If the parabola of the form [tex]y=a(x-h)^2+k[/tex] is always shifted horizontally [tex]h[/tex] units and vertically [tex]k[/tex] units, then its vertex is always
[tex](-h,-k)[/tex]
[tex](h, k)[/tex]
[tex](-h, k)[/tex]
[tex](h,-k)[/tex]

The vertex form of a parabola is given by y = a ( x − h ) 2 + k .
The vertex of the parabola in this form is ( h , k ) .
Therefore, the vertex is at the point ( h , k ) .
The final answer is ( h , k ) ​ .

Explanation

Understanding the Vertex Form The given equation of the parabola is in vertex form: y = a ( x − h ) 2 + k . In this form, the vertex of the parabola is represented by the point ( h , k ) . The question asks us to identify the vertex of the parabola based on the horizontal and vertical shifts.

Identifying the Vertex The vertex form of a parabola is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. The parameter h represents the horizontal shift, and the parameter k represents the vertical shift.

Conclusion Therefore, the vertex of the parabola y = a ( x − h ) 2 + k is ( h , k ) .


Examples
Understanding the vertex form of a parabola is useful in various real-world applications. For example, engineers designing parabolic reflectors for satellite dishes or solar ovens use the vertex form to determine the optimal focal point. Similarly, in physics, when analyzing projectile motion, the vertex of the parabolic trajectory represents the maximum height reached by the projectile. Knowing the vertex allows for precise calculations and efficient designs in these scenarios.

Answered by GinnyAnswer | 2025-07-08