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Question 1209270: Find the vertex of the graph of the equation x - y^2 + 8y = -4y^2 + 15y + 16.
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! **1. Rewrite the Equation**
* Start by rearranging the equation to isolate 'x':
x = y² - 8y + 4y² - 15y + 16
x = 3y² - 23y + 16
**2. Complete the Square**
* **Factor out the coefficient of y²:**
x = 3(y² - (23/3)y) + 16
* **Inside the parentheses, add and subtract the square of half the coefficient of y:**
x = 3(y² - (23/3)y + (23/6)² - (23/6)²) + 16
* **Rewrite as a perfect square trinomial:**
x = 3[(y - 23/6)² - 529/36] + 16
* **Distribute the 3:**
x = 3(y - 23/6)² - 529/12 + 16
* **Simplify:**
x = 3(y - 23/6)² - 145/12
**3. Identify the Vertex**
* The equation is now in vertex form: x = a(y - k)² + h
* Where (h, k) represents the vertex of the parabola.
* In this case:
* h = -145/12
* k = 23/6
**Therefore, the vertex of the graph is (-145/12, 23/6).**
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