Which function has a vertex at $(2,-9)$?
$f(x)=-(x-1)(x-5)$
$f(x)=(x+8)^2$
$f(x)=-(x-3)^2$
$f(x)=(x-5)(x+1)$
Answer (1)
Expand each function to the standard form if necessary.
Complete the square to convert each function to vertex form f ( x ) = a ( x − h ) 2 + k , where (h, k) is the vertex.
Identify the vertex (h, k) for each function.
The function with vertex (2, -9) is f ( x ) = ( x − 5 ) ( x + 1 ) .
f ( x ) = ( x − 5 ) ( x + 1 )
Explanation
Understanding the Problem We are looking for a function with its vertex at the point (2, -9). The vertex form of a quadratic function is given by f ( x ) = a ( x − h ) 2 + k , where (h, k) represents the vertex of the parabola. We need to check each of the given functions to see which one has the vertex (2, -9).
Analyzing the First Function Let's analyze the first function: f ( x ) = − ( x − 1 ) ( x − 5 ) . First, we expand the expression: f ( x ) = − ( x 2 − 5 x − x + 5 ) = − ( x 2 − 6 x + 5 ) = − x 2 + 6 x − 5 Now, we complete the square to find the vertex form: f ( x ) = − ( x 2 − 6 x ) − 5 = − ( x 2 − 6 x + 9 − 9 ) − 5 = − ( x 2 − 6 x + 9 ) + 9 − 5 = − ( x − 3 ) 2 + 4 The vertex of this function is (3, 4), which is not (2, -9).
Analyzing the Second Function Now, let's analyze the second function: f ( x ) = ( x + 8 ) 2 . This is already in vertex form: f ( x ) = ( x + 8 ) 2 + 0 . The vertex of this function is (-8, 0), which is not (2, -9).
Analyzing the Third Function Next, let's analyze the third function: f ( x ) = − ( x − 3 ) 2 . This is already in vertex form: f ( x ) = − ( x − 3 ) 2 + 0 . The vertex of this function is (3, 0), which is not (2, -9).
Analyzing the Fourth Function Finally, let's analyze the fourth function: f ( x ) = ( x − 5 ) ( x + 1 ) . First, we expand the expression: f ( x ) = x 2 + x − 5 x − 5 = x 2 − 4 x − 5 Now, we complete the square to find the vertex form: f ( x ) = x 2 − 4 x − 5 = ( x 2 − 4 x + 4 ) − 4 − 5 = ( x − 2 ) 2 − 9 The vertex of this function is (2, -9), which matches the given vertex.
Conclusion Therefore, the function with a vertex at (2, -9) is f ( x ) = ( x − 5 ) ( x + 1 ) .
Examples
Understanding the vertex form of a quadratic equation is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or a satellite dish, knowing the vertex helps in focusing light or signals efficiently. Similarly, in physics, the trajectory of a projectile under gravity follows a parabolic path, and the vertex represents the maximum height reached by the projectile. By identifying the vertex, engineers and scientists can optimize designs and predict outcomes in these scenarios.
