Graph the polynomial function [tex]f(x)=-2(x-3)^2\left(x^2-25\right)[/tex] using parts (a) through (e).
(a) Determine the end behavior of the graph of the function.
The graph of [tex]f[/tex] behaves like [tex]y=-2 x^4[/tex] for large values of [tex]|x|[/tex].
(b) Find the [tex]x[/tex]- and [tex]y[/tex]-intercepts of the graph of the function.
The [tex]x[/tex]-intercept(s) is/are
Answer (1)
Find x-intercepts by solving f ( x ) = 0 , which gives x = − 5 , 3 , 5 .
Find y-intercept by evaluating f ( 0 ) = − 2 ( 0 − 3 ) 2 ( 0 2 − 25 ) .
Calculate f ( 0 ) = − 2 ( 9 ) ( − 25 ) = 450 .
The x-intercepts are − 5 , 3 , 5 and the y-intercept is 450 .
Explanation
Problem Analysis We are given the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) and asked to find its x - and y -intercepts.
Finding x-intercepts To find the x -intercepts, we need to solve the equation f ( x ) = 0 . This means we need to find the values of x for which − 2 ( x − 3 ) 2 ( x 2 − 25 ) = 0 .
Solving for x We have − 2 ( x − 3 ) 2 ( x 2 − 25 ) = 0 . Since − 2 e q 0 , we must have either ( x − 3 ) 2 = 0 or x 2 − 25 = 0 .
Repeated Root If ( x − 3 ) 2 = 0 , then x − 3 = 0 , so x = 3 . This is a repeated root, meaning the graph touches the x-axis at x = 3 but doesn't cross it.
Finding Other Roots If x 2 − 25 = 0 , then x 2 = 25 , so x = ± 5 . Thus, x = 5 and x = − 5 are also x -intercepts.
X-intercepts Therefore, the x -intercepts are x = − 5 , 3 , 5 .
Finding y-intercept To find the y -intercept, we need to evaluate f ( 0 ) .
Calculating f(0) f ( 0 ) = − 2 ( 0 − 3 ) 2 ( 0 2 − 25 ) = − 2 ( − 3 ) 2 ( − 25 ) = − 2 ( 9 ) ( − 25 ) = − 18 ( − 25 ) = 450 .
Y-intercept Therefore, the y -intercept is y = 450 .
Final Answer The x -intercepts are − 5 , 3 , 5 and the y -intercept is 450 .
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 can indicate the initial position of an object. By finding intercepts, we can analyze key aspects of a function's behavior and apply it to practical scenarios.
