Verified Answer :

  • Recall the vertex form of a parabola: \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
  • Substitute the target vertex \((-2, 0)\) into the vertex form to find the target equation structure: \(y = a(x + 2)^2\).
  • Analyze each option to find that only \(y = (x + 2)^2\) matches the required form.
  • The final answer is \(\boxed{y = (x + 2)^2}\).

Explanation

  1. Understanding the Vertex Form of a Parabola

The goal is to identify which of the provided equations represents a parabola with its vertex at the point \((-2, 0)\). The standard 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 \(a eq 0\). Given the vertex \((-2, 0)\), we have \(h = -2\) and \(k = 0\). Substituting these values into the vertex form formula gives: \[y = a(x - (-2))^2 + 0 = a(x + 2)^2\] for some non-zero constant \(a\). Any equation that can be written in the form \(y = a(x + 2)^2\) will have its vertex at \((-2, 0)\).

  1. Evaluating the Given Equations

We test each of the given options to see which matches the form \(y = a(x + 2)^2\):

  1. For \(y = -2x^2\), the vertex form is \(y = -2(x - 0)^2 + 0\), so its vertex is \((0, 0)\).
  2. For \(y = (x + 2)^2\), this is already in the form \(y = a(x + 2)^2\) with \(a = 1\), so its vertex is \((-2, 0)\).
  3. For \(y = (x - 2)^2\), the vertex form is \(y = 1(x - 2)^2 + 0\), so its vertex is \((2, 0)\).
  4. For \(y = x^2 - 2\), the vertex form is \(y = 1(x - 0)^2 - 2\), so its vertex is \((0, -2)\).

Only the second equation, \(y = (x + 2)^2\), corresponds to a parabola with a vertex at \((-2, 0)\).

Examples

Understanding vertex form is essential in fields like physics and engineering for modeling projectile motion or designing curved structures like satellite dishes. For example, if an engineer is designing a bridge arch, identifying the vertex of a parabolic support beam helps determine the lowest or highest point of the span, ensuring structural integrity and precise placement relative to the ground. By mastering how parameters in \(y = a(x - h)^2 + k\) shift and scale the parabola, you can accurately model these physical systems.

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