Investigation of the Standard Quadratic Equation
Dr. Jim Wilson
This is the graph of the values of ÔbÕ within the set of -2 ³ b £ 2 for the base equation:
The locus is the set of points which satisfies a given condition, in this case the given condition is the set if points on the x-axis of particular values where -2 ³ b £ 2 on the standard quadratic equation . The locus is a function that contains all of the vertices for the condition of functions. In this example, the locus equation must pass through -1 and 1 on the x-axis because the quadratic solutions lay within this domain. Also, all the parabolas pass through 1 on the y-axis, which would make the point (0,1) the vertex of the locus equation.
This equation must satisfy all of the requirements, thus the equation must concave downward to have itÕs highest point as the vertex and must have -1 and 1 as the roots. Therefore, the equation would be the product of (-x+1) and (-x+1) which equals:
, resulting in the graph below where the locus is represented by the aqua-colored graph.
Now consider the overlay of graphs where there are fixed ÔbÕ and ÔcÕ values and we vary the values -2 ³ a £ 2, for the equation where b=1 and c=1.
As we vary the values for ÔaÕ, when it is negative, the parabola concaves downward, when it is positive, it concaves upward, but when it is Ô0Õ, the equation is linear. As the absolute value of the ÔaÕ terms become larger, the vertices get closer to the y-axis. All of the quadratic equations have a common tangent line which is the line when a=0.
Now lets explore the graphs as we vary ÔcÕ and have fixed constants for ÔaÕ and ÔbÕ in the quadratic equation , where has a domain of -2 ³ c £ 2. In graph 4, by varying the ÔcÕ values, the vertices have a vertical increase and the shape of the parabolas narrow.
As the ÔcÕ values get larger, the graphs also cross the y-axis at a higher point. The vertices all move along a common line of x=-0.5, which in this case is the locus of vertices.