How to find the equation of a parabola with vertex on the line y = -3x?

Question

Its axis are parallel to the y-axis and passing through (-7,13) and (5,1).

Answer 1

With the vertex given as $ \ (X \ , \ -3X) \ , $ and the information that the symmetry axis of the parabola is "vertical" (parallel to the $ \ y-$ axis), we can write the standard-form equation of the parabola as

$$ 4p \ (y + 3X) \ = \ (x - X)^2 \ \ . $$

For the points we are told lie on this curve, we obtain two simultaneous equations,

$$ 4p \ (13 + 3X) \ = \ (-7 - X)^2 \ \ \text{and} \ \ 4p \ (1 + 3X) \ = \ (5 - X)^2 \ \ . $$

This looks a bit messier than it actually is, since some terms cancel when we subtract one equation from the other. We will get a simple relation between $ \ p \ $ and $ \ X \ $ which we can then substitute into either of these equations to find $ \ X \ $ , and thus the coordinates of the vertex and the value of $ \ p \ $ .

There are, in fact, two solutions for $ \ X \ $ and, correspondingly, two parabolas through the specified points with their vertices on $ \ y = -3x \ $ . The graph below presents the situation.

The two parabolas intersect at the points $ \ (-7,13) \ \ $ and $ \ (5,1) \ $ , as required. The red line is $ \ y = -3x \ , $ on which the vertices of both parabolas lie.