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Write the equation of the parabola that passes through the points (−3,−5)(-3,-5), (−2,0)(-2,0), (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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Q. Write the equation of the parabola that passes through the points (−3,−5)(-3,-5), (−2,0)(-2,0), (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 (−3,−5)(-3,-5), (−2,0)(-2,0), and (3,0)(3,0). The points where y=0y=0 are the x-intercepts, so p=−2p = -2 and q=3q = 3.
  2. Substitute into equation: Use the form y=a(x−p)(x−q)y = a(x - p)(x - q) and substitute p=−2p = -2 and q=3q = 3. This gives y=a(x+2)(x−3)y = a(x + 2)(x - 3).
  3. Find value of 'a': Substitute the point (−3,−5)(-3, -5) into the equation to find 'a'. Plugging in x=−3x = -3 and y=−5y = -5, we get −5=a(−3+2)(−3−3)-5 = a(-3 + 2)(-3 - 3).
  4. Simplify equation: Simplify the equation: −5=a(−1)(−6)=6a-5 = a(-1)(-6) = 6a. Solving for aa gives a=−56a = -\frac{5}{6}.
  5. Write final equation: Write the final equation using the values of aa, pp, and qq. Substitute a=−56a = -\frac{5}{6}, p=−2p = -2, and q=3q = 3 into y=a(x−p)(x−q)y = a(x - p)(x - q). This results in y=−56(x+2)(x−3)y = -\frac{5}{6}(x + 2)(x - 3).

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