Teacher Notes
The purpose of this activity is to al low the students to
explore the vertex form of the
parabola and discover how the vertex, direction, and width of the parabola can
be
de termined by looking at the parameters.
Students will be able to make predictions about the location of the vertex of a
parabola expressed in vertex form. They will not have had as much practice
understanding the effect of A. They will be able to make statements about
location
based on the quadratic function .
The concept of translation is basic to the general study of functions, and the
activity
ends with a look at translation in general. These closing questions on
translation are
not really part of the study of the vertex form; you can easily leave them out.
They
serve as a overview of translation for those students who encounter translation
as
part of their study of function.
The work with translation at the end of the activity encourages students to make
hypotheses and to verify them with their graphing handhelds .
Answers
The Graph of a Quadratic Function in Vertex Form
3. The vertex is at (2, 1). The parabola opens up.
Studying the Effect of B: Questions for Discussion
1. The curve moves in the x-direction. This moving of the curve is called a
translation in the x-direction, or horizontal translation.
2. If B=3, the vertex is at (3, y). If B=5, the vertex is (5, y).
If B=-1, the vertex is at
(-1, y).
3. The value of B indicates the x- coordinate of the vertex of the parabola.
Studying the Effect of C
3. Changes in the value of C create a vertical translation of the curve. When
the
value of C increases, the curve moves up. When it decreases, the curve moves
down. The value of C is the y-coordinate of the vertex.
Check Your Understanding So Far
y = (x – 2)2
Vertex is at (2, 0). |
 |
| |
y = (x – 2)2 + 3
Vertex is at (2, 3). |
 |
| |
y = (x + 1)2 +3
Vertex is at (-1, 3). |
 |
| |
y = (x + 1)2 – 2
Vertex is at (-1, -2). |
 |
| |
y = (x + 5)2
Vertex is at (-5, 0). |
 |
| |
y = x2 – 2
Vertex is at (0, -2). |
 |
| |
What is the equation of the parabola (quadratic function)?
y = (x – 4)2 + 2
Studying the Effect of A: Question for Discussion
1. The value of A determines the direction of the parabola and its width. The
larger
the magnitude of A, the narrower the curve. The smaller the magnitude of A, the
wider the curve. A positive sign means that the parabola is opening up. A
negative sign means that the parabola is opening down.
Check Your Understanding
1. b
2. d
3. a
4. e
5. c
Use Your New Skill
Equation
|
Opens up/
down
|
Function has a
maximum/
minimum |
Maximum/
minimum
value |
 |
up |
minimum |
2 |
| down |
maximum |
10 |
| down |
maximum |
-100 |
| up |
minimum |
-36 |
A Quick Application
The maximum height is 259 feet. It takes 4 seconds to reach the maximum. The
vertex is the maximum or minimum value of the function. Because A is negative,
the
parabola opens down and has a maximum.
Student Worksheet
1. The second function (y = x + 3) can be viewed
as a translation either 3 units up or 3 units to
the left. If it is viewed as a translation 3 units
up, you would be viewing the equation as
y = x + 3. If you view it as a translation 3 units
to the left (x-direction), you would be seeing
the equation as y = (x + 3).
|
 |
2. The graph of y = x3 + 2 is a vertical
translation
of y = x3 two units up. The point at the origin
should be moved to (0, 2). |
| |
