Sarah Donaldson
EMAT 6680
9/25/06
Assignment 2 Write-Up
Problem 6: y = 2x2 + 3x – 4
The accompanying Graphing Calculator File (click here) shows 4 different
parabolas. The first (in blue) shows y = 2x2 + 3x – 4. This parabola is concave up and has
a vertex in the third quadrant.
i. Overlay a new graph replacing each x by (x - 4).
The second graph, in red, shows a new parabola in which each x in the previous
equation is replaced by (x – 4). The result is a parabola whose shape is the same as the
blue one, but whose position is different. The effect of replacing x with x – 4 is that the
vertex shifts 4 units to the right and is therefore in the fourth quadrant.
ii. Change the equation to move the vertex of the graph into the second
quadrant.
There are several options for a change that could be made in y = 2x2 + 3x – 4 in
order for the vertex to move to the second quadrant. I have chosen a simple sign change:
y = 2x2 + 3x + 4. The graph of this equation is the purple parabola in the picture. Again,
its concavity is the same as the first two parabolas. Only the vertex has shifted: up 8 units
from the original graph. (Eight units is the difference between -4 and 4, which are the
constants in the two equations.)
iii. Change the equation to produce a graph concave down that shares the
same vertex.
In the quadratic function y = ax2 + bx + c, the value of a determines the parabola’s
concavity (both width and direction of opening). In the first 3 parabolas I have graphed,
the values for a are positive so the parabolas are concave upward. In order for the
parabola to be concave downward, a must be less than zero. I have chosen a = -2. And in
order to write the equation of a parabola with the same vertex as y = 2x2 + 3x – 4, the
vertex of y = 2x2 + 3x – 4 must be determined. One method is to change the equation into
the form y = a(x – h)2 + k in which the vertex is the point (h, k). This change requires
completing the square:
y = 2x2 + 3x – 4
y + 4 = 2x2 + 3x
3
y + 4 = 2(x2 + x)
2
9
3
9
y + 4 + = 2(x2 + x +
)
8
2
16
41
3
y +  = 2(x + )2
8
4
3 2 41
y = 2(x +  ) –
4
8


3 41
The vertex of this original parabola is (- , - ). The following equation produces a
4 8

 with this vertex, but is concave down:
parabola
y = -2(x +
3 2 41
) –
4
8


25
Some more algebraic manipulation will result in y = -2x2 – 3x –
. This parabola is
4


shown
in green in the Graphing Calculator file.

Ideas for classroom use:
This activity would fit well into an Algebra class in which students were working
on graphing quadratic equations. Part (i) is a good example of the horizontal shift that
happens to the parabola when replacing each x with (x – h) in a quadratic equation. If
used in a lesson, students should explore many examples (not just (x – 4)). A goal would
be to get students to a point where they could accurately predict what the graph of y = a(x
– h)2 + b(x – h) + c looks like, given the graph of y = ax2 + bx + c. Additionally, students
should be challenged to extend that knowledge to writing equations of parabolas that
have been translated by horizontal shift. In other words, they should be able to write the
equation of a parabola, given its graph and its “parent” equation and graph.
The focus for Part (ii) is the location of the vertex of a parabola. Teachers should
be interested in students having reasons for their choices of equations. It is possible, for
example, that a student might simply guess at a change in the equation of y = 2x2 + 3x – 4
to move its vertex into the second quadrant. This could be done by having students
explore several different kinds of changes to the equation, and then examine the results.
Then, once again, students could be challenged to predict the graph of a particular
equation, given the “parent” equation and graph. In other words, students should be
familiar with the effect (on the parabola) of changing the value of c in the equation y =
ax2 + bx + c.
The purpose of Part (iii) is twofold: it highlights the role of the a in the standard
quadratic equation (a determines width and direction of opening of the parabola). The
second purpose is to look again at the vertex—this time it will be useful to change the
form of the equation into “parabola form” in order to easily identify the vertex. If this
activity were to be used in a lesson, both of these purposes deserve attention. Again,
students’ own exploration of the graphs when a varies is a good way to help them see
these changes in concavity (or confirm what they already know). As before, this
knowledge should be extended by having students predict what the graph will look like
when given an equation, as well as having them write the equation for a graph that is
given.
Seeing the ease with which the vertex is identified when the quadratic equation is
in “parabola form” (that is, y = a(x – h)2 + k) is a good motivator for the tedious process
of completing the square. When students are well-skilled in this procedure, they should
have no trouble completing the task in Part (iii).