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Write the equation of the parabola that passes through the points (−6,0)(-6,0), (1,0)(1,0), and (−5,−42)(-5, -42). 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 (−6,0)(-6,0), (1,0)(1,0), and (−5,−42)(-5, -42). 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 Points: Identify the values of pp and qq from the given points where the parabola crosses the x-axis, which are the points (−6,0)(-6,0) and (1,0)(1,0). Thus, p=−6p = -6 and q=1q = 1.
  2. Write General Form: Write the general form of the equation using the identified pp and qq: y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting p=−6p = -6 and q=1q = 1, we get y=a(x+6)(x−1)y = a(x + 6)(x - 1).
  3. Find Value of a: Use the third point (−5,−42)(-5, -42) to find the value of a. Substitute x=−5x = -5 and y=−42y = -42 into the equation: −42=a(−5+6)(−5−1)-42 = a(-5 + 6)(-5 - 1).
  4. Solve for aa: Simplify and solve for aa: −42=a(1)(−6)-42 = a(1)(-6), −42=−6a-42 = -6a, a=7a = 7.

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