Graphing Calculator Lesson
Completing the Square
TASEL-M
August Institute
2006
Margaret Kidd
mkidd@fullerton.edu
Completing the Square Using Calculators
Overview
CA Standard:
22.0 Students use the quadratic formula or factoring techniques or both to determine
whether the graph of a quadratic function will intersect the x-axis in zero, one, or two
points.
Curriculum Placement:
This lesson can be used at any time after students have been introduced to graphing
quadratic equations. The lesson presented here assumes that students have done some
work with completing the square using traditional methods.
Previous Knowledge and Connections to It:
Students should know how to graph y = x and how the slope and y-intercept affect the
graph.
Students should be familiar with quadratic equations and how to graph y = x2.
Warm Ups (in the weeks prior to introducing this lesson):
Should include problems such as: graph y = x, y = 2 x, y = 3 x, etc. They should also
include problems in the form of y = x, y = x + 2, y = x + 3, etc. After quadratics have
been introduced, warm ups should include y = x 2, y = x 2 + 2, y = x 2 + 3, etc. and y = x 2,
y = (x + 2)2 , y = (x + 3)2.
Additional Suggestions:
A method for solving equations that students seem to find fairly easy is called arrow
diagrams. It can be used to solve any linear equation and for quadratic equations that are
perfect squares. This method has the added benefit of giving students a reason to put
equations in the form of a perfect square. There is an example of this method in Handout
3 (attached).
Although transformations are not emphasized in many classes, point slope form for linear
equations provides a link from linear to quadratic as well as exponential and absolute
value equations.
Quick recap:
[ y – y1 = m(x – x1) is written as y – k = m(x - h) in later courses and is used here.]
Linear:
y – k = m(x - h) where (h,k) is a point on the line.
Quadratic: y – k = a(x - h)2 where (h,k) is the vertex and a is the same as m in linear
Absolute Value: y – k = a| x - h | where (h,k) is the vertex and a is the same as m in linear
When students understand they do not need to memorize a “new” formula for each
function and can see the connection it makes more sense to them and they do not find it
as complicated.
Follow up activity:
After students are familiar with quadratic equations, they can hone their skills of writing
equations from graphs using the calculator program “Quad”. This gives students random
graphs that they have to “guess” the equation using vertex form.
Handouts:
Included in this packet are four handouts adapted from Mathematical Models with
Applications developed by COMAP. Handout 1 is the only one needed for this lesson
and should be copied for each student. The other two are background information for
teachers that may or may not necessarily be duplicated for students.
Handout 1 contains the keystrokes necessary to have calculators cleared and ready for
use.
Handout 2 explains how to graph equations on the calculator
Handout 3 has suggestions for solving equations using both arrow diagrams and
graphing.
Handout 4 TI83 program code for Quad.
TASEL-M 06 Institute
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Completing the Square Using a Calculator
Lesson Plan
Introduction:
This lesson is designed to be used grouped in pairs or groups of four with each student or
pair of students having a graphing calculator.
It can also be done if only the overhead calculator is available (if this is the case, have a
student operate the overhead so that the teacher can walk around the classroom). Notes
in parentheses will indicate this modification throughout the lesson.
[The teacher should realize that calculators are a new toy for the students. If they have
not been previously used with the class, give students simple instructions on how to turn
them on and off and where the “y =” is. On the first day, give students a few minutes to
get used to them. As with whiteboards, paper and pencil, or any other tool students will
want to write messages etc. The same classroom rules should be applied as with any
other tool.]
Handout 1 explains the “housekeeping” that is necessary each time the student uses a
calculator. This will ensure that all students are on the same screen and that extraneous
material is not present. The teacher will save time by making this a routine for students.
(If only the overhead is being used, these steps should also be taken and explained.)
CA Standard: 22.0 Students use the quadratic formula or factoring techniques or both
to determine whether the graph of a quadratic function will intersect the x-axis in zero,
one, or two points.
Objective: Students understand how to complete the square using a calculator.
Warm Up: [This is to be done only the first time quadratic graphing on the calculator is
introduced to students.]
In y1 have students enter the parent parabola x2 (or teacher enters on overhead). Using
ZOOM 6, the graph will be on a 10 x 10 plane. (In order to distinguish this graph from
the others that are entered, use a bold line when graphing (found to the right of y1).
Have students enter the following (or teacher enters on overhead) and graph after each
entry. (These specific numbers do not need to be used but positive and negative values
should be included.)
Start with the change on the y-axis only as it is intuitive for the students.
