Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x² but with the given point as the vertex. (5, 3)

Verified step by step guidance

1. Recall that the vertex form of a parabola is given by the equation \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola and \(a\) determines the shape (width and direction) of the parabola.
2. Identify the value of \(a\) from the given function \(f(x) = 2x^2\). Here, \(a = 2\), which means the parabola opens upward and is narrower than the standard parabola \(y = x^2\).
3. Use the given vertex point \((5, 3)\) to substitute \(h = 5\) and \(k = 3\) into the vertex form equation, so it becomes \(y = 2(x - 5)^2 + 3\).
4. This new equation represents a parabola with the same shape as \(f(x) = 2x^2\) but shifted so that its vertex is at \((5, 3)\).
5. You can verify the correctness by expanding the vertex form and comparing it to the standard form or by plotting the graph to see the vertex at the correct point.

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and understand the graph's shape and position.

Effect of the 'a' Coefficient on Parabola Shape

The coefficient 'a' in a quadratic function affects the parabola's width and direction. If |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider. A positive 'a' opens upward, while a negative 'a' opens downward.

Using a Given Vertex to Write the Equation

When given a vertex and the shape of a parabola, substitute the vertex coordinates (h, k) into the vertex form and use the known 'a' value to write the equation. This ensures the parabola has the correct shape and position.