The vertex of this parabola is at $(-3,-2)$. Which of the following could be its equation?
A. $y=-2(x-3)^2-2$
B. $y=-2(x+3)^2-2$
C. $y=-2(x-3)^2+2$
D. $y=-2(x+3)^2+2$
Answer (1)
The vertex form of a parabola is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex.
Substitute the given vertex ( − 3 , − 2 ) into the vertex form: y = a ( x + 3 ) 2 − 2 .
Compare the resulting equation with the given options.
The correct equation is y = − 2 ( x + 3 ) 2 − 2 , which corresponds to option B.
B
Explanation
Understanding the Vertex Form The vertex of a parabola in vertex form y = a ( x − h ) 2 + k is given by the point ( h , k ) . In this problem, we are given that the vertex is at ( − 3 , − 2 ) . Therefore, we have h = − 3 and k = − 2 .
Substituting the Vertex Coordinates Substituting these values into the vertex form, we get: y = a ( x − ( − 3 ) ) 2 + ( − 2 ) Simplifying, we have: y = a ( x + 3 ) 2 − 2
Comparing with the Options Now, we compare this equation with the given options: A. y = − 2 ( x − 3 ) 2 − 2 B. y = − 2 ( x + 3 ) 2 − 2 C. y = − 2 ( x − 3 ) 2 + 2 D. y = − 2 ( x + 3 ) 2 + 2 We are looking for an equation in the form y = a ( x + 3 ) 2 − 2 . Option B matches this form with a = − 2 .
Identifying the Correct Equation Therefore, the equation of the parabola could be y = − 2 ( x + 3 ) 2 − 2 .
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. Knowing the vertex of the parabola allows us to determine the maximum height the projectile reaches and how far it travels. Similarly, the shape of satellite dishes and reflecting telescopes is parabolic, focusing incoming signals or light to a single point. The vertex plays a key role in optimizing the design and performance of these devices.
