What is the $y$-intercept of the quadratic function $f(x)=(x-6)(x-2)$?
A. $(2,0)$
B. $(0,12)$
C. $(-8,0)$
D. $(0,-6)$
Answer (1)
To find the y -intercept of the quadratic function f ( x ) = ( x − 6 ) ( x − 2 ) , we set x = 0 .
Substitute x = 0 into the function: f ( 0 ) = ( 0 − 6 ) ( 0 − 2 ) .
Calculate f ( 0 ) = ( − 6 ) ( − 2 ) = 12 .
The y -intercept is ( 0 , 12 ) .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = ( x − 6 ) ( x − 2 ) and asked to find its y -intercept. 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 f ( 0 ) . We substitute x = 0 into the function:
Substituting x=0 f ( 0 ) = ( 0 − 6 ) ( 0 − 2 ) = ( − 6 ) ( − 2 )
Calculating f(0) Now we multiply − 6 and − 2 :
Result ( − 6 ) ( − 2 ) = 12
Conclusion Therefore, the y -intercept is 12 , which corresponds to the point ( 0 , 12 ) .
Examples
Understanding the y-intercept of a quadratic function is useful in various real-world applications. For example, if we model the height of a projectile over time with a quadratic function, the y-intercept represents the initial height of the projectile when time is zero. Similarly, in business, if a quadratic function models the profit of a company as a function of the number of units sold, the y-intercept represents the profit when no units are sold (which could be a loss or an initial investment). Knowing the y-intercept provides a crucial starting point for analyzing the behavior of the function and making informed decisions.
