I need to develope a formula and show how to come to that conclusion Please READ

Question

Consider the graphs of equations of the quadratic form y=ax²+bx+c that have two x-intercepts.  Let d denote the shortest distance from the axis of the parabola to either of the x-intercepts, and let h denote the value of the y-coordinate of the vertex.  Explore the relationship between d and h and then develop a formula for this relationship.

Answer 1

At the vertex of the parabola, the slope is zero (becomes horizontal straight line). But slope is just the first derivative of y with respect to x, thus:

dy/dx = 0

2ax+b=0

x=-b/2a

 

At this x-value, the corresponding y-value which is just the value of h, is given by:

y = h = c

 

Now, the x coordinate of the vertex is -b/2a and the x-intercepts are where y=0=ax²+bx+c. Solving this quadratic equation, we get:

x = -b/2a +- sqrt(b²-4ac)/2a

 

Thus the distance between axis of the parabola (which of course, goes through the vertex) and any of the two intercepts is:

d = sqrt(b²-4ac)/2a

d = sqrt(b²-4ah)/2a

 

Which is a relationship between d and h.