Find the $x$-intercepts and the $y$-intercept of the graph of $f(x)=(x-3)(x-2)(x-1)$ without using technology. Show all work.
Answer (2)
The x -intercepts of the function f ( x ) = ( x − 3 ) ( x − 2 ) ( x − 1 ) are 1 , 2 , and 3 . The y -intercept is − 6 .
;
To find the x -intercepts, set f ( x ) = 0 and solve for x , which gives x = 1 , 2 , 3 .
To find the y -intercept, set x = 0 and evaluate f ( 0 ) , which gives f ( 0 ) = − 6 .
The x -intercepts are 1 , 2 , 3 .
The y -intercept is − 6 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x − 3 ) ( x − 2 ) ( x − 1 ) and we need to find the x -intercepts and the y -intercept of its graph. The x -intercepts are the points where the graph intersects the x -axis, which means f ( x ) = 0 . The y -intercept is the point where the graph intersects the y -axis, which means x = 0 .
Finding the x-intercepts To find the x -intercepts, we set f ( x ) = 0 and solve for x :
( x − 3 ) ( x − 2 ) ( x − 1 ) = 0 This equation is satisfied when any of the factors are equal to zero. Thus, we have: x − 3 = 0 ⇒ x = 3 x − 2 = 0 ⇒ x = 2 x − 1 = 0 ⇒ x = 1 So, the x -intercepts are x = 1 , 2 , 3 .
Finding the y-intercept To find the y -intercept, we set x = 0 and evaluate f ( 0 ) :
f ( 0 ) = ( 0 − 3 ) ( 0 − 2 ) ( 0 − 1 ) = ( − 3 ) ( − 2 ) ( − 1 ) = − 6 So, the y -intercept is − 6 .
Final Answer Therefore, the x -intercepts are 1 , 2 , 3 and the y -intercept is − 6 .
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in business, the x-intercept can represent the break-even point where costs equal revenue. In physics, the y-intercept of a motion graph might indicate the initial position of an object. By finding intercepts, we gain valuable insights into the behavior and characteristics of functions and their practical implications.
