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Write the equation of the parabola that passes through the points shown in the table. (−7,0)(−6,2)(−1,0)(-7,0)(-6,2)( -1,0) 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. (−7,0)(−6,2)(−1,0)(-7,0)(-6,2)( -1,0) 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 (−7,0)(-7,0) and (−1,0)(-1,0) are x-intercepts, so p=−7p = -7 and q=−1q = -1.
  2. Write parabola equation: Write the equation of the parabola using the form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting pp and qq, we get y=a(x+7)(x+1)y = a(x + 7)(x + 1).
  3. Use point to find aa: Use the point (−6,2)(-6,2) to find the value of aa. Substitute x=−6x = -6 and y=2y = 2 into the equation: 2=a(−6+7)(−6+1)2 = a(-6 + 7)(-6 + 1).
  4. Solve for a: Simplify and solve for a: 2=a(1)(−5)2 = a(1)(-5), 2=−5a2 = -5a, a = - rac{2}{5}.
  5. Final equation: Write the final equation of the parabola substituting a=−25a = -\frac{2}{5} into y=a(x+7)(x+1)y = a(x + 7)(x + 1). The equation becomes y=−25(x+7)(x+1)y = -\frac{2}{5}(x + 7)(x + 1).

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