To find an equation of a parabola with a specific vertex, we can use the vertex form of a quadratic equation, which is given by:
y = a(x – h)² + k
In this equation, (h, k) represents the vertex of the parabola. Here, the vertex is given as (5, 3), so we can substitute these values into the equation:
y = a(x – 5)² + 3
The next step is to determine the value of ‘a’. The value of ‘a’ affects the width and direction (opening upwards or downwards) of the parabola. For instance, if ‘a’ is positive, the parabola opens upwards, and if ‘a’ is negative, it opens downwards. You can choose any non-zero value for ‘a’ depending on how narrow or wide you want the parabola to be.
For example, if we select a = 1, the equation becomes:
y = (x – 5)² + 3
This graph will be a parabola that opens upwards with its vertex at the point (5, 3). If instead we choose a = -1, the equation would transform to:
y = -(x – 5)² + 3
This would create a parabola that opens downwards, also with its vertex at (5, 3).
In summary, the general form for the equation of a parabola with a vertex at (5, 3) can be written as:
y = a(x – 5)² + 3
where ‘a’ can be any non-zero real number to determine the direction and width of the parabola.