To derive the equation of a parabola that has its vertex at a certain point and opens in a specific direction, we can use the vertex form of a parabola’s equation. The vertex form is given as:
y = a(x – h)² + k
In this equation, (h, k) represents the coordinates of the vertex, and ‘a’ indicates the direction in which the parabola opens. If ‘a’ is positive, the parabola opens upwards; if ‘a’ is negative, it opens downwards.
In our case, the vertex is at (1, 0), so h = 1 and k = 0. Since the parabola opens downwards, we will use a negative value for ‘a’. Hence, our equation looks like this:
y = a(x – 1)²
To determine ‘a’, we need additional information, such as another point through which the parabola passes. Without this point, we can only express the equation in terms of ‘a’. Therefore, the equation of the parabola is:
y = a(x – 1)² where ‘a’ is a negative constant.
If a specific point is given, we can substitute the coordinates of that point into the equation to solve for ‘a’ and find the exact equation of the parabola.