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Write the equation of the parabola that passes through the points (−5,0)(-5,0), (1,6)(1,6), (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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Full solution

Q. Write the equation of the parabola that passes through the points (−5,0)(-5,0), (1,6)(1,6), (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 (−5,0)(-5,0) and (3,0)(3,0). These points indicate where the parabola crosses the x-axis, so p=−5p = -5 and q=3q = 3.
  2. Write general form: Write the general form of the parabola using the identified x-intercepts: y=a(x+5)(x−3)y = a(x + 5)(x - 3).
  3. Use point to find 'a': Use the point (1,6)(1, 6) to find the value of 'a'. Substitute x=1x = 1 and y=6y = 6 into the equation: 6=a(1+5)(1−3)6 = a(1 + 5)(1 - 3).
  4. Simplify and solve: Simplify and solve for aa: 6=a(6)(−2)6 = a(6)(-2), 6=−12a6 = -12a, a=−12a = -\frac{1}{2}.
  5. Substitute 'a' back: Substitute the value of 'a' back into the equation to get the final form of the parabola: y=−12(x+5)(x−3)y = -\frac{1}{2}(x + 5)(x - 3).

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