What is the equation of a parabola whose vertex is (-6,0) and whose graph is half as tall as y = x^2 and opens upside down?
The equation of a parabola whose vertex is at (h,k) is given by
y = a(x - h)² + k
So many parabolas have the vertex at (-6,0) and
open down. We'll find one.
Substitute (h,k) = (-6,0) in the standard equation
y = a(x - h)^2 + k
y = a(x - (-6) )^2 + 0
y = a(x + 6)^2 = ax2 +12ax+36a
But then choose any negative number for a, and its
graph will open downward. For instance, letting
a = -1 gives this parabola:
y = -(x + 6)^2
As for the height of the parabola y = x^2, it passes through the origin and opens upward. The question of tallness of the parabola is thus meaningless.
Assuming its width to be double that of the parabola y = x^2 (thus wider and shorter in appearance), the equation then takes the form:
y = -2(x + 6)^2