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Write the equation of the parabola that passes through the points (−5,28(-5,28), (2,0)(2,0), and (3,0)(3,0). 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 (−5,28(-5,28), (2,0)(2,0), and (3,0)(3,0). 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 (−5,28(-5,28), (2,0)(2,0), and (3,0)(3,0). The points (2,0)(2,0) and (3,0)(3,0) are the x-intercepts of the parabola, so they give us the values of pp and qq directly.\newlinep=2p = 2 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 p=2p = 2 and q=3q = 3, we get y=a(x−2)(x−3)y = a(x - 2)(x - 3).
  3. Find Value of a: Now we need to find the value of aa. We can use the point (−5,28)(-5,28) to do this. Substituting x=−5x = -5 and y=28y = 28 into the equation y=a(x−2)(x−3)y = a(x - 2)(x - 3), we get:\newline28=a(−5−2)(−5−3)28 = a(-5 - 2)(-5 - 3).
  4. Solve for aa: Solving the equation 28=a(−7)(−8)28 = a(-7)(-8), we find:\newline28=a(56)28 = a(56).\newlineTo find aa, we divide both sides by 5656:\newlinea=2856a = \frac{28}{56}.
  5. Final Equation: Simplifying the fraction 2856\frac{28}{56}, we get:\newlinea=12a = \frac{1}{2}.
  6. Final Equation: Simplifying the fraction 2856\frac{28}{56}, we get:\newlinea=12a = \frac{1}{2}.Now that we have found a=12a = \frac{1}{2}, we can write the final equation of the parabola:\newliney=(12)(x−2)(x−3)y = \left(\frac{1}{2}\right)(x - 2)(x - 3).

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