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Write the equation of the parabola that passes through the points (2,0)(2,0), (−1,0)(-1,0), and (3,8)(3,8). 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), (−1,0)(-1,0), and (3,8)(3,8). 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 Points: Identify the values of pp and qq from the given points where the parabola crosses the x-axis, which are the points (2,0)(2,0) and (−1,0)(-1,0). These points give us the x-intercepts of the parabola, so p=2p = 2 and q=−1q = -1.
  2. Write Parabola Equation: Using the values of pp and qq, we can write the equation of the parabola in the form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting p=2p = 2 and q=−1q = -1, we get y=a(x−2)(x−(−1))y = a(x - 2)(x - (-1)) or y=a(x−2)(x+1)y = a(x - 2)(x + 1).
  3. Find Value of aa: Now we need to find the value of aa. We can use the third point (3,8)(3,8) to do this. Substitute x=3x = 3 and y=8y = 8 into the equation y=a(x−2)(x+1)y = a(x - 2)(x + 1) to solve for aa.8=a(3−2)(3+1)8 = a(3 - 2)(3 + 1)
  4. Simplify Equation: Simplify the equation to find the value of aa.8=a(1)(4)8 = a(1)(4)8=4a8 = 4aa=84a = \frac{8}{4}a=2a = 2
  5. Final Equation: Now that we have found a=2a = 2, we can write the final equation of the parabola. Substitute a=2a = 2 into the equation y=a(x−2)(x+1)y = a(x - 2)(x + 1).\newliney=2(x−2)(x+1)y = 2(x - 2)(x + 1)

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