The vertex of this parabola is at $(-5,4)$. Which of the following could be its equation?
A. $y=-(x-5)^2-4$
B. $y=-(x+5)^2+4$
C. $y=-(x-5)^2+4$
D. $y=-(x+5)^2-4

The vertex form of a parabola is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Substitute the given vertex ( − 5 , 4 ) into the vertex form: y = a ( x + 5 ) 2 + 4 .
Compare the resulting equation with the given options.
Option B, y = − ( x + 5 ) 2 + 4 , matches the vertex ( − 5 , 4 ) , so the answer is B ​ .

Explanation

Understanding the Problem We are given that the vertex of the parabola is at ( − 5 , 4 ) . We need to determine which of the given equations could represent this parabola. The general vertex form of a parabola is given by: y = a ( x − h ) 2 + k where ( h , k ) is the vertex of the parabola and a determines the direction and width of the parabola.

Substituting the Vertex Substitute the given vertex ( − 5 , 4 ) into the vertex form of the parabola: y = a ( x − ( − 5 ) ) 2 + 4 y = a ( x + 5 ) 2 + 4 Now, we compare this equation with the given options to find the correct one.

Comparing with the Options Let's analyze each option:


Option A: y = − ( x − 5 ) 2 − 4 . The vertex would be ( 5 , − 4 ) , which does not match the given vertex ( − 5 , 4 ) .
Option B: y = − ( x + 5 ) 2 + 4 . The vertex is ( − 5 , 4 ) , which matches the given vertex. The value of a is − 1 .
Option C: y = − ( x − 5 ) 2 + 4 . The vertex would be ( 5 , 4 ) , which does not match the given vertex ( − 5 , 4 ) .
Option D: y = − ( x + 5 ) 2 − 4 . The vertex would be ( − 5 , − 4 ) , which does not match the given vertex ( − 5 , 4 ) .
Therefore, the correct equation is the one that matches the vertex ( − 5 , 4 ) .

Conclusion Based on the analysis above, option B, y = − ( x + 5 ) 2 + 4 , is the only equation with the correct vertex ( − 5 , 4 ) .

Examples
Understanding parabolas is crucial in various real-world applications. For instance, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The vertex of this parabola represents the maximum height the ball reaches. Similarly, satellite dishes and reflecting telescopes use parabolic reflectors to focus signals or light to a specific point. Knowing the vertex and equation of the parabola helps engineers design these systems efficiently.

Answered by GinnyAnswer | 2025-07-05