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Polynomial functions are expressions comprised of terms, each containing a variable raised to a non-negative integer power, such as the function \( C(q) = -2q^3 + 5q^2 - 3q + 7 \). In such functions, the behavior of the graph is primarily influenced by the leading term, which is the term with the highest exponent. For example, in \(-2q^3 + 5q^2 - 3q + 7\), the leading term is \(-2q^3\), dictating the overall behavior of the polynomial for large values of \( q \).
Understanding the leading term allows us to determine the end behavior of the polynomial. This involves assessing how the function behaves as \( q \) approaches positive or negative infinity. For odd-degree polynomials like \(-2q^3\), with a negative leading coefficient, the function will fall towards \(-\infty\) as \( q \) increases towards \( \infty \) and rise towards \( \infty \) as \( q \) decreases towards \(-\infty \).
This characteristic pattern helps in sketching graphs and predicting polynomial behavior, making polynomial functions a foundational concept in calculus and algebra.

Limit notation is a mathematical way of describing the behavior of a function as the input approaches a certain value. For polynomial functions, limit notation is vital in expressing end behavior as the input heads towards positive or negative infinity.
As in our specific function \( C(q) = -2q^3 + 5q^2 - 3q + 7 \), the end behavior can be clearly articulated using limits. The expression \( \lim_{{q \to \infty}} C(q) = -\infty \) shows that as \( q \) increases without bound, the output of \( C(q) \) decreases indefinitely. Conversely, \( \lim_{{q \to -\infty}} C(q) = \infty \) indicates that as \( q \) decreases without bound, \( C(q) \) increases indefinitely.
Mastering the use of limit notation can greatly aid in analyzing how functions behave at extreme values, providing a concise language to express complex behavior in calculus.

Horizontal asymptotes are lines that the graph of a function approaches but never touches. They are crucial in understanding the long-term behavior of rational functions, which have both numerators and denominators.
However, in polynomial functions like \( C(q) = -2q^3 + 5q^2 - 3q + 7 \), horizontal asymptotes do not exist because there are no denominators involved. Instead, the polynomial’s degree dictates its end behavior rather than approaching a specific horizontal line.
The lack of horizontal asymptotes emphasizes the power of the polynomial’s degree in controlling the function's ultimate direction and behavior. Understanding this aspect helps in distinguishing polynomial functions from rational functions, where horizontal or oblique asymptotes frequently appear.