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Write the equation of the parabola that passes through the points (−1,0)(-1,0), (2,0)(2,0), and (1,5)(1,5). 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 (−1,0)(-1,0), (2,0)(2,0), and (1,5)(1,5). 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: We have the points (−1,0)(-1,0), (2,0)(2,0), and (1,5)(1,5). The points where the parabola crosses the x-axis, (−1,0)(-1,0) and (2,0)(2,0), give us the values of pp and qq directly, since these are the x-intercepts of the parabola.\newlinep=−1p = -1 and q=2q = 2.
  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 −1-1 for pp and 22 for qq, we get: y=a(x+1)(x−2)y = a(x + 1)(x - 2).
  3. Find value of a: Now we need to find the value of aa. We will use the third point (1,5)(1,5) for this. Substituting x=1x = 1 and y=5y = 5 into the equation y=a(x+1)(x−2)y = a(x + 1)(x - 2) gives us:\newline5=a(1+1)(1−2)5 = a(1 + 1)(1 - 2).
  4. Solve for aa: Solving the equation 5=a(2)(−1)5 = a(2)(-1) to find the value of aa:5=−2a5 = -2aa=−52a = -\frac{5}{2}.
  5. Final equation: Now that we have found a=−52a = -\frac{5}{2}, we can write the final equation of the parabola.\newlineSubstituting −52-\frac{5}{2} for aa, −1-1 for pp, and 22 for qq into the equation y=a(x−p)(x−q)y = a(x - p)(x - q), we get:\newliney=−52(x+1)(x−2)y = -\frac{5}{2}(x + 1)(x - 2).

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