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Write the equation of the parabola that passes through the points (−3,−14(-3,-14), (−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 (−3,−14(-3,-14), (−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: Identify the x-intercepts from the given points.\newlineThe x-intercepts are the x-values of the points where the y-values are zero.\newlinex-intercepts: −5-5, 22
  2. Write equation in form: Use the x-intercepts to write the equation in the form y=a(x−p)(x−q)y = a(x - p)(x - q).\newlinep=−5p = -5\newlineq=2q = 2\newliney=a(x−(−5))(x−2)y = a(x - (-5))(x - 2)\newliney=a(x+5)(x−2)y = a(x + 5)(x - 2)
  3. Find value of 'a': Use the third point (−3,−14)(-3,-14) to find the value of 'a'.\newlineSubstitute x=−3x = -3 and y=−14y = -14 into the equation y=a(x+5)(x−2)y = a(x + 5)(x - 2).\newline−14=a(−3+5)(−3−2)-14 = a(-3 + 5)(-3 - 2)\newline−14=a(2)(−5)-14 = a(2)(-5)\newline−14=−10a-14 = -10a
  4. Solve for 'a': Solve for 'a'.\newline−14=−10a-14 = -10a\newline−14/−10=a-14 / -10 = a\newlinea=14/10a = 14/10\newlinea=7/5a = 7/5
  5. Write final equation: Write the final equation of the parabola using the value of aa and the x-intercepts.a=75a = \frac{7}{5}p=−5p = -5q=2q = 2y=a(x−p)(x−q)y = a(x - p)(x - q)y=(75)(x+5)(x−2)y = \left(\frac{7}{5}\right)(x + 5)(x - 2)

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