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 one end approaching positive infinity and the other approaching negative infinity.
Therefore, the ends of the graph will extend in opposite directions: $\boxed{The ends of the graph will extend in opposite directions.}
Explanation
Analyzing the Polynomial Let's analyze the behavior of the given polynomial y = 8 x 4 − 2 x 3 + 5 and see how it changes when we add the term 2 x 5 . This involves understanding the concept of end behavior, which is determined by the term with the highest power of x (the leading term).
Determining End Behavior of Original Polynomial The original polynomial is y = 8 x 4 − 2 x 3 + 5 . The leading term is 8 x 4 . Since the exponent is even (4) and the coefficient is positive (8), both ends of the graph will go towards positive infinity. This means as x becomes very large (positive or negative), y becomes very large and positive.
Determining End Behavior of New Polynomial Now, let's add the term 2 x 5 to the polynomial. The new polynomial becomes y = 2 x 5 + 8 x 4 − 2 x 3 + 5 . The leading term is now 2 x 5 . Since the exponent is odd (5) and the coefficient is positive (2), the end behavior changes. As x approaches positive infinity, y also approaches positive infinity. However, as x approaches negative infinity, y approaches negative infinity.
Comparing End Behaviors Comparing the end behaviors, we see that the original polynomial had both ends going towards positive infinity. The new polynomial has one end going towards positive infinity and the other end going towards negative infinity. This means the ends of the graph now extend in opposite directions.
Conclusion Therefore, the statement that describes how the graph of the given polynomial would change if the term 2 x 5 is added is: The ends of the graph will extend in opposite directions.
Examples
Understanding how polynomial functions change with the addition of terms is crucial in various real-world applications. For instance, in physics, the trajectory of a projectile can be modeled by a polynomial. Adding a term might represent an additional force acting on the projectile, altering its path. Similarly, in economics, polynomial functions can model cost and revenue curves. Adding a term could represent a change in market conditions, affecting the overall profitability. By analyzing these changes, we can make informed decisions and predictions in these fields. For example, the initial polynomial might be y = x 2 , and adding x 3 results in y = x 3 + x 2 . This significantly changes the behavior of the function, especially as x becomes very large or very small.