A.
y2 = x2 + 2
y3 = x2 + 4
y4 = x2 – 2
y5 = x2 – 4
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Give students time to discover how the constant “k” changes the graph. They should see
that the constant moves the graph up and down the y-axis and the vertex is at (0,k). If
students do not see this, ask questions such as
What is the difference in these graphs?
How does the “+k” or “–k” affect the graph of y1 = x2?
Once students are comfortable with this and understand the effect of adding a constant (it
will be quite obvious to most of them), have them enter the following (or teacher enters
on overhead):
B.
y1 = x2
y2 = (x + 2) 2
y3 = (x + 4) 2
y4 = (x – 2) 2
y5 = (x – 4) 2
Again, give students time to discover how the constant “h” changes the graph. They
should see that this constant moves the graph along the x-axis and the vertex is at (h,0).
If they do not see this, ask questions such as:
What is the difference in these graphs?
How does the “+h” or “–h” affect the graph of y1 = x2?
This step is not as intuitive to the students as the “+h” moves the graph to the left and the
“–h” moves it to the right. Spend a few minutes discussing this if it has not been
discussed previously. Use the formula y = a(x – h)2 + k
Now, students are ready to put the two pieces together. Have them enter the following
(or teacher enters on overhead):
Discuss the change in the graph after each entry and put it together after all of the entries
are complete.
C.
y1 = x2
y2 = (x + 2) 2 + 3
y3 = (x + 4) 2 – 5
y4 = (x – 2) 2 + 3
y5 = (x – 4) 2 – 5
If students have difficulty with this, graph the corresponding y2 = entries from parts A, B,
and C before moving on to the remaining equations. Give students time to talk with their
partners before discussing it as a whole class. Finally, explain to them the changes if it is
still not clear to them.
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Lesson Activity:
Graph the equations in Part B. of the warm up on the overhead. Ask students what they
notice about all of these graphs. [Accept all reasonable answers and guide them to the
fact that they are all tangent to the x-axis.] Referring to the equations, emphasize they are
all perfect squares. Conclusion: graphs of perfect squares touch the x-axis at exactly one
point.
Using the expressions in Part B. have students expand them:
(x + 2) 2 = [x2 + 4x + 4]
(x + 4) 2 = [x 2 + 16 x + 16]
(x – 2) 2 = [x 2 – 4 x + 4]
(x – 4) 2 = [x 2 – 16 x + 16]
Discuss the relationship between the squared form and the expanded form.
Have students graph x2 – 4x by entering:
y1 = x2 – 4x
“What do we need to make this a perfect square or what number do we need to add to
make it touch the x-axis exactly once?”
[Give students time to add and discuss various values until they find the 16.]
Graph the final result on the overhead and discuss how this is the same value they found
by hand.
Next, have them enter and graph y1 = x2 – 6x
“What do we need to make this a perfect square or what number do we need to add to
make it touch the x-axis exactly once?”
[9; Give students time to add and discuss various values until they find the necessary
value.]
Hint: How far below the x-axis is the minimum value for this graph?
When they find that 9 must be added, graph it on the overhead.
Ask how they would write this equation as a perfect square. [(x – 3)2]
Use at least one, but as many as necessary, of the following for the students to grasp what
value needs to be added to make the graph touch the x-axis exactly once or to make the
expression a perfect square. Then have them rewrite it as a perfect square.
x2 + 6x [+9 (x + 3)2]
x2 – 8x [+16 (x – 4)2]
x2 + 10x [+25 (x + 5)2]
x2 – 9x [+20.25 (x – 4.5)2]
(Make certain to use both positive and negative linear coefficients as well as odd ones.)
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Using the calculator, they can “count” how many spaces the vertex of the graph is from
the x-axis. Give students sufficient examples until they realize that the value they must
add is half of the linear coefficient squared in every case.
Extra: Have students make up their own problems and either challenge their partner or
the whole class.
Ask students how can they can tell that (x – 5)2 is not equal to x2 + 25?
Hint: Graph both. (x – 5)2 is tangent to the x-axis whereas x2 + 25 is 10 units above it.
Follow up question: What do you need to add to x2 + 25 to make it equal to (x – 5)2?
Emphasize the relationship between the constant term and the linear coefficient again as
this is goes from a constant given to finding the linear term.
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Extension:
Solving Quadratic Equations by Graphing
Have students graph x2 – 2x = 3 by entering
y1 = x2 – 2x
y2 = 3
Then ask the following questions:
1. Where do the x-values of the graph intersect?
[-1 and 3]
2. What do we need to add to make the left side a perfect square or what number do we
need to add to the left side to make it touch the x-axis exactly once?
