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Write the equation of the parabola that passes through the points (2,0)(2,0), (−2,−24)(-2, -24), and (−3,0)(-3,0). 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 (2,0)(2,0), (−2,−24)(-2, -24), and (−3,0)(-3,0). 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: Identify the x-intercepts from the given points (2,0)(2,0) and (−3,0)(-3,0). These points indicate where the parabola crosses the x-axis, so p=2p = 2 and q=−3q = -3.
  2. Write general form: Write the general form of the equation using the identified x-intercepts. The equation becomes y=a(x−2)(x+3)y = a(x - 2)(x + 3).
  3. Substitute point: Substitute the point (−2,−24)(-2, -24) into the equation to find the value of aa. Plugging in x=−2x = -2 and y=−24y = -24 gives: −24=a(−2−2)(−2+3)-24 = a(-2 - 2)(-2 + 3).
  4. Simplify and solve: Simplify and solve for aa. The equation becomes −24=a(−4)(1)-24 = a(-4)(1), so −24=−4a-24 = -4a. Dividing both sides by −4-4 gives a=6a = 6.

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