Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 3x² or g(x) = -3x², but with the given maximum or minimum. Maximum = 4 at x = -2

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 and direction (up or down) of the parabola.
2. Identify the vertex from the problem: the maximum value is 4 at \(x = -2\), so the vertex is \((-2, 4)\), meaning \(h = -2\) and \(k = 4\).
3. Determine the value of \(a\) by considering the shape of the parabola. Since the parabola has the same shape as either \(f(x) = 3x^2\) or \(g(x) = -3x^2\), the value of \(a\) is either \$3\( or \)-3\(. Because the parabola has a maximum, it opens downward, so \)a = -3$.
4. Substitute the values of \(a\), \(h\), and \(k\) into the vertex form equation: \(y = -3(x - (-2))^2 + 4\).
5. Simplify the expression inside the parentheses to get the final vertex form equation: \(y = -3(x + 2)^2 + 4\).

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 maximum or minimum point and the direction the parabola opens, based on the value of a.

Effect of the Leading Coefficient on Parabola Shape

The leading coefficient 'a' in a quadratic function determines the parabola's shape and direction. If a is positive, the parabola opens upward with a minimum vertex; if negative, it opens downward with a maximum vertex. The absolute value of a affects the width or steepness of the parabola.

Using Given Vertex to Write Equation

When given a vertex and a shape condition, you substitute the vertex coordinates (h, k) into the vertex form and use the known 'a' value from the original function to maintain the shape. This allows writing the new quadratic equation that matches the required maximum or minimum.