Exploring a, b, c for a parabola.

Joshua DuMont

Parabolas with vertical lines of symmetry can be written in the form y=ax2+bx+c , a≠0.

Let’s fix a=1 and b=1 in this equation and look at the parabola for a few values of c. notice the parabola passes though the point (0,c) each time.

The y-intercept of a parabola in our form is the point (0,c)

We can see that this is the case by setting x=0. Then, y=a(0)2+b(0)+c=c.

From our pictures we can also guess that changing the value of c in the equation translates the graph vertically. This is shown to be true by taking a parabola yoriginal= ax2+bx+c1 and changing the value of c by Δc= c2-c1

The equation now reads: ynew= ax2+bx+c2= ax2+bx+c1+ c2-c1=ax2+bx+c1+ Δc= yoriginal+ Δc. Thus each point in the original graph is shifted up Δc units.

Parabolas are symmetric across a line called the axis of symmetry.

Looking at the graphs as a and b change, we can make a few observations.

If we look at the graph y= ax2+2x+1 for a couple a values, the graphs seem to be symmetric around the line x=-1/a

When we look at the graphs of y= x2+bx+1 for a couple b values it looks like the graphs are symmetric around the line x=-b/2

We can guess parabolas in the form y=ax2+bx+c have an axis of symmetry: x=.

If this is the case, then any line y=k (horizontal lines are perpendicular to vertical ones) should intersect the parabola at two points with x-coordinates the same distance from x=.

Setting y=k yields:

a(=0

=0

=0

=

=

=

=

There are two x-coordinates which give a y-coordinate of y=k and they are  units to the right and left of x=.

{Using a y-value of k=0 above will find the x-intercepts of the graph at

= =   .

Parabolas have a highest or lowest point called a vertex. This point can't have a reflection across the axis of symmetry or that point would be at the same height and our choice would not be highest or lowest. This lets us know that the vertex must be on the axis of symmetry. If we set x= then:

y= a2+b()+c= +c=

The vertex of a parabola in our form is at the point:

(   ,   ).