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Write the equation of the parabola that passes through the points shown in the table. (−2,0)(-2,0), (2,−11)(2,-11), (4,0)(4,0) 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 shown in the table. (−2,0)(-2,0), (2,−11)(2,-11), (4,0)(4,0) 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. Points (−2,0)(-2,0) and (4,0)(4,0) are x-intercepts, so p=−2p = -2 and q=4q = 4.
  2. Write equation in form: Write the equation in the form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting pp and qq, we get y=a(x+2)(x−4)y = a(x + 2)(x - 4).
  3. Use point to find aa: Use the point (2,−11)(2, -11) to find the value of aa. Substitute x=2x = 2 and y=−11y = -11 into the equation: −11=a(2+2)(2−4)-11 = a(2 + 2)(2 - 4).
  4. Simplify and solve for aa: Simplify and solve for aa: −11=a(4)(−2)-11 = a(4)(-2), −11=−8a-11 = -8a, a=−11/−8a = -11 / -8, a=11/8a = 11/8.
  5. Write final equation: Write the final equation using the value of aa: y=(118)(x+2)(x−4)y = \left(\frac{11}{8}\right)(x + 2)(x - 4).

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