Exploring Loci of Parabolas
We will look at the graph of y = ax2 + bx + c
(b is varied and a and c are held constant)
First, we will graph the equation where a, b, and c are equal to 1.
Like any parabola the graph is concave upward when a is positive. Therefore the graph will be concave downward when a is negative. Because c = 1 the graph is not symmetrical around the y-axis. The parabola passes through the points (0.1) because c = 1. When x = 0, the graph would be symmetrical around the y-axis. Next, we will graph the equation where b is varied and a and c remain 1.
The following graphs intersect the y-axis at (0,1) because c = 1. The vertex is always left of the graph when b is greater than 0. When b = 0 the vertex is on the x-axis. The real root of the equation occurs where the parabola intersects the x-axis. The graph of y = x2 + 2x + 1 have a real root because it intersect the x-axis. The graph of y = x2 + 1x + 1 does not have any real root because it does not intersect the x-axis. When b is positive the real roots occur on the negative side of the y-axis.
When b is negative the graphs intersect the y-axis at (o,1). The vertex is to the right of the graph and the vertex is on the y-axis when b = 0. The graph have no real root when b = -1.
Purple y = x2 + 4x + 2
Red y = 2x2 + 4x + 2
Blue y = 3x2 + 4x + 2
Green y = - x2 + 4x + 2
Teal y = -2x2 + 4x + 2
Yellow y = -3x2 + 4x + 2
From the picture above, we can conclude that each parabola passes through the point (0,2). When a is positive the parabola is concave up and the parabola is concave down when a is negative. The x-coordinate of the vertex is x = -b/2a.