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Write the equation of the parabola that passes through the points (−1,12(-1,12), (−2,0(-2,0), and (1,0(1,0). Write your answer in the form y=a(x−p)(x−q)y = a(x - p)(x - q), where aa, pp, and qq are integers, decimals, or simplified fractions.

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Q. Write the equation of the parabola that passes through the points (−1,12(-1,12), (−2,0(-2,0), and (1,0(1,0). Write your answer in the form y=a(x−p)(x−q)y = a(x - p)(x - q), where aa, pp, and qq are integers, decimals, or simplified fractions.
  1. Identify x-intercepts: We have the points (−1,12)(-1,12), (−2,0)(-2,0), and (1,0)(1,0). The points (−2,0)(-2,0) and (1,0)(1,0) are the x-intercepts of the parabola, so they give us the values of pp and qq directly. Therefore, p=−2p = -2 and q=1q = 1.
  2. Write parabola equation: Using the values of pp and qq, we can write the equation of the parabola in the form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting −2-2 for pp and 11 for qq, we get y=a(x+2)(x−1)y = a(x + 2)(x - 1).
  3. Find value of a: Now we need to find the value of aa. We can use the point (−1,12)(-1,12) to do this. Substituting −1-1 for xx and 1212 for yy in the equation y=a(x+2)(x−1)y = a(x + 2)(x - 1), we get 12=a(−1+2)(−1−1)12 = a(-1 + 2)(-1 - 1).
  4. Solve for aa: Solving the equation 12=a(1)(−2)12 = a(1)(-2), we find that 12=−2a12 = -2a, which means a=−6a = -6.
  5. Final equation of parabola: Now that we have found a=−6a = -6, we can write the final equation of the parabola. Substituting −6-6 for aa, −2-2 for pp, and 11 for qq in the equation y=a(x−p)(x−q)y = a(x - p)(x - q), we get y=−6(x+2)(x−1)y = -6(x + 2)(x - 1).

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