- Recognize the vertex form of a quadratic equation: \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex.
- Examine each given pair of equations to identify their respective vertices \((h, k)\).
- Identify that the pair \(y = -4x^2\) and \(y = 4x^2\) both have their vertex at \((0, 0)\).
- Conclude that the correct pair is \(\boxed{y = -4x^2 \text{ and } y = 4x^2}\).
Explanation
- Analyzing the Vertex Form of Quadratic Equations
The vertex form of a quadratic equation is given by the expression \(y = a(x-h)^2 + k\), where the point \((h, k)\) represents the vertex of the parabola. Our goal is to identify which of the provided pairs of equations share the exact same vertex \((h, k)\). We will examine each option by extracting the vertex for each function in the pair.
- Evaluating Each Option for Shared Vertices
- Option 1: \(y = -(x+4)^2\) and \(y = (x-4)^2\).
For the first equation, \(y = -1(x - (-4))^2 + 0\), the vertex is \((-4, 0)\).
For the second equation, \(y = 1(x - 4)^2 + 0\), the vertex is \((4, 0)\).
Since \((-4, 0)
eq (4, 0)\), this pair does not have the same vertex.
- Option 2: \(y = -4x^2\) and \(y = 4x^2\).
For the first equation, \(y = -4(x - 0)^2 + 0\), the vertex is \((0, 0)\).
For the second equation, \(y = 4(x - 0)^2 + 0\), the vertex is \((0, 0)\).
Both equations share the vertex \((0, 0)\).
- Option 3: \(y = -x^2-4\) and \(y = x^2+4\).
For the first equation, \(y = -1(x - 0)^2 - 4\), the vertex is \((0, -4)\).
For the second equation, \(y = 1(x - 0)^2 + 4\), the vertex is \((0, 4)\).
Since \((0, -4)
eq (0, 4)\), this pair does not have the same vertex.
- Option 4: \(y = (x-4)^2\) and \(y = x^2+4\).
For the first equation, \(y = 1(x - 4)^2 + 0\), the vertex is \((4, 0)\).
For the second equation, \(y = 1(x - 0)^2 + 4\), the vertex is \((0, 4)\).
Since \((4, 0)
eq (0, 4)\), this pair does not have the same vertex.
- Conclusion and Final Answer
After evaluating all four given pairs of equations, we have determined that the pair \(y = -4x^2\) and \(y = 4x^2\) consists of two parabolas that both share the vertex \((0, 0)\). Therefore, this is the correct choice.
Examples
Understanding the vertex of a parabola is fundamental in fields like physics and engineering. For example, when launching a projectile, the vertex of its parabolic trajectory represents the maximum height it reaches. If two different projectiles, described by equations with the same vertex, are launched, it means they share the same peak height at the same horizontal displacement. Recognizing shared geometric properties helps in designing systems where specific spatial constraints must be met, such as in trajectory optimization or structural design.