The vertex of this parabola is at $(1,3)$. Which of the following could be its equation?
A. $y=3(x+1)^2-3$
B. $x=3(y-3)^2-1$
C. $y=3(x+1)^2+3$
D. $x=3(y-3)^2+1$

The equation for the parabola with vertex (1,3) is found by analyzing each option. The correct equation is Option D: x = 3 ( y − 3 ) 2 + 1 , as it has the vertex at (1,3). Options A, B, and C do not match the vertex provided.
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Answered by Anonymous | 2025-07-05

The vertex form of a parabola is y = a ( x − h ) 2 + k or x = a ( y − k ) 2 + h , where ( h , k ) is the vertex.
Option A: y = 3 ( x + 1 ) 2 − 3 has vertex ( − 1 , − 3 ) .
Option B: x = 3 ( y − 3 ) 2 − 1 has vertex ( − 1 , 3 ) .
Option C: y = 3 ( x + 1 ) 2 + 3 has vertex ( − 1 , 3 ) .
Option D: x = 3 ( y − 3 ) 2 + 1 has vertex ( 1 , 3 ) .
Therefore, the correct answer is D ​ .

Explanation

Understanding the Problem We are given that the vertex of the parabola is at ( 1 , 3 ) . We need to determine which of the given options could be the equation of this parabola. Recall that the vertex form of a parabola is given by y = a ( x − h ) 2 + k for a vertical parabola with vertex ( h , k ) , and x = a ( y − k ) 2 + h for a horizontal parabola with vertex ( h , k ) . We will analyze each option to see which one has the vertex at ( 1 , 3 ) .

Analyzing Option A Option A: y = 3 ( x + 1 ) 2 − 3 . This is a vertical parabola with vertex at ( − 1 , − 3 ) . This does not match the given vertex ( 1 , 3 ) .

Analyzing Option B Option B: x = 3 ( y − 3 ) 2 − 1 . This is a horizontal parabola with vertex at ( − 1 , 3 ) . This does not match the given vertex ( 1 , 3 ) .

Analyzing Option C Option C: y = 3 ( x + 1 ) 2 + 3 . This is a vertical parabola with vertex at ( − 1 , 3 ) . This does not match the given vertex ( 1 , 3 ) .

Analyzing Option D Option D: x = 3 ( y − 3 ) 2 + 1 . This is a horizontal parabola with vertex at ( 1 , 3 ) . This matches the given vertex ( 1 , 3 ) .

Conclusion Therefore, the equation of the parabola could be x = 3 ( y − 3 ) 2 + 1 .


Examples
Understanding parabolas is essential 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 highest point the ball reaches. Similarly, satellite dishes and reflecting telescopes use parabolic reflectors to focus signals or light onto a specific point. The location of the vertex and the shape of the parabola are crucial in determining the efficiency and effectiveness of these devices. By understanding the vertex form of a parabola, we can easily identify the key features and apply this knowledge to practical scenarios.

Answered by GinnyAnswer | 2025-07-05