the common factors are removed, m(-2)
is 4. The removable discontinuity is at
(-2, 4).
Removable Discontinuity
Removable discontinuity at (-2, 4)
The line connecting the top dot with the
bottom dot is not a part of the graph.
The graph should have a vertical
asymptote at x = -1.
Try calculating the values of m(-2) and
m(-1). What are they?