What are the coordinates of the vertex of the parabola described by the equation below?

[tex]$y=-7(x-4)^2-5$[/tex]

A. (-5,-4)
B. (5,4)
C. (4,-5)
D. (-4,5)

Recognize the vertex form of a parabola: y = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Compare the given equation y = − 7 ( x − 4 ) 2 − 5 to the vertex form.
Identify h = 4 and k = − 5 from the equation.
State the vertex coordinates as ( 4 , − 5 ) ​ .

Explanation

Understanding the Equation We are given the equation of a parabola in vertex form: y = − 7 ( x − 4 ) 2 − 5 . We need to identify the coordinates of the vertex.

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

Identifying h and k Comparing the given equation y = − 7 ( x − 4 ) 2 − 5 with the general vertex form y = a ( x − h ) 2 + k , we can identify the values of h and k . In this case, h = 4 and k = − 5 .

Finding the Vertex Therefore, the coordinates of the vertex are ( 4 , − 5 ) .


Examples
Understanding parabolas is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The vertex of this parabola represents the highest point the ball reaches. Similarly, satellite dishes and reflecting telescopes use parabolic reflectors to focus signals or light onto a single point, which is related to the properties of the parabola's vertex. Knowing how to find the vertex helps in optimizing the design and performance of these systems.

Answered by GinnyAnswer | 2025-07-05