__Category__: Analytic Geometry, Algebra"Published in Newark, California, USA"

Find the equation of a parabola with horizontal axis, vertex on y axis, and passing through the points (2, 4) and (8, -2).

__Solution:__

To illustrate the problem, let's plot all the given items and sketch the parabola in the rectangular coordinate system as follows

Photo by Math Principles in Everyday Life |

As you can see in the figure, we can have two equations of parabola based on the given items in the problem. Since the axis of the parabola is horizontal and it opens to the right, the equation of a parabola in standard form is

The vertex of a parabola is located in y-axis, the coordinates of the vertex is (0, k). The above equation becomes

If (2, 4) is one of the points of a parabola, substitute the values of x and y to the above equation

If (8, -2) is one of the points of a parabola, substitute the values of x and y to the above equation

Equate

Multiply both sides of the equation by 32 and solve for the value of k

Divide both sides of the equation by 3

Equate each factor to zero and solve for the value of k

If

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is

If

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is