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Polynomial functions are mathematical expressions involving a variable raised to various powers. They can be written in the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each coefficient \( a_i \) is a constant. In the context of our problem, the given polynomial function:\[p(x) = 0.0987x^4 - 2.1729x^3 + 17.027x^2 - 55.023x + 72.133\]represents the average price of natural gas as a function of time (in years). The variable \( x \) indicates the number of years since 2000.Polynomial functions can be classified by their degree, which is determined by the highest power of \( x \). In this case, the degree is 4. Such functions are called quartic polynomials.
They feature various behaviors, such as turning points and inflection points, which can be identified using calculus. These points are essential for understanding changes in trends over time.

The second derivative test helps determine local maxima, minima, or inflection points of a function.
By taking the second derivative of a polynomial and setting it to zero, we can find where the concavity changes, indicating potential inflection points.Within our exercise, calculating the second derivative from the first derivative: \[p''(x) = 1.1844x^2 - 13.0374x + 34.054 \]This formula allows us to identify these critical points on the graph of the function by solving the equation \( p''(x) = 0 \).
This reveals values of \( x \) that may correspond to inflection points, special points where the rate of increasing or decreasing might shift.

The concept of rate of change highlights how a function's value changes over time or across different inputs.
In polynomial functions, this can be examined by looking at the first derivative of the function \( p'(x) \). For our problem:\[p'(x) = 0.3948x^3 - 6.5187x^2 + 34.054x - 55.023 \]By evaluating \( p'(x) \) at specific \( x \) values (in this case, the inflection points), we determine how steep the slope of the graph is at those points.
A positive rate indicates an increasing trend, while a negative value shows a decreasing trend.
In other words, the rate of change at an inflection point describes how the natural gas price is increasing or decreasing at that particular time, offering insights into its overall trend.

The quadratic formula is a method used to find the solutions for quadratic equations of the form \( ax^2 + bx + c = 0 \).The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This is crucial in solving for inflection points in our exercise.
For the polynomial function provided, the second derivative produces a quadratic equation:\[1.1844x^2 - 13.0374x + 34.054 = 0 \]Applying the quadratic formula to this equation helps us find the \( x \) values \( \approx 4.47 \) and \( \approx 6.41 \), which are the points where inflection occurs.
Recognizing these solutions is fundamental in identifying shifts in the trend of the gas price, signaling significant changes in its behavior.