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Write the equation of the parabola that passes through the points (7,0)(7,0), (6,−1)(6,-1), and (−1,0)(-1,0). 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 (7,0)(7,0), (6,−1)(6,-1), and (−1,0)(-1,0). 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 Intercepts: We have the points (7,0)(7,0), (6,−1)(6,-1), and (−1,0)(-1,0). The points (7,0)(7, 0) and (−1,0)(-1, 0) are xx-intercepts of the parabola, so they give us the values of pp and qq directly.\newlinep=7p = 7 and q=−1q = -1.
  2. Write Parabola Equation: Now we have 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=7p = 7 and q=−1q = -1, we get y=a(x−7)(x−(−1))y = a(x - 7)(x - (-1)) or y=a(x−7)(x+1)y = a(x - 7)(x + 1).
  3. Find Value of a: Next, we need to find the value of aa. We will use the point (6,−1)(6, -1) for this purpose.\newlineSubstitute x=6x = 6 and y=−1y = -1 into the equation y=a(x−7)(x+1)y = a(x - 7)(x + 1) to find aa.\newline−1=a(6−7)(6+1)-1 = a(6 - 7)(6 + 1)\newline−1=a(−1)(7)-1 = a(-1)(7)\newline−1=−7a-1 = -7a
  4. Solve for aa: Solve the equation −1=−7a-1 = -7a to find the value of aa.\newlineDivide both sides by −7-7 to isolate aa.\newlinea=−1−7a = \frac{-1}{-7}\newlinea=17a = \frac{1}{7}
  5. Final Parabola Equation: Now that we have found a=17a = \frac{1}{7}, we can write the final equation of the parabola.\newlineSubstitute a=17a = \frac{1}{7}, p=7p = 7, and q=−1q = -1 into the equation y=a(x−p)(x−q)y = a(x - p)(x - q).\newliney=(17)(x−7)(x+1)y = \left(\frac{1}{7}\right)(x - 7)(x + 1)

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