Parabolic Investigations
Mary F. Klein
Topic
: Using technology to Explore the general equation of a parabola.
Level
: Introductory Algebra, Algebra, Second Year Algebra
Links to The Standards
: (The page numbers refer to the 1989 edition.)
Standard 2: Mathematics as Communication
reflect upon and clarify their thinking about mathematical ideas and relationships;
formulate mathematical definitions and express generalizations discovered through
investigations;
express mathematical ideas orally and in writing; (Page 137)
Standard 3: Mathematics as reasoning
make and test conjectures; (Page 140)
Standard 6: Functions
analyze the effects of parameter changes on the graphs of functions; (Page 154)
Technology Used:
I used the TI 82 or else the function plotter in the computer Green Globs
. However, any graphing calculator or program could be used.
The General Idea
: This lesson is one thread in a set of lessons used throughout the year. The student
is encouraged to take the general form of an equation and create a prototype by changing
to 1
those parameters that are coefficients of the variable and changing to 0
those parameters that are added to the variable. In this case,
y = a(x - b)^2 + c becomes
y = 1(x -0)2 + 0
or y = x^2
The student graphs the prototype and then compares the graphs of other equations to
the graph of the prototype. He investigates each parameter alone and then in combination
with other parameters. To investigate the parameter a
, he might graph
y = 2x^2, y = 3x^2, y = .2x^2, y = .3x^2, y = -2x^2, y = -3x^2, etc. Notice, the student uses negative and positive values as well as larger and smaller
absolute values. The student then writes conjectures (a very guided conjecture perhaps)
about the effect of changing the parameter, a
. Then the student proceeds to b
, c
, and d
. Each time he begins by changing one parameter and comparing the graphs to the graph
of the prototype.
This lesson is a lesson for the general form of a parabola. I decided to put instruction
for using the calculator (TI 82) on a separate sheet. This way only the cover sheet
would need to be changed if different equipment were used.
I would follow this lesson with some time playing the game Green Globs
on the computer. Green Globs,
is available from Sunburst.
Graphing Parabolas. Name ______________
The general form for the equation of a parabola is y = a(x - b)2 + c. Each constant a, b, and c has a predictable effect on the graph. Changing the
value of a constant changes the graph. Our goal is to discover exactly what change
in the graph is produced by each variable so that we can quickly graph any equation
of this type.
To accomplish our goal, we compare the graph of y = x^2 , the simplest version of our equation, to the graph of other equations. (Note: y = x^2 is equivalent to y = 1(x - 0)2 + 0. Coefficientrs of x were changed to 1, and constants added to x were changed
to 0.) The graph is the basic shape for all parabolas.
Enter and graph the equation. Sketch it here:
Part 1:
Equations of the form y = ax
2
.
a.) Enter the following equations.
Graph
and sketch them, then compare to the graph of y = x^2.
y = 2x^2
| y = 3x^2
| y = 4x^2
|
Tell how multiplying x^2 by a constant greater than one changed the graph. Did the graph get fatter or thinner?______
b.) Clear all but the first equation, then enter the following equations.
Graph
and sketch them.
y = .2x^2
| y = .3x^2
| y = .4x^2
|
Tell how multiplying x^2 by a constant less than one changed the graph of y = x^2. Did the graph get fatter or thinner?______
c.) Clear all but the first equation, then enter some equations where the coefficient
of x^2 is negative. Use numbers with absolute value greater than one and numbers with absolute
value less than one. List the equations, sketch the graphs, and tell what you discovered.
Part 2.
Equations of the form y = x^2+c
a.) Clear all but the first equation, then enter the following equations.
Graph
and sketch them. Try out some equations of your own, too.
Compare to y = x^2.
y = x^2 + 1
y = x^2 + 4
What did you notice?
When a graph appears to move from one place to another, we say the graph has been
translated.
In which direction was it translated? ____________ How many units was it translated?
_____
b.) Clear all but the first equation, then enter the following equations.
Graph
and sketch them. Try out some equations of your own, too.
Compare to y = x^2.
y = x^2 - 1
y = x^2 - 3 ____________
Tell what kind of translation is produced when we use negative values of c?
c.) Make predictions about each of the following equations. Tell how it compares to
the equation y = x^2. Then check your work by graphing.
wider or narrower smile or frown translated up/down
y = 2x^2 - 3 ___________ ____________ _______________
y = .4x^2 + 3 ___________ ____________ _______________
y = .1x^2 - 3 ___________ ____________ _______________
y = 2x^2 + 3 ___________ ____________ _______________
Part 3.
Adding a constant to x
.
Equations of the form y = (x - b)
2
a.) Clear all but the first equation, then enter the following equations.
Graph
and sketch them. Try out some equations of your own, too. Compare to y = x^2.
y = (x - +2)^2 y = (x - +4)^2 What did you notice?
What kind of translation?
How many units was it translated?____ In what direction?
b.) Repeat problem a using negative values for b. (Note "y = (x - -2)2 " is the
same as " y = (x + 2)2 " ) Tell what you noticed. Compare to y = x^2.
c.) Make predictions about each of the following equations. Tell how it compares
to the equation y = x^2. Then check your work by graphing.
wide / narrow smile / frown translated up or down left or right
y = 3(x - 2)^2 - 3 ___________ ____________ _________________
y = .4(x + 1)^2 + 3 ___________ ____________ _________________
y = .1(x - 1)^2 ___________ ____________ _________________
y = .2x^2 + 3 ___________ ____________ _________________
Part 4. Summary.
The general form of the equation of a parabola is:
y = a
(x - b)^2 + c.
a
When the absolute value of
a
is greater than 1, the graph is (wider/narrower)
than the graph of y = x^2. When
a
is less than 1, the graph is (wider/narrower)
than the graph of y = x^2. When
a
is negative the graph opens (at the top; at the bottom) but when
a
is positive, it opens (at the top; at the bottom).
b
Making
b
positive, translates (or slides) the graph b units to the (left, right.)
Making
b
negative, translates (or slides) the graph b units to the (left, right.)
c
Making
c
is positive, translates (or slides) the graph b units (up, down.)
Making
c
is positive, translates (or slides) the graph b units (up, down.)
What other things did you notice?
Part 5. Extension.
Make a picture on your screen, using only parabolas. Use at least 4 different parabolas
in the picture. List the equations of your parabolas in the space below. below.