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. (−10, −5)

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 original 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 \((-10, -5)\) to substitute \(h = -10\) and \(k = -5\) into the vertex form equation, so it becomes \(y = 2(x - (-10))^2 + (-5)\).
4. Simplify the expression inside the parentheses: \(x - (-10)\) becomes \(x + 10\), so the equation is \(y = 2(x + 10)^2 - 5\).
5. This equation represents a parabola with the same shape as \(f(x) = 2x^2\) but shifted so that its vertex is at \((-10, -5)\).

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 parabola's position and shape.

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 a shape defined by 'a', substitute the vertex coordinates (h, k) into the vertex form and use the given 'a' value to write the equation. This ensures the parabola has the correct shape and vertex location.