[1]
3. Add that number to both sides and graph the two new expressions:
[x2 – 2x + 1 = 3 + 1]
4. How has the graph of the parabola changed?
[The graph is now tangent to the x-axis.]
5. What do you notice about the x-values for the points of intersection?
[They stay the same.]
6. Why was it necessary to add to both expressions?
[Must move the line also, so that points of intersection don’t get farther apart
or closer together when the parabola moves.]
7. Now write the squared form of the left side and simplify the right side.
[(x – 1)2 = 4]
8. Draw an arrow diagram and use it to solve.
square
–1
x
4
+1
square root
9. Write the solutions.
[-1 and 3]
10. Where have you seen the solutions before?
[They are the x-values of the points of intersection.]
B. Use the same method as above for the equation:
x2 – 3x = 10
and x2 – 4x = 10
C. Using a similar method solve:
2 = 1 + 3x – x2
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Handout 1
Housekeeping (TI-83)
To be performed each time the calculator is turned on.
Operation name: Go "Home" (exits to "home" screen; clears off old display)
Buttons to push: 2nd MODE CLEAR
Screen Display: nothing
Operation name: Set Correct Modes
Buttons to push
MODE
Screen Display: All entries down the right side should be highlighted.
Operation name: Format Calculator
Buttons to push:
2nd
ZOOM
Screen Display: All entries down the right side should be highlighted.
Operation name: Clear Equations (Clears equations from memory)
Buttons to push: Y= CLEAR  CLEAR etc., until they are all cleared.
Operation name: Clear StatPlots (Turns plots off)
Buttons to push: 2nd
Repeat for Plot 2, Plot 3
Y=
1
Use the arrow keys and <ENTER> to select 'OFF'
Screen Display:
Operation name: Clear Lists (Clears lists L1 and L2 from memory)
Buttons to push: STAT
4
2nd
1
,
2nd
2
Enter
Screen Display: ClrList L1 , L2
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute
Handout 2
Equations and Graphs on the TI-83
Step 1: Make sure the calculator is setup correctly. Press
the MODE button; your screen should look like what is
shown in Figure 1. Make changes by using the arrow keys,
positioning the cursor at the beginning of a line, and then
pressing <ENTER>. Then check the FORMAT menu
(press 2nd and then ZOOM); the screen should look like
Figure 2. Make changes in the same way, if necessary.
Figure 1.
Figure 2.
Step 2: Enter in the equation to be graphed. First press the Y=
button; if there are already equations in that menu, you can either
throw them away (make sure the cursor is on that line, then press the
CLEAR button) or keep them from being seen (put the cursor on the
"equal sign" and then press <ENTER>). When entering the variable
for the equation, use the X,T,,n key. When done, your calculator
screen should have an equation that looks like Y1 in Figure 3.
Step 3: Press WINDOW to set the part of the graph that
you want to see. By using a small, negative number for
Xmin and Ymin, the two axes will show up in the picture.
Xmax, Xscl, Ymax and Yscl are chosen to match the original
axes that were used in drawing the graph in Activity 1.
When done, your calculator screen should look like what
is in Figure 4. Press GRAPH, and your screen should
change to look like Figure 5.
Step 4: Before looking at table values, first check make
sure that they are set up correctly. Press 2nd and then
WINDOW to get the TBLSET menu. To match the
table used in Activity 1, make the values match what is
shown in Figure 6. Then, go to the TABLE menu by
pressing 2nd and then GRAPH. You should see the
same numbers as what is shown in Figure 7.
Step 5: Press TRACE and the graph should re-appear
with a "cross-hair" on the line and coordinates displayed
at the bottom, like what is shown in Figure 8. To check
the location of specific points (while the TRACE is still
active), type an x-value and press <ENTER>. If you use
a value of '4', your screen should look like what is in
Figure 9.
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.
TASEL-M 06 Institute
Handout 3
Solving Equations
One of the basic skills of algebra is to solve an equation -- to find the value of a variable that
makes a sentence "true". That means that if the solution was substituted in place of every
occurrence of the variable, the left and right sides of the equation would have the same exact
value. While "guess-and-check" can eventually provide an answer in most cases, arrow
diagrams and algebra will be more reliable, accurate and faster.
Example 1: Direct Variation
25x = 480
The equation describes this "order", or relationship: start with the value
for x. When you multiply it by 25, the answer is equal to 480. Figure 1
communicates that order, if you follow the arrow from left to right.
