$f(x)=(x-4)(x+2)$ ?
A. $(0,2)$
B. $(-2,0)$
C. $(-4,0)$
D. $(4,-2)$
Answer (1)
We are given the function f ( x ) = ( x − 4 ) ( x + 2 ) and need to check which of the given points satisfy the equation.
We substitute the x value of each point into the function and check if it equals the y value.
f ( 0 ) = − 8 , f ( − 2 ) = 0 , f ( − 4 ) = 16 , f ( 4 ) = 0 .
The only point that satisfies the equation is ( − 2 , 0 ) . So the answer is ( − 2 , 0 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x − 4 ) ( x + 2 ) and four points: ( 0 , 2 ) , ( − 2 , 0 ) , ( − 4 , 0 ) , and ( 4 , − 2 ) . Our goal is to determine which of these points lie on the graph of the function. A point ( x , y ) lies on the graph of f ( x ) if and only if f ( x ) = y . We will evaluate f ( x ) for each given x value and compare the result to the corresponding y value.
Checking the point (0, 2) For the point ( 0 , 2 ) , we have x = 0 and y = 2 . We calculate f ( 0 ) = ( 0 − 4 ) ( 0 + 2 ) = ( − 4 ) ( 2 ) = − 8 . Since − 8 e q 2 , the point ( 0 , 2 ) is not on the graph of f ( x ) .
Checking the point (-2, 0) For the point ( − 2 , 0 ) , we have x = − 2 and y = 0 . We calculate f ( − 2 ) = ( − 2 − 4 ) ( − 2 + 2 ) = ( − 6 ) ( 0 ) = 0 . Since 0 = 0 , the point ( − 2 , 0 ) is on the graph of f ( x ) .
Checking the point (-4, 0) For the point ( − 4 , 0 ) , we have x = − 4 and y = 0 . We calculate f ( − 4 ) = ( − 4 − 4 ) ( − 4 + 2 ) = ( − 8 ) ( − 2 ) = 16 . Since 16 e q 0 , the point ( − 4 , 0 ) is not on the graph of f ( x ) .
Checking the point (4, -2) For the point ( 4 , − 2 ) , we have x = 4 and y = − 2 . We calculate f ( 4 ) = ( 4 − 4 ) ( 4 + 2 ) = ( 0 ) ( 6 ) = 0 . Since 0 e q − 2 , the point ( 4 , − 2 ) is not on the graph of f ( x ) .
Final Answer Therefore, the only point among the given options that lies on the graph of f ( x ) = ( x − 4 ) ( x + 2 ) is ( − 2 , 0 ) .
Examples
Understanding functions and their graphs is crucial in many real-world applications. For example, engineers use functions to model the behavior of circuits, and determining if a specific input will result in a desired output is equivalent to checking if a point lies on the function's graph. Similarly, economists use functions to model supply and demand curves, and finding equilibrium points involves identifying where these curves intersect. In physics, projectile motion can be modeled by a quadratic function, and knowing the function allows us to predict the trajectory of a projectile and determine if it will hit a specific target.
