What is the $y$-value of the vertex of the function $f(x)=-(x+8)(x-14)$?
Answer (1)
Expand the given function f ( x ) = − ( x + 8 ) ( x − 14 ) to get f ( x ) = − x 2 + 6 x + 112 .
Find the x -coordinate of the vertex using the formula x v = − 2 a b , which gives x v = 3 .
Substitute x v = 3 into the function to find the y -coordinate of the vertex: y v = f ( 3 ) = − 3 2 + 6 ( 3 ) + 112 = 121 .
The y -value of the vertex is 121 .
Explanation
Understanding the Problem We are given the function f ( x ) = − ( x + 8 ) ( x − 14 ) and asked to find the y -value of the vertex. The vertex of a quadratic function is the point where the function reaches its maximum or minimum value.
Expanding the Function First, we need to expand the function to the form f ( x ) = a x 2 + b x + c . So, we have
f ( x ) = − ( x + 8 ) ( x − 14 ) = − ( x 2 − 14 x + 8 x − 112 ) = − ( x 2 − 6 x − 112 ) = − x 2 + 6 x + 112 .
Thus, a = − 1 , b = 6 , and c = 112 .
Finding the x-coordinate of the Vertex Next, we find the x -coordinate of the vertex using the formula x v = − 2 a b .
x v = − 2 ( − 1 ) 6 = − − 2 6 = 3 .
Finding the y-coordinate of the Vertex Now, we substitute x v = 3 into the function f ( x ) to find the y -coordinate of the vertex, y v = f ( x v ) .
y v = f ( 3 ) = − ( 3 ) 2 + 6 ( 3 ) + 112 = − 9 + 18 + 112 = 9 + 112 = 121 .
Final Answer Therefore, the y -value of the vertex is 121.
Examples
The vertex of a parabola is a crucial point in many real-world applications. For example, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if a quadratic function models profit, the vertex indicates the maximum profit achievable. Understanding how to find the vertex helps in optimizing various scenarios, from physics to economics. For instance, if the function f ( x ) = − ( x + 8 ) ( x − 14 ) represents the profit margin of a product, then finding the vertex helps determine the optimal production level to maximize profit.
