What is the $y$-intercept of the quadratic function $f(x)=(x-6)(x-2)$?
A. $(0,-6)$
B. $(0,12)$
C. $(-8,0)$
D. $(2,0)$
Answer (2)
The y -intercept of a function is the point where the graph intersects the y -axis, which occurs when x = 0 .
Substitute x = 0 into the quadratic function f ( x ) = ( x − 6 ) ( x − 2 ) .
Calculate f ( 0 ) = ( 0 − 6 ) ( 0 − 2 ) = ( − 6 ) ( − 2 ) = 12 .
The y -intercept is ( 0 , 12 ) , so the answer is ( 0 , 12 ) .
Explanation
Understanding the Problem The problem asks us to find the y -intercept of the quadratic function f ( x ) = ( x − 6 ) ( x − 2 ) . 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 f ( x ) at x = 0 . So, we need to calculate f ( 0 ) .
Calculating f(0) We substitute x = 0 into the function: f ( 0 ) = ( 0 − 6 ) ( 0 − 2 ) = ( − 6 ) ( − 2 ) = 12.
The y-intercept Therefore, the y -intercept is the point ( 0 , 12 ) .
Examples
Understanding the y-intercept of a quadratic function is useful in many real-world applications. For example, if you are modeling the height of a ball thrown into the air with a quadratic function, the y-intercept represents the initial height of the ball when it was thrown. Similarly, in business, if you are modeling the profit of a company with a quadratic function, the y-intercept might represent the initial investment or startup costs. Knowing the y-intercept gives you a starting point or a baseline value in these types of scenarios. In this case, the y-intercept is ( 0 , 12 ) , which means at x = 0 , the value of the function is 12.
The y -intercept of the function f ( x ) = ( x − 6 ) ( x − 2 ) is found by substituting x = 0 into the function, resulting in f ( 0 ) = 12 . Therefore, the y -intercept is the point ( 0 , 12 ) . The correct option is ( 0 , 12 ) .
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