What's the $y$-intercept of the function $f(x)=(x-2)^2(x+3)(x+1)$?
Answer (2)
To find the y -intercept of the function, set x = 0 .
Substitute x = 0 into the function: f ( 0 ) = ( 0 − 2 ) 2 ( 0 + 3 ) ( 0 + 1 ) .
Calculate f ( 0 ) = ( 4 ) ( 3 ) ( 1 ) = 12 .
The y -intercept is ( 0 , 12 ) .
Explanation
Understanding the Problem We are asked to find the y -intercept of the function f ( x ) = ( x − 2 ) 2 ( x + 3 ) ( x + 1 ) . The y -intercept is the point where the graph of the function intersects the y -axis. This occurs when x = 0 .
Finding the y-intercept To find the y -intercept, we need to evaluate the function at x = 0 . So we need to calculate f ( 0 ) .
Calculating f(0) Substitute x = 0 into the function: f ( 0 ) = ( 0 − 2 ) 2 ( 0 + 3 ) ( 0 + 1 ) f ( 0 ) = ( − 2 ) 2 ( 3 ) ( 1 ) f ( 0 ) = ( 4 ) ( 3 ) ( 1 ) f ( 0 ) = 12 So, the y -intercept is at the point ( 0 , 12 ) .
Final Answer The y -intercept of the function f ( x ) = ( x − 2 ) 2 ( x + 3 ) ( x + 1 ) is ( 0 , 12 ) .
Examples
Understanding the y-intercept of a function is crucial in many real-world applications. For example, in business, if f ( x ) represents the profit of a company as a function of the number of products sold ( x ), the y-intercept f ( 0 ) would represent the profit when no products are sold. This could indicate the initial investment or fixed costs of the company. Similarly, in physics, if f ( x ) represents the height of an object as a function of time ( x ), the y-intercept f ( 0 ) would represent the initial height of the object at time zero. Knowing the y-intercept provides a starting point for analyzing the behavior of the function and making predictions.
The y -intercept of the function f ( x ) = ( x − 2 ) 2 ( x + 3 ) ( x + 1 ) is found by evaluating f ( 0 ) , which equals 12. Thus, the y -intercept is the point ( 0 , 12 ) .
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