Make the equation of a parabola with an axis of symmetry parallel to the Oy

Make the equation of a parabola with an axis of symmetry parallel to the Oy axis if the parabola passes through the point M (2; 0) and has a vertex A (-2; 4).

Solution:

The parabola equation has the form: a * x ^ 2 + b * x + c.
The abscissa of the vertex of the parabola is found by the formula: -b / 2a. It means:
-b / 2a = -2;

-b = -4a;

b = 4a;

The expression for the ordinate of the vertex of the parabola looks like this:

4a – 2b + c = 4;

4a – 2 * 4a + c = 4;

-4a + c = 4;

Let’s compose an expression for the ordinate of point M:
4a + 2b + c = 0;

4a + 2 * 4a + c = 0;

12a + c = 0;

We got the following system of equations:
-4a + c = 4;

12a + c = 0;

Express the value of c from the first equation and substitute it into the second:
c = 4 + 4a;

12a + 4 + 4a = 0;

12a + 4a = -4;

16a = -4;

a = -4 / 16;

a = -0.25;

If a = -0.25, then c = 4 + 4a = 4 + 4 * (-0.25) = 3;

If a = -0.25, then b = 4a = 4 * (-0.25) = -1;

Answer: the equation of the parabola is: -0.25 * x ^ 2 – x + 3.