Question

write the equation of the parabola that passes through the points (2, 0), (-1, 0write your answer in the form y = a(x - p)(x - q), where a, p, and q are integeor simplified fractions.

Explanation:

Step1: Identify roots from points

The parabola passes through \((2, 0)\) and \((-1, 0)\), so roots are \(x = 2\) and \(x=-1\). In factored form \(y = a(x - p)(x - q)\), \(p = 2\), \(q=-1\) (or vice versa). So equation is \(y = a(x - 2)(x + 1)\).

Step2: Find value of \(a\)

Wait, maybe there's a third point missing? Wait, the original problem might have a third point (maybe a typo in the user's input). Wait, if we assume a common case, maybe the third point is, say, \((0, -2)\) (common in such problems). Let's check: if \(x = 0\), \(y=-2\). Substitute into \(y = a(x - 2)(x + 1)\): \(-2=a(0 - 2)(0 + 1)\) → \(-2=a(-2)(1)\) → \(-2=-2a\) → \(a = 1\). So equation is \(y=(x - 2)(x + 1)\) or \(y = 1(x - 2)(x - (-1))\). But since the user's problem might have a third point (maybe the user missed it), but assuming a typical problem, if we take \(a = 1\) (or another integer), but let's confirm. Wait, maybe the original problem has three points, but the user's input is cut. Wait, the given points are \((2,0)\), \((-1,0)\), and maybe another. Wait, if we proceed with the two roots, the factored form is \(y = a(x - 2)(x + 1)\). If we assume a third point, say \((0, -2)\) (common), then \(a = 1\). So the equation is \(y=(x - 2)(x + 1)\) or \(y = 1(x - 2)(x - (-1))\).

Answer:

\(y = (x - 2)(x + 1)\) (assuming \(a = 1\) and the third point leads to \(a = 1\); if there's a different third point, \(a\) changes, but with the given info, this is a common solution)