def idealized_arc_curve(width, index):
    """
    Returns the value at x for the parabola of width n and height n / 2.
    Formula taken from https://www.desmos.com/calculator/awtnrxh6rk.

    Parameters
    ----------
    width : int
        Length of the time series to calculate the parabola for.
    index : int
        location to compute the parabola value at.

    Returns
    -------
    float : y
        The value at index for the parabola.

    """
    height = width / 2
    c = width / 2
    b = height
    a = height / (width / 2) ** 2
    y = -(a * (index - c) ** 2) + b
    return y
