The vertex of this parabola is at $(-1,-3)$. Which of the following could be the equation?

The vertex form of a parabola is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Substitute the given vertex ( − 1 , − 3 ) into the vertex form: y = a ( x − ( − 1 ) ) 2 + ( − 3 ) .
Simplify the equation: y = a ( x + 1 ) 2 − 3 .
A possible equation for the parabola is y = a ( x + 1 ) 2 − 3 ​ .

Explanation

Understanding the Problem The vertex of the parabola is given as ( − 1 , − 3 ) . We need to find a possible equation for this parabola.

Recalling the Vertex Form The general vertex form of a parabola is given by the equation: y = a ( x − h ) 2 + k where ( h , k ) represents the coordinates of the vertex.

Substituting the Vertex Coordinates In our case, the vertex is ( − 1 , − 3 ) , so we have h = − 1 and k = − 3 . Substituting these values into the vertex form, we get: y = a ( x − ( − 1 ) ) 2 + ( − 3 ) Simplifying this, we have: y = a ( x + 1 ) 2 − 3

Finding a Possible Equation This equation represents a parabola with vertex at ( − 1 , − 3 ) . The problem asks for a possible equation, which means any non-zero value of a will give a valid equation. For example, if a = 1 , the equation becomes: y = ( x + 1 ) 2 − 3 y = x 2 + 2 x + 1 − 3 y = x 2 + 2 x − 2

Final Answer Therefore, a possible equation for the parabola with vertex at ( − 1 , − 3 ) is: y = a ( x + 1 ) 2 − 3 ​ or, equivalently, y = a x 2 + 2 a x + a − 3 ​


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. Knowing the vertex of this parabola allows us to determine the maximum height the projectile reaches and how far it travels. Similarly, satellite dishes and reflecting telescopes use parabolic shapes to focus signals or light onto a specific point. The vertex plays a key role in the design and alignment of these devices to ensure optimal performance.

Answered by GinnyAnswer | 2025-07-05