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Write the equation of the parabola that passes through the points (−4,0)(-4,0), (−2,0)(-2,0), and (−1,−18)(-1,-18). 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 (−4,0)(-4,0), (−2,0)(-2,0), and (−1,−18)(-1,-18). 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: Identify the x-intercepts from the given points (−4,0)(-4,0) and (−2,0)(-2,0). These points indicate where the parabola crosses the x-axis, so p=−4p = -4 and q=−2q = -2.
  2. Formulate parabola equation: Formulate the equation of the parabola using the standard form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting pp and qq, we get y=a(x+4)(x+2)y = a(x + 4)(x + 2).
  3. Find value of aa: Use the third point (−1,−18)(-1, -18) to find the value of aa. Substitute x=−1x = -1 and y=−18y = -18 into the equation: −18=a(−1+4)(−1+2)-18 = a(-1 + 4)(-1 + 2).
  4. Solve for a: Simplify and solve for a: −18=a(3)(1)-18 = a(3)(1), −18=3a-18 = 3a, a=−18/3a = -18 / 3, a=−6a = -6.
  5. Write final equation: Write the final equation of the parabola substituting a=−6a = -6 into y=a(x+4)(x+2)y = a(x + 4)(x + 2). The equation becomes y=−6(x+4)(x+2)y = -6(x + 4)(x + 2).

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