Without using technology, describe the end behavior of [tex]$f(x)=3 x^4+8 x^2-22 x+43$[/tex].
A. Down on the left, down on the right
B. Down on the left, up on the right
C. Up on the left, down on the right
D. Up on the left, up on the right
Answer (2)
The leading term of the polynomial f ( x ) = 3 x 4 + 8 x 2 − 22 x + 43 is 3 x 4 .
Since the leading coefficient is positive (3) and the degree is even (4), the end behavior is the same in both directions.
As x approaches positive or negative infinity, f ( x ) approaches positive infinity.
Therefore, the end behavior is up on the left and up on the right: Up on the left, up on the right
Explanation
Analyze the polynomial We are given the polynomial function f ( x ) = 3 x 4 + 8 x 2 − 22 x + 43 . We want to determine its end behavior, which means we want to know what happens to the function as x approaches positive infinity and negative infinity.
Identify the leading term The end behavior of a polynomial is determined by its leading term, which is the term with the highest power of x . In this case, the leading term is 3 x 4 . The coefficient of the leading term is 3, which is positive, and the exponent of the leading term is 4, which is even.
Determine the end behavior When the leading coefficient is positive and the exponent is even, the function will go to positive infinity as x goes to both positive and negative infinity. In other words, as x becomes very large (either positive or negative), the value of f ( x ) becomes very large and positive.
State the conclusion Therefore, the end behavior of the function f ( x ) = 3 x 4 + 8 x 2 − 22 x + 43 is up on the left and up on the right.
Examples
Understanding the end behavior of polynomial functions is useful in many real-world applications. For example, if you are modeling the cost of production as a function of the number of units produced, the end behavior of the cost function can tell you whether the cost will increase without bound as you produce more and more units. Similarly, in physics, the end behavior of a potential energy function can tell you about the stability of a system. In general, understanding the behavior of functions as their inputs become very large or very small is an important tool for making predictions and understanding the world around us. For instance, consider a simple model where the profit P of a company depends on the number of items x sold, and the relationship is modeled by a polynomial P ( x ) = a x 4 + b x 2 + c . Knowing the end behavior (determined by the sign of a ) helps predict if the profit will eventually increase or decrease as sales grow significantly.
The polynomial f ( x ) = 3 x 4 + 8 x 2 − 22 x + 43 has a leading term of 3 x 4 which is positive and has an even degree. Hence, the end behavior of the function is that it goes to positive infinity on both the left and the right. The correct answer is: Up on the left, up on the right.
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