Ask AI: A parabola has its vertex at (2,1) and crosses the x-axis at(1,0) and at(3,0) . Which of the equations below has this parabola for its graph?

Given the vertex of the parabola is at (2,1) and it crosses the x-axis at (1,0) and (3,0), we can derive the equation of the parabola in two different forms and then compare it with the given options (though none are shown here). ### Vertex Form A standard form for the equation of a parabola given its vertex \((h, k)\) is: \[ y = a(x-h)^2 + k \] For the given vertex (2,1), the equation becomes: \[ y = a(x-2)^2 + 1 \] Since the parabola crosses the x-axis at (1,0) and (3,0), we can substitute one of these points into the equation to find \( a \). Let's use (1,0): \[ 0 = a(1-2)^2 + 1 \] \[ 0 = a(1)^2 + 1 \] \[ 0 = a + 1 \] \[ a = -1 \] So the vertex form of the equation is: \[ y = -(x-2)^2 + 1 \] ### Factored Form Another way to express the parabola is in factored form: \[ y = a(x-r_1)(x-r_2) \] Given the roots \(r_1 = 1\) and \(r_2 = 3\), the equation becomes: \[ y = a(x-1)(x-3) \] To find \( a \), we use the fact that the vertex form y should be equal at \( x = 2 \): When \(x = 2\): \[ y = -(2-2)^2 + 1 = 1 \] Checking the factored form: \[ y = a(2-1)(2-3) \] \[ y = a(1)(-1) \] \[ y = -a \] Given y (vertex y-coordinate) is 1. \[ 1 = -a \] \[ a = -1 \] Thus, the factored form is: \[ y = -1(x-1)(x-3) \] ### Conclusion Both the forms vertex and factored reduced to the same parabola equation: \[ y = -(x-2)^2 + 1 \] or \[ y = -1(x-1)(x-3) \] Given this, the correct equation of the parabola is: \[ y = -(x-2)^2 + 1 \] Please compare with the equations provided (none are visible to me) and choose the one matching this. If one of the options is: \[ y = -(x-2)^2 + 1 \] or \[ y = -(x-1)(x-3) \] That would be your equation.