Which function has two $x$-intercepts, one at $(0,0)$ and one at $(4,0)$?
A. $f(x)=x(x-4)$
B. $f(x)=x(x+4)$
C. $f(x)=(x-4)(x-4)$
D. $f(x)=(x+4)(x+4)$
Answer (2)
The problem asks for a function with x-intercepts at (0,0) and (4,0).
This means the function must equal zero when x=0 and x=4.
Test each function by plugging in x=0 and x=4.
The function f ( x ) = x ( x − 4 ) is the only one that satisfies both conditions, so the answer is f ( x ) = x ( x − 4 ) .
Explanation
Understanding the Problem We need to find the function that equals zero when x = 0 and x = 4 . This means that when we plug in these values for x , the function should output 0. Let's test each option.
Analyzing the first function Let's analyze the first function: f ( x ) = x ( x − 4 ) .
When x = 0 , f ( 0 ) = 0 ( 0 − 4 ) = 0 . This satisfies the first condition. When x = 4 , f ( 4 ) = 4 ( 4 − 4 ) = 4 ( 0 ) = 0 . This satisfies the second condition.
Analyzing the second function Now let's analyze the second function: f ( x ) = x ( x + 4 ) .
When x = 0 , f ( 0 ) = 0 ( 0 + 4 ) = 0 . This satisfies the first condition. When x = 4 , f ( 4 ) = 4 ( 4 + 4 ) = 4 ( 8 ) = 32 . This does not satisfy the second condition.
Analyzing the third function Now let's analyze the third function: f ( x ) = ( x − 4 ) ( x − 4 ) .
When x = 0 , f ( 0 ) = ( 0 − 4 ) ( 0 − 4 ) = ( − 4 ) ( − 4 ) = 16 . This does not satisfy the first condition. When x = 4 , f ( 4 ) = ( 4 − 4 ) ( 4 − 4 ) = ( 0 ) ( 0 ) = 0 . This satisfies the second condition.
Analyzing the fourth function Now let's analyze the fourth function: f ( x ) = ( x + 4 ) ( x + 4 ) .
When x = 0 , f ( 0 ) = ( 0 + 4 ) ( 0 + 4 ) = ( 4 ) ( 4 ) = 16 . This does not satisfy the first condition. When x = 4 , f ( 4 ) = ( 4 + 4 ) ( 4 + 4 ) = ( 8 ) ( 8 ) = 64 . This does not satisfy the second condition.
Conclusion The only function that satisfies both conditions is f ( x ) = x ( x − 4 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, in physics, the x-intercepts of a projectile's trajectory equation represent the points where the projectile lands. Similarly, in economics, the x-intercepts of a cost function can indicate break-even points where costs equal revenue. By identifying these intercepts, we can make informed decisions and predictions in diverse fields.
The function that has x-intercepts at (0,0) and (4,0) is f ( x ) = x ( x − 4 ) . This function equals zero when x is either 0 or 4, satisfying the given conditions. Thus, the chosen option is A. f ( x ) = x ( x − 4 ) .
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