The vertex of this parabola is at $(5,-4)$. Which of the following could be its equation?
A. $y=2(x-5)^2-4$
B. $y=2(x-5)^2+4$
C. $y=2(x+5)^2+4$
D. $y=2(x+5)^2-4$
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 ( 5 , − 4 ) into the vertex form: y = a ( x − 5 ) 2 − 4 .
Compare the resulting equation with the given options.
The correct option is y = 2 ( x − 5 ) 2 − 4 , which matches the vertex form with a = 2 . The final answer is y = 2 ( x − 5 ) 2 − 4 .
Explanation
Understanding the Problem We are given that the vertex of the parabola is at ( 5 , − 4 ) . We need to determine which of the given equations could represent 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 ) is the vertex of the parabola and a is a constant that determines the direction and width of the parabola.
Substituting the Vertex Coordinates In our case, the vertex is ( 5 , − 4 ) , so we have h = 5 and k = − 4 . Substituting these values into the vertex form, we get: y = a ( x − 5 ) 2 − 4
Comparing with the Options Now, we compare this equation with the given options:
A. y = 2 ( x − 5 ) 2 − 4 B. y = 2 ( x − 5 ) 2 + 4 C. y = 2 ( x + 5 ) 2 + 4 D. y = 2 ( x + 5 ) 2 − 4
Option A has the form y = a ( x − 5 ) 2 − 4 with a = 2 , which matches our equation.
Eliminating Incorrect Options Option B has the form y = a ( x − 5 ) 2 + 4 , which does not match our equation because the k value is + 4 instead of − 4 .
Option C has the form y = a ( x + 5 ) 2 + 4 , which does not match our equation because the h value is − 5 instead of 5 , and the k value is + 4 instead of − 4 .
Option D has the form y = a ( x + 5 ) 2 − 4 , which does not match our equation because the h value is − 5 instead of 5 .
Conclusion Therefore, the only equation that could represent the parabola with vertex ( 5 , − 4 ) is option A.
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. By knowing the vertex and a few other points, we can determine the equation of the parabola and predict where the projectile will land. Similarly, the shape of suspension bridge cables and satellite dishes are also parabolic, allowing engineers to design structures that efficiently distribute weight or focus signals.
