Problem 39 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(39-44,\) (a) find a power function end behavior model for \(f .\) (b) Identify any horizontal asymptotes. $$f(x)=3 x^{2}-2 x+1$$

Step-by-Step Solution

Answer

The power function end behavior model is \(+\infty\) and the function does not have a horizontal asymptote.

Step 1: Determining the power function end behavior model

Consider the highest power term of the function \(f(x) = 3x^2 - 2x + 1\). That is \(3x^2\). The coefficient is positive and the power of \(x\) is even. So as \(x\) approaches \(+\infty\) and \(-\infty\), \(3x^2\) approaches \(+\infty\). Therefore, the power function end behavior model is \(+\infty\)

Step 2: Identifying any horizontal asymptotes

As for the horizontal asymptotes, since the highest power of the function is \(2\) which is a finite number, and as \(x\) goes to \(+\infty\) or \(-\infty\), \(3x^2\) tends to \(+\infty\) and doesn't stay close to a constant number. Therefore, the function does not have a horizontal asymptote

Key Concepts

Power Function

In calculus, understanding the concept of a power function is crucial. A power function is typically in the form of \( f(x) = ax^n \), where \( a \) is a nonzero constant and \( n \) is a real number. In our given function \( f(x) = 3x^2 - 2x + 1 \), the power function is extracted by identifying the term with the highest power of \( x \), which is \( 3x^2 \). This means that as \( x \) becomes extremely large or very small, \( 3x^2 \) primarily dictates the behavior of the function.

• The presence of lower-degree terms, such as \(-2x + 1\), becomes negligible when compared to the term \(3x^2\) for large values of \(x\).
• This is because they do not grow as fast as the higher power term when \(x\) increases or decreases without bound.

Thus, the power function is the major player in determining how the function behaves at infinity.

End Behavior

End behavior of a function refers to the direction that the graph of a function heads as \( x \) approaches positive or negative infinity. In this case, we are focusing on the polynomial function \( f(x) = 3x^2 - 2x + 1 \). The most telling part of the end behavior comes from examining the power function, \( 3x^2 \). Since this is a quadratic function (the degree is 2 and is even), it will behave in a characteristic way:

• For positive values of \( x \) (as \( x \to +\infty \)), \( 3x^2 \to +\infty \), indicating that the function's graph opens upwards.
• For negative values of \( x \) (as \( x \to -\infty \)), \( 3x^2 \to +\infty \) as well, indicating the same upward direction.

This consistent upward direction on both ends means there is no switch in direction, reflecting how power functions with even exponents behave.

Horizontal Asymptotes

The concept of horizontal asymptotes involves analyzing the behavior of a function as \( x \) approaches infinity. Specifically, horizontal asymptotes are straight lines that the graph of a function approaches but never touches as \( x \) tends towards \( +\infty \) or \( -\infty \). In considering our function \( f(x) = 3x^2 - 2x + 1 \), we check if there are any horizontal asymptotes by looking at the growth rate of the terms:

• Since the term \( 3x^2 \) grows much faster than \(-2x\) and \(1\) as \( x \) gets large, it dominates the behavior of \( f(x) \).
• A horizontal asymptote exists when the function approaches a constant value at infinity, which does not occur here as \( 3x^2 \to +\infty \).

Therefore, for \( f(x) = 3x^2 - 2x + 1 \), there are no horizontal asymptotes because the function does not approach a constant value at infinity.