Graph the polynomial function [tex]f(x)=-2(x-3)^2(x^2-25)[/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=[/tex] $\square$ for large values of [tex]|x|[/tex].
Answer (1)
Expand the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) .
Identify the leading term of the expanded polynomial.
Determine the end behavior based on the leading term.
The graph of f behaves like $y =
− 2 x 4 for large values of ∣ x ∣ .
Explanation
Understanding the Problem We want to determine the end behavior of the polynomial function f ( x ) = − 2 ( x − 3 ) 2 ( x 2 − 25 ) . The end behavior of a polynomial is determined by its leading term, which is the term with the highest power of x . To find the leading term, we need to expand the polynomial.
Expanding the terms First, let's expand ( x − 3 ) 2 . ( x − 3 ) 2 = ( x − 3 ) ( x − 3 ) = x 2 − 3 x − 3 x + 9 = x 2 − 6 x + 9 Next, let's expand ( x 2 − 25 ) . This is already expanded.
Multiplying the Quadratics Now, we multiply the two quadratic expressions: ( x 2 − 6 x + 9 ) ( x 2 − 25 ) = x 2 ( x 2 − 25 ) − 6 x ( x 2 − 25 ) + 9 ( x 2 − 25 ) & = x 4 − 25 x 2 − 6 x 3 + 150 x + 9 x 2 − 225& = x 4 − 6 x 3 − 16 x 2 + 150 x − 225
Multiplying by -2 Finally, we multiply the result by − 2 : − 2 ( x 4 − 6 x 3 − 16 x 2 + 150 x − 225 ) = − 2 x 4 + 12 x 3 + 32 x 2 − 300 x + 450
Determining the End Behavior The leading term of the polynomial is − 2 x 4 . Therefore, for large values of ∣ x ∣ , the graph of f behaves like y = − 2 x 4 .
Final Answer The graph of f behaves like y = − 2 x 4 for large values of ∣ x ∣ .
Examples
Understanding the end behavior of polynomial functions is crucial in various fields, such as physics and engineering, where polynomials are used to model real-world phenomena. For example, in projectile motion, the height of a projectile can be modeled by a quadratic function. Knowing the end behavior of this function helps us understand the overall trajectory of the projectile and predict its behavior over time. Similarly, in electrical engineering, polynomials are used to model the behavior of circuits, and understanding their end behavior helps in designing stable and efficient systems. By analyzing the leading term of the polynomial, we can determine the long-term trends and make informed decisions about the system's performance.
