In Exercises 49–52, write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x^2 but with the given point as the vertex. (5, 3)

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 and 'a' determines the width and direction of the parabola. This form is particularly useful for graphing and understanding the transformations of parabolas.

Parabola Shape and Coefficient 'a'

The coefficient 'a' in the vertex form affects the shape of the parabola. If 'a' is positive, the parabola opens upwards, while a negative 'a' indicates it opens downwards. The absolute value of 'a' also influences the width; larger values result in a narrower parabola, while smaller values create a wider one.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. In this context, moving the vertex of the parabola to a new point (5, 3) requires applying a horizontal and vertical shift to the standard form of the parabola, which is essential for writing the equation in vertex form.