Quadratic Function
y = x^{2} + bx + 1
(varying ÔbÕ)
Karyn Carson
LetÕs start by looking at the
graphs of the function y = x^{2} + bx + 1, and vary b from
3 to 3:
What IÕd
like to look at is the vertices of each of the parabolas. To figure these out, IÕll use the
vertex formula and an Excel
spreadsheet:
b 
xcoordinate 
ycoordinate 
3 
1.5 
1.25 
2 
1 
0 
1 
0.5 
0.75 
0 
0 
1 
1 
0.5 
0.75 
2 
1 
0 
3 
1.5 
1.25 
And now, letÕs
plot these points on the graphs of the other functions:
It seems
as though a pattern is formed – the vertices of the parabolas look like
they form another parabola! If we
Ôconnect the dotsÕ, we do get a parabola:
This
parabola is a reflection of the graph of x^{2} +1, so itÕs equation
must be y = x^{2} +1. If
we plug the xvalues from the vertices into the equation, we can see that the
points do fit the equation.
xcoordinate 

1.5 
1.25 
1 
0 
0.5 
0.75 
0 
1 
0.5 
0.75 
1 
0 
1.5 
1.25 