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. Maximum = 4 at x = -2

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. This concept is crucial for maintaining the same shape as the given parabolas.

Finding Vertex from Given Conditions

To write the equation of a parabola in vertex form based on given conditions, one must identify the vertex coordinates. In this case, the maximum is given as 4 at x = -2, which means the vertex is (-2, 4). This information allows us to substitute h and k into the vertex form equation, ensuring that the new parabola has the desired maximum while retaining the shape defined by the original parabolas.