Which statement describes how the graph of the given polynomial would change if the term [tex]$2 x^5$[/tex] is added?
[tex]$y=8 x^4-2 x^3+5$[/tex]
A. Both ends of the graph will approach negative infinity.
B. The ends of the graph will extend in opposite directions.
C. Both ends of the graph will approach positive infinity.
D. The ends of the graph will approach zero.
Answer (1)
The original polynomial y = 8 x 4 − 2 x 3 + 5 has both ends approaching positive infinity.
Adding the term 2 x 5 results in the polynomial y = 2 x 5 + 8 x 4 − 2 x 3 + 5 .
The new polynomial has ends extending in opposite directions due to the odd degree of the leading term.
Therefore, the graph's end behavior changes, and the ends extend in opposite directions: The ends of the graph will extend in opposite directions.
Explanation
Analyze original polynomial Let's analyze the original polynomial y = 8 x 4 − 2 x 3 + 5 . The highest degree term is 8 x 4 . Since the degree is even (4) and the leading coefficient is positive (8), both ends of the graph will approach positive infinity. This means as x goes to positive infinity, y goes to positive infinity, and as x goes to negative infinity, y also goes to positive infinity.
Analyze new polynomial Now, let's consider the new polynomial with the added term: y = 2 x 5 + 8 x 4 − 2 x 3 + 5 . The highest degree term is now 2 x 5 . Since the degree is odd (5) and the leading coefficient is positive (2), the ends of the graph will extend in opposite directions. As x goes to positive infinity, y goes to positive infinity, and as x goes to negative infinity, y goes to negative infinity.
Compare the end behaviors Comparing the two polynomials, the original polynomial had both ends approaching positive infinity. The new polynomial has ends extending in opposite directions. Therefore, adding the term 2 x 5 changes the end behavior of the graph so that the ends extend in opposite directions.
Final Answer The correct statement is: The ends of the graph will extend in opposite directions.
Examples
Understanding the end behavior of polynomials is crucial in various fields. For instance, in physics, when modeling the trajectory of a projectile, the polynomial's end behavior can predict the long-term path of the object. In economics, polynomial functions can model growth or decay, and analyzing their end behavior helps forecast long-term trends. By understanding how the leading term affects the graph's behavior, we can make informed predictions in real-world scenarios.
