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Write the equation of the parabola that passes through the points (−4,0)(-4,0), (3,0)(3,0), and (−1,−6)(-1,-6). 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 (−4,0)(-4,0), (3,0)(3,0), and (−1,−6)(-1,-6). 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: We have the points (−4,0)(-4,0), (3,0)(3,0), and (−1,−6)(-1,-6). The points where the parabola crosses the xx-axis, (−4,0)(-4,0) and (3,0)(3,0), give us the values of pp and qq directly. Therefore, p=−4p = -4 and q=3q = 3.
  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 −4-4 for pp and 33 for qq, we get y=a(x+4)(x−3)y = a(x + 4)(x - 3).
  3. Find Value of a: Now we need to find the value of aa. We will use the third point (−1,−6)(-1,-6) to do this. Substituting x=−1x = -1 and y=−6y = -6 into the equation y=a(x+4)(x−3)y = a(x + 4)(x - 3), we get −6=a(−1+4)(−1−3)-6 = a(-1 + 4)(-1 - 3).
  4. Solve for aa: Solving the equation −6=a(3)(−4)-6 = a(3)(-4), we find that −6=−12a-6 = -12a. Dividing both sides by −12-12, we get a=−6−12=12a = \frac{-6}{-12} = \frac{1}{2}.
  5. Final Equation: Now that we have found a=12a = \frac{1}{2}, we can write the final equation of the parabola. Substituting 12\frac{1}{2} for aa, −4-4 for pp, and 33 for qq into the equation y=a(x−p)(x−q)y = a(x - p)(x - q), we get y=(12)(x+4)(x−3)y = \left(\frac{1}{2}\right)(x + 4)(x - 3).

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