Solving the equation means working "backwards" -- starting with the
answer, 480, and determining what the number would have to be prior to
carrying out the calculation. To do that, you do the "opposite" operation.
Figure 1 illustrates that, if you follow the arrow from right to left.
In algebra, you learn that adding, subtracting, multiplying or dividing both
sides of an equation does not affect the solution to the equation, as long as
the operation is done to all terms. Algebra can allow you to solve the
same equation by canceling the "25 times" in the equation. Again, you do
the opposite to cancel, and the steps are shown in Figure 2. In multiplying
and dividing by 25, you simply get x. Another way to think about this is
that 2525=1, and 1x is simply x; that's why the left side reduces to x. The
right side reduces to the solution by actually dividing 480 by 25.
25
x
480
25
Figure 1.
25x = 480
25 x 480

,
25
25
so x = 19.2
Figure 2.
Example 2: Slope-Intercept (Two-Step) Equation
5x + 32 = 187
While more steps are involved, solving an equation of this form is really
similar to the first example. Figure 3 shows how the arrow diagram for the
equation is set up, and how to work backwards to "solve for x". Figure 4
shows the algebraic steps involved, and the "simplified" equations that are
formed by going through those steps. You should see a strong resemblance
between the two approaches:
+32
5
x
5x + 32 = 187
5x + 32 = 187
 32  32
5x = 155
5x
155
=
5
5
x
= 31
187
32
5
Figure 4.
Figure 3.
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute
Almost all other equations that are encountered are variations on a theme, and are solved in a
similar manner.
12 5

x 32
Cross-multiply to eliminate the fractions. Then (12)(32) = (5)(x), or 384 = 5x. It looks like
Example 1 now, so you know what to do!
Example 3: Proportion
Example 4: Parentheses
4(2x  7) = 63
You could add a third step to the arrow diagram used

in Example 2, which would be to multiply by 4.
However, the Distributive Property comes in handy
4
2x  7
here. After distributing the multiplication, the
equation now looks like a "two-step" equation, so it's
So, 8x  28 = 63
nothing new!

Example 5: Two-step expressions on both sides
This is a little more complicated, and having
the variable in two places makes the arrow
diagram impossible to use here. However,
subtract one variable term from both sides of
the equation (to cancel it from one side), and
you're back in business!

2x + 15 = 8x  39
2x + 15 =
2x + 15 =
2x
15 =
8x  39
8x  39
2x
6x  39
A really nice way to solve an equation involves using the graphing calculator. Since you want
the condition under which the left-side of the equation equals the right-side of the equation, enter
each expression as a separate expression in the Y= menu, and see where the two graphs meet.
Figures 5-7 show the equations, possible WINDOW settings and a graph of the situation which
corresponds to solving the equation 5x + 32 = 187.
Figure 5.
Figure 6.
Figure 7.
Use the TRACE, TABLE or INTERSECT feature of the graphing calculator to determine the
solution to the equation.
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute
Handout 4
Program QUAD
This program was written for use with a TI-83 graphing calculator. Either type in the
program listing manually, link with your teacher and SEND/RECEIVE a copy, or use the
TI Graph Link software to download a copy from the computer.
PROGRAM:QUAD
:Lbl Y
:PlotsOff:Func
:ClrHome
:GridOn
:"A(XH)^2+K"Y2
:"D(XE)^2+F"Y1
:iPart(16*rand8)E
:iPart(10*rand5)F
:Lbl R
:iPart(8*rand4)/2D
:If D=0
:Goto R
:0A:0H:0K
:ZStandard
:DispGraph
:Pause
:Lbl Z
:Prompt A,H,K
:If A=D
:Text(1,1,"A IS OK")
:If AD
:Text(1,1,"A IS WRONG")
:If H=E and K=F
:Text(1,50,"VERTEX IS OK")
:If HE or KF
:Text(1,50,"VERTEX IS WRONG")
:DispGraph :Pause
:If AD or HE or KF
:Goto Z
:Goto Y
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute
Handout 4a
QUAD Program Record Sheet
Trial #1
Move
1
2
3
4
a
h
k
Reason for those Particular Values
10
-10
10
-10
Trial #2
Move
1
2
3
4
a
h
k
Reason for those Particular Values
10
-10
10
-10
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute
Trial #3
Move
1
2
3
4
a
h
k
Reason for those Particular Values
10
-10
10
-10
Summary
Write three ideas that you discovered from completing your trials using the QUAD
program.
Adapted from Math Models Course
"Modeling IS Mathematics" Chapter
TASEL-M 06 Institute