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Write the equation of the parabola that passes through the points (−6,−15)(-6, -15), (−1,0)(-1,0), and (−7,0)(-7,0). 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 (−6,−15)(-6, -15), (−1,0)(-1,0), and (−7,0)(-7,0). 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 roots of parabola: Identify the values of pp and qq from the given points where the parabola crosses the x-axis, which are the roots of the parabola. Points (−1,0)(-1, 0) and (−7,0)(-7, 0) indicate that p=−1p = -1 and q=−7q = -7.
  2. Formulate standard form equation: Formulate the equation of the parabola using the standard form y=a(x−p)(x−q)y = a(x - p)(x - q). Substituting p=−1p = -1 and q=−7q = -7, we get y=a(x+1)(x+7)y = a(x + 1)(x + 7).
  3. Substitute point to find aa: Substitute the point (−6,−15)(-6, -15) into the equation to find the value of aa. Plugging in x=−6x = -6 and y=−15y = -15, we get −15=a(−6+1)(−6+7)-15 = a(-6 + 1)(-6 + 7).
  4. Solve for aa: Simplify and solve for aa. The equation becomes –15=a(–5)(–1)–15 = a(–5)(–1), which simplifies to –15=5a–15 = 5a. Solving for aa gives a=–15/5=–3a = –15 / 5 = –3.
  5. Write final equation: Write the final equation of the parabola using the values of aa, pp, and qq. Substituting a=−3a = -3, p=−1p = -1, and q=−7q = -7 into y=a(x−p)(x−q)y = a(x - p)(x - q) gives y=−3(x+1)(x+7)y = -3(x + 1)(x + 7).

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