How Does the Vertex Location of a Parabola Change?

​
plot y = a
2
x
+ b x + c
a
b
c
show
label
grid
reset
The purpose of this visualization is to help students understand the relationship between the location of the vertex of the parabolic curve described by
y=a
2
x
+bx+c
and the values of the coefficients
a
,
b
,
c
.

Details

It is understood the vertex of a parabolic curve described by the quadratic function
f(x)=a
2
x
+bx+c
is located where
x=-
b
2a
and hence,
y=f-
b
2a
=-
2
b
-4ac
4a
. It is insightful to view the location of the vertex as being the intersection point of the line
y=
b
2
x+c
and the parabola
y=-a
2
x
+c
to better understand how the coefficients
a
,
b
,
c
affect the location of the vertex.
If the linear coefficient
b≠0
and the constant coefficient
c
remains fixed while the quadratic coefficient
a
varies, then you can observe the vertex of the parabola sliding along the line
y=
b
2
x+c
as the parabola
y=-a
2
x
+c
changes. If the quadratic coefficient
a≠0
and the constant coefficient
c
remains fixed while the linear coefficient
b
varies, then you can observe the vertex of the parabola sliding along the parabola
y=-a
2
x
+c
as the line
y=-
b
2
x+c
changes.

External Links

Parabola (Wolfram MathWorld)
Quadratic Polynomial (Wolfram MathWorld)

Permanent Citation

Eric Schulz
​
​"How Does the Vertex Location of a Parabola Change?"​
​http://demonstrations.wolfram.com/HowDoesTheVertexLocationOfAParabolaChange/​
​Wolfram Demonstrations Project​
​Published: October 3, 2007