Which graph is represented by this equation? [tex]$x+y^2+6 y+3=0$[/tex]

The equation x + y 2 + 6 y + 3 = 0 represents a parabola that opens to the left with its vertex at the point ( 6 , − 3 ) . This is determined by completing the square for the y terms. The resulting graph is found in the form x = − ( y + 3 ) 2 + 6 .
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Answered by Anonymous | 2025-07-03

Complete the square for the y terms: x = − y 2 − 6 y − 3 .
Rewrite the equation: x = − ( y + 3 ) 2 + 6 .
Identify the vertex and direction: vertex is ( 6 , − 3 ) , opens to the left.
The graph is a parabola: x = − ( y + 3 ) 2 + 6 ​ .

Explanation

Analyzing the Equation We are given the equation x + y 2 + 6 y + 3 = 0 and asked to identify the graph it represents. This equation relates x and y , and by completing the square, we can rewrite it in a standard form that reveals the type of conic section it represents.

Completing the Square To determine the type of graph, we complete the square for the y terms. The equation is x + y 2 + 6 y + 3 = 0 . We can rewrite this as


x = − y 2 − 6 y − 3 .
To complete the square for y 2 + 6 y , we need to add and subtract ( 6/2 ) 2 = 3 2 = 9 . So we have
x = − ( y 2 + 6 y + 9 − 9 ) − 3
x = − ( y 2 + 6 y + 9 ) + 9 − 3
x = − ( y + 3 ) 2 + 6 .

Identifying the Parabola Now we rewrite the equation in the standard form of a parabola:

x = − ( y + 3 ) 2 + 6
x − 6 = − ( y + 3 ) 2
− ( x − 6 ) = ( y + 3 ) 2
This is the equation of a parabola that opens to the left. The vertex of the parabola is ( 6 , − 3 ) .

Conclusion The equation x + y 2 + 6 y + 3 = 0 represents a parabola that opens to the left with vertex at ( 6 , − 3 ) .

Examples
Parabolas are more than just abstract math; they appear everywhere in the real world! Imagine a skateboarder riding a half-pipe. The path they take as they go up and down follows a parabolic curve. Similarly, satellite dishes and radio telescopes are shaped like parabolas to focus incoming signals to a single point, improving reception. Even the trajectory of a ball thrown in the air (ignoring air resistance) traces a parabola. Understanding the equation of a parabola helps us analyze and design these systems effectively.

Answered by GinnyAnswer | 2025-07-03