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Write the equation of the parabola that passes through the points (−7,−7)(-7,-7), (−5,0)(-5,0), and (−2,0)(-2,0). 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,−7)(-7,-7), (−5,0)(-5,0), and (−2,0)(-2,0). 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: We have the points (−7,−7)(-7,-7), (−5,0)(-5,0), and (−2,0)(-2,0). The points (−5,0)(-5,0) and (−2,0)(-2,0) are the x-intercepts of the parabola, so they correspond to pp and qq in the equation y=a(x−p)(x−q)y = a(x - p)(x - q).
  2. Substitute values into equation: Since the x-intercepts are given by the points where y=0y = 0, we can identify p=−5p = -5 and q=−2q = -2 from the points (−5,0)(-5,0) and (−2,0)(-2,0).
  3. Find value of aa: Now we have p=−5p = -5 and q=−2q = -2. We can substitute these values into the equation to get y=a(x−(−5))(x−(−2))y = a(x - (-5))(x - (-2)) or y=a(x+5)(x+2)y = a(x + 5)(x + 2).
  4. Substitute third point: Next, we need to find the value of aa. We can use the third point (−7,−7)(-7,-7) to do this. We substitute x=−7x = -7 and y=−7y = -7 into the equation y=a(x+5)(x+2)y = a(x + 5)(x + 2) to solve for aa.
  5. Solve for aa: Substituting the point (−7,−7)(-7,-7) into the equation gives us −7=a(−7+5)(−7+2)-7 = a(-7 + 5)(-7 + 2). Simplifying the right side, we get −7=a(−2)(−5)-7 = a(-2)(-5).
  6. Write final equation: Now we solve for aa: −7=a(10)-7 = a(10). Dividing both sides by 1010 gives us a=−710a = -\frac{7}{10}.
  7. Write final equation: Now we solve for aa: −7=a(10)-7 = a(10). Dividing both sides by 1010 gives us a=−710a = -\frac{7}{10}.We have found a=−710a = -\frac{7}{10}. We can now write the final equation of the parabola by substituting aa, pp, and qq into y=a(x−p)(x−q)y = a(x - p)(x - q). This gives us y=(−710)(x+5)(x+2)y = \left(-\frac{7}{10}\right)(x + 5)(x + 2).

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