To find the equation of a parabola from the given points and vertex, we can use the vertex form of a parabola's equation. The vertex form is: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex

To find the correct equation for the given parabola, we need to use the fact that the parabola has its vertex at \((2, 1)\) and crosses the x-axis at \((1, 0)\) and \((3, 0)\). The standard form of the equation of a parabola with its vertex at \((h, k)\) is given by: \[ y = a(x - h)^2 + k \] In this case, the vertex \((h, k)\) is \((2, 1)\), so the equation becomes: \[ y = a(x - 2)^2 + 1 \] We also know the parabola crosses the x-axis at \((1, 0)\) and \((3, 0)\). The x-intercepts provide a way to find \(a\). Let's use one of the intercepts, \((1, 0)\), to find \(a\): \[ 0 = a(1 - 2)^2 + 1 \] \[ 0 = a(1) + 1 \] \[ 0 = a + 1 \] \[ a = -1 \] So now we have the equation: \[ y = -1(x - 2)^2 + 1 \] \[ y = 1 - (x - 2)^2 \] Given this, the correct equation for the parabola is: \[ y = 1 - (x - 2)^2 \] Therefore, the correct answer is: B. \[ y = 1 - (x - 2)^2 \]