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Write the equation of the parabola that passes through the points (−2,0(-2,0), (2,0)(2,0), and (3,−15)(3,-15). 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), (2,0)(2,0), and (3,−15)(3,-15). 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 pp and qq: Identify the values of pp and qq from the given xx-intercepts of the parabola.\newlineSince the parabola passes through (−2,0)(-2,0) and (2,0)(2,0), these points are the xx-intercepts of the parabola. Therefore, p=−2p = -2 and q=2q = 2.
  2. Write general form: Write the general form of the parabola using the identified values of pp and qq. \newlineThe general form of the parabola is y=a(x−p)(x−q)y = a(x - p)(x - q). \newlineSubstituting p=−2p = -2 and q=2q = 2, we get y=a(x+2)(x−2)y = a(x + 2)(x - 2).
  3. Find value of aa: Use the third point (3,−15)(3, -15) to find the value of aa. \newlineSubstitute x=3x = 3 and y=−15y = -15 into the equation y=a(x+2)(x−2)y = a(x + 2)(x - 2) to solve for aa.\newline −15=a(3+2)(3−2)-15 = a(3 + 2)(3 - 2) −15=a(5)(1)-15 = a(5)(1) \newline−15=5a-15 = 5aa=−3a=-3
  4. Write final equation: Write the final equation of the parabola using the found value of aa.\newlineSubstitute a=−3a = -3 into the equation y=a(x+2)(x−2)y = a(x + 2)(x - 2) to get the final equation of the parabola. \newliney=−3(x+2)(x−2)y = -3(x + 2)(x - 2)

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