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The zeros of the function [tex]$f(x)=-(x+1)(x-3)(x+2)$[/tex] are -1, 3, and $\boxed{}$ and the [tex]$y$[/tex]-intercept of the function is located at (0, $\boxed{}$ ).
Answer (1)
The zeros of the function f ( x ) = − ( x + 1 ) ( x − 3 ) ( x + 2 ) are found by setting each factor to zero: x + 1 = 0 , x − 3 = 0 , and x + 2 = 0 .
Solving these equations gives the zeros x = − 1 , x = 3 , and x = − 2 .
The y -intercept is found by evaluating f ( 0 ) = − ( 0 + 1 ) ( 0 − 3 ) ( 0 + 2 ) .
Calculating this gives f ( 0 ) = 6 , so the y -intercept is at ( 0 , 6 ) . The final answer is − 2 , 6 .
Explanation
Understanding the problem We are given the function f ( x ) = − ( x + 1 ) ( x − 3 ) ( x + 2 ) . We need to find the zeros of the function and the y -intercept.
Finding the zeros The zeros of the function are the values of x for which f ( x ) = 0 . So we need to solve the equation − ( x + 1 ) ( x − 3 ) ( x + 2 ) = 0 . This equation is satisfied when x + 1 = 0 , x − 3 = 0 , or x + 2 = 0 .
Identifying the zeros Solving these equations, we get x = − 1 , x = 3 , and x = − 2 . Thus, the zeros of the function are − 1 , 3 , and − 2 .
Finding the y-intercept The y -intercept is the point where the graph of the function intersects the y -axis. This occurs when x = 0 . To find the y -intercept, we need to evaluate f ( 0 ) .
Calculating the y-intercept We have f ( 0 ) = − ( 0 + 1 ) ( 0 − 3 ) ( 0 + 2 ) = − ( 1 ) ( − 3 ) ( 2 ) = − ( − 6 ) = 6 . So the y -intercept is located at ( 0 , 6 ) .
Final Answer Therefore, the zeros of the function are − 1 , 3 , and − 2 , and the y -intercept is located at ( 0 , 6 ) .
Examples
Understanding zeros and intercepts of functions is crucial in many real-world applications. For instance, in physics, the zeros of a projectile's height function tell us when the projectile hits the ground. The y-intercept represents the initial height of the projectile. Similarly, in economics, the zeros of a cost function can represent break-even points, and the y-intercept can represent fixed costs. By analyzing these key points, we can gain valuable insights into the behavior of various systems and make informed decisions.
