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Write the equation of the parabola that passes through the points (3,0)(3,0), (5,0)(5,0), and (6,−12)(6,-12). 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 (3,0)(3,0), (5,0)(5,0), and (6,−12)(6,-12). 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 (3,0)(3,0), (5,0)(5,0), and (6,−12)(6,-12). The points (3,0)(3,0) and (5,0)(5,0) are the x-intercepts of the parabola, so they give us the values of pp and qq directly. Therefore, p=3p = 3 and q=5q = 5.
  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 p=3p = 3 and q=5q = 5, we get y=a(x−3)(x−5)y = a(x - 3)(x - 5).
  3. Find value of aa: Now we need to find the value of aa. We will use the third point (6,−12)(6,-12) to do this. Substituting x=6x = 6 and y=−12y = -12 into the equation y=a(x−3)(x−5)y = a(x - 3)(x - 5) gives us −12=a(6−3)(6−5)-12 = a(6 - 3)(6 - 5).
  4. Solve for aa: Solving the equation −12=a(3)(1)-12 = a(3)(1) to find the value of aa, we get −12=3a-12 = 3a, which means a=−12/3=−4a = -12 / 3 = -4.
  5. Final equation of parabola: Now that we have found a=−4a = -4, we can write the final equation of the parabola. Substituting a=−4a = -4, p=3p = 3, and q=5q = 5 into y=a(x−p)(x−q)y = a(x - p)(x - q), we get y=−4(x−3)(x−5)y = -4(x - 3)(x - 5).

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