Understanding the Formula for a Vertical Parabola: Equation, Vertex, Coefficient

Vertical Parabola Formula

The formula for a vertical parabola is given by the equation:

y = a(x-h)^2 + k

In this equation, (h, k) represents the vertex of the parabola, and “a” represents the coefficient that determines the shape of the parabola

The formula for a vertical parabola is given by the equation:

y = a(x-h)^2 + k

In this equation, (h, k) represents the vertex of the parabola, and “a” represents the coefficient that determines the shape of the parabola.

The value of “a” determines whether the parabola opens upwards or downwards:
– If “a” is positive, the parabola opens upwards.
– If “a” is negative, the parabola opens downwards.

The vertex of the parabola is given by the coordinates (h, k), which represents the point where the parabola reaches its minimum or maximum value.

If the equation of the parabola is given in a different form, such as in general form (ax^2 + bx + c = 0), you can convert it to the vertex form using the process of completing the square.

Let’s go through an example to illustrate the use of the formula:

Example: Find the equation of a vertical parabola with a vertex at (2, -3) that opens upwards.

Solution:
We already have the vertex (h, k), which is (2, -3).
To find the value of “a,” we can use another point on the parabola. Let’s use the point (4, 1).

Substituting the values into the equation, we get:
-3 = a(2-2)^2 – a
1 = a(4-2)^2 – 3a

Expanding the equations, we have:
-3 = a(0)^2 – a = 0 – a = -a
1 = a(2)^2 – 3a = 4a – 3a = a

Since we have derived two equations for “a” with opposite signs, this means that “a” must be zero. But a parabola with a = 0 is simply a straight line, not a parabola. Therefore, there is no vertical parabola that opens upwards with a vertex at (2, -3).

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