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A parabola graphed in the xy-plane has equation y=-3x^(2)+9x-2. What is the y-coordinate of the vertex of the parabola?

A parabola graphed in the xy x y -plane has equation y=−3x2+9x−2 y=-3 x^{2}+9 x-2 . What is the y y -coordinate of the vertex of the parabola?

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Full solution

Q. A parabola graphed in the xy x y -plane has equation y=−3x2+9x−2 y=-3 x^{2}+9 x-2 . What is the y y -coordinate of the vertex of the parabola?
  1. Write quadratic equation: Write down the given quadratic equation.\newlineThe given quadratic equation is y=−3x2+9x−2y = -3x^2 + 9x - 2.
  2. Convert to vertex form: Convert the quadratic equation into vertex form.\newlineThe vertex form of a quadratic equation is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.\newlineTo convert the given equation into vertex form, we need to complete the square.
  3. Factor out xx terms: Factor out the coefficient of x2x^2 from the xx terms.\newlineFactor out −3-3 from −3x2-3x^2 and 9x9x.\newliney=−3(x2−3x)−2y = -3(x^2 - 3x) - 2
  4. Find (b/2)2(b/2)^2: Find the value of (b/2)2(b/2)^2 for x2−3xx^2 - 3x. The coefficient of xx is −3-3. (−3/2)2=(−1.5)2=2.25(-3/2)^2 = (-1.5)^2 = 2.25
  5. Complete the square: Add and subtract (b2)2(\frac{b}{2})^2 inside the parentheses to complete the square.\newliney=−3(x2−3x+2.25−2.25)−2y = -3(x^2 - 3x + 2.25 - 2.25) - 2\newliney=−3((x2−3x+2.25)−2.25)−2y = -3((x^2 - 3x + 2.25) - 2.25) - 2
  6. Rewrite as perfect square trinomial: Rewrite the equation as a perfect square trinomial.\newliney=−3((x−1.5)2−2.25)−2y = -3((x - 1.5)^2 - 2.25) - 2
  7. Distribute and simplify: Distribute the −3-3 and simplify the equation.\newliney=−3(x−1.5)2+6.75−2y = -3(x - 1.5)^2 + 6.75 - 2\newliney=−3(x−1.5)2+4.75y = -3(x - 1.5)^2 + 4.75
  8. Identify the vertex: Identify the vertex of the parabola.\newlineThe vertex form of the equation is now y=−3(x−1.5)2+4.75y = -3(x - 1.5)^2 + 4.75, so the vertex is at (h,k)=(1.5,4.75)(h, k) = (1.5, 4.75).\newlineThe yy-coordinate of the vertex is 4.754.75.

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