Question

write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 4x² or g(x) = -4x², but with the given minimum. minimum = 0 at x = 8

Explanation:

Step1: Recall Vertex Form of Parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Since the parabola has a minimum, \(a>0\) (opens upward). The minimum value is \(0\) at \(x = 8\), so the vertex \((h, k)=(8, 0)\). The shape is the same as \(f(x)=4x^2\) or \(g(x)=-4x^2\), so \(|a| = 4\). Since it has a minimum, \(a = 4\).

Step2: Substitute Vertex and \(a\) into Vertex Form

Substitute \(a = 4\), \(h = 8\), and \(k = 0\) into \(y = a(x - h)^2 + k\). We get \(y = 4(x - 8)^2 + 0\), which simplifies to \(y = 4(x - 8)^2\).

Answer:

\(h(x)=4(x - 8)^2\)