this exploration, I will look at graphs of the quadratic for various
values of b and discover a
relationship among the vertices.
Below are the graphs of , where b is equal to integers ranging from -3 to 3, on one grid.
When , the graph is fuschia. When , the graph is red. When , the graph is blue. When , the graph is green. When , the graph is light blue. When , the graph is yellow. When , the graph is gray.
Notice that when and , the graphs cross the x-axis at exactly two places, when and , the graphs touch the x-axis at precisely one place, and for the other values of x, the graphs do not touch the x-axis.
A conjecture is that the vertices of the above
quadratics all lie along the quadratic . Below is a
picture of the quadratic with the
previous quadratics and it appears that the conjecture is correct. To be more certain, I will calculate
the vertices of the quadratics and determine whether they lie along the
To find the vertex of a parabola, I can calculate the
derivative of the quadratic, set that derivative equal to zero, solve for x and then plug that x-value into the original equation to find the
corresponding y-value. I will then have the vertex.
I will find the vertex of .
Find the derivative.
Set it equal to zero and solve for x.
Plug that value into the original equation and solve for y.
the value of the vertex is .
Likewise, using the method above you can get that the