To determine which equation describes the parabola correctly, we can start by noting that the vertex form of a parabola's equation is: \[ y = a(x-h)^2 + k \] where \((h, k)\) is the vertex of the pa

To determine which equation represents the given parabola, we need to consider the features provided: 1. The vertex of the parabola is at \((2, 1)\). 2. The parabola crosses the x-axis at \((1, 0)\) and \((3, 0)\). Given these features, the standard form of the equation of a parabola with vertex \((h, k)\) is: \[ y = a(x - h)^2 + k \] For this problem, the vertex \((h, k)\) is \((2, 1)\), so the equation form should look like: \[ y = a(x - 2)^2 + 1 \] Given that the parabola crosses the x-axis at \((1, 0)\) and \((3, 0)\), this means these points are the roots of the equation. Substitute the point \((1, 0)\) into the general equation to find \(a\): \[ 0 = a(1 - 2)^2 + 1 \] \[ 0 = a(1) + 1 \] \[ a = -1 \] So, we substitute \(a\) and the vertex into the equation: \[ y = -1(x - 2)^2 + 1 \] \[ y = -(x - 2)^2 + 1 \] Now, let's match this with the given choices: A. \( y = 1 + (x + 2)^2 \) B. \( y = 1 - (x - 2)^2 \) C. \( y = -1 - (x-2)^2 \) D. \( y = -1 - (x + 2)^2 \) Clearly, option B \[ y = 1 - (x - 2)^2 \] matches our derived form. Thus, the correct equation is: **B. \( y = 1 - (x - 2)^2 \)**