In Exercises 53–56, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x^2 or g(x) = -3x^2, but with the given maximum or minimum. Minimum = 0 at x = 11

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is expressed as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form is particularly useful for identifying the vertex directly and understanding the transformations applied to the basic parabola y = x². The parameter 'a' determines the width and direction of the parabola, with positive values opening upwards and negative values opening downwards.

Effect of 'a' on Parabola Shape

The coefficient 'a' in the vertex form of a parabola affects its width and direction. If |a| > 1, the parabola is narrower than the standard parabola, while |a| < 1 makes it wider. Additionally, if 'a' is positive, the parabola opens upwards, indicating a minimum point, whereas a negative 'a' indicates it opens downwards, signifying a maximum point. Understanding this helps in determining the shape of the parabola based on the given function.

Finding the Vertex

The vertex of a parabola is a critical point that represents either the maximum or minimum value of the function. In the context of the question, the vertex is given as (11, 0), indicating that the parabola has a minimum value of 0 at x = 11. To write the equation in vertex form, one must substitute the vertex coordinates into the vertex form equation and adjust the coefficient 'a' to match the desired shape of the parabola, either from f(x) = 3x² or g(x) = -3x².