Complete the square to find the vertex of this parabola.

[tex]y^2-4 x-8 y-12=0[/tex]
([?], [ ])

Rearrange the equation: y 2 − 8 y = 4 x + 12 .
Complete the square: ( y − 4 ) 2 = 4 x + 28 .
Factor: ( y − 4 ) 2 = 4 ( x + 7 ) .
Identify the vertex: ( − 7 , 4 ) ​ .

Explanation

Understanding the Problem We are given the equation of a parabola: y 2 − 4 x − 8 y − 12 = 0 . Our goal is to find the vertex of this parabola by completing the square.

Isolating y Terms First, we rearrange the equation to isolate the terms involving y on one side:


y 2 − 8 y = 4 x + 12

Completing the Square Next, we complete the square for the y terms. To do this, we take half of the coefficient of the y term (which is -8), square it ((-4)^2 = 16), and add it to both sides of the equation:

y 2 − 8 y + 16 = 4 x + 12 + 16
( y − 4 ) 2 = 4 x + 28

Factoring Now, we factor out the coefficient of x on the right side:

( y − 4 ) 2 = 4 ( x + 7 )

Identifying the Vertex The equation is now in the standard form of a horizontal parabola: ( y − k ) 2 = 4 p ( x − h ) , where ( h , k ) is the vertex of the parabola. Comparing our equation ( y − 4 ) 2 = 4 ( x + 7 ) with the standard form, we can identify the vertex.

In this case, h = − 7 and k = 4 .

Final Answer Therefore, the vertex of the parabola is ( − 7 , 4 ) .

Examples
Understanding parabolas and their vertices is crucial in various real-world applications. For example, engineers use parabolas to design satellite dishes and suspension bridges. The vertex represents the optimal point, such as the deepest point of a satellite dish focusing signals or the lowest point of a bridge cable bearing the load. By finding the vertex, we can determine key parameters for these designs, ensuring efficiency and stability. Similarly, in projectile motion, the vertex helps determine the maximum height reached by a projectile.

Answered by GinnyAnswer | 2025-07-05