Problem 125 - Precalculus Essentials

Question

The function $$f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95$$ models the number of annual physician visits, \(f(x),\) by a person of age \(x .\) Graph the function in a [0,100,5] by [0,40,2] viewing rectangle. What does the shape of the graph indicate about the relationship between one's age and the number of annual physician visits? Use the \([\mathrm{TABLE}]\) or minimum function capability to find the coordinates of the minimum point on the graph of the function. What does this mean?

Step-by-Step Solution

Answer

Upon observation, the graph shows that younger people tend to have more visits due to obligatory medical check-ups, after which visits decrease but start increasing once again after a certain age due to age-related health problems. The minimum point on the graph occurs when the annual physician's visits are the least, typically in early adulthood.

Step 1 - Graph the Function

Start by plotting the function \(f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x+6.95\) using a graphing tool. Be certain to adjust the viewing window to [0,100,5] by [0,40,2] to correctly visualize the data.

Step 2 - Analyze the Graph

Upon observing the graph, the relationship between age and annual doctor visits can be recognized. Initially, the visits are relatively high at younger ages (due to requirements for children's medical check-ups) and then it decreases as the age gradually increases but after reaching a certain age, it begins to rise again. This indicates more visits to the doctor with increasing age due to potential age-related health problems.

Step 3 - Find the Minimum Point

Use the TABLE or a minimum function capability of a graphing tool to determine the coordinates of the minimum point on the graph. This point corresponds to the age at which the number of annual physician visits is the lowest.

Step 4 - Interpret the Minimum Point

The minimum point on the graph corresponds to the age when the number of annual physician visits is the least. This also indicates, when people in their early adulthood are usually healthier and hence, tend to visit the physician less.

Step 5 - Conclusion

Summarize the findings that as one grows old, initially the annual visits to a physician decrease to the minimum and later they tend to increase as seen from the shape of the function graph. The age where the annual visits is minimum is given by the minimum point on the graph.

Key Concepts

Polynomial Function Analysis

Polynomial functions are a type of mathematical function that include terms with variables raised to whole number exponents. The function in our exercise, \(f(x) = -0.00002x^3 + 0.008x^2 - 0.3x + 6.95\), is a cubic polynomial because the highest exponent is 3. Analyzing such a function involves looking at its graph to understand its behavior.

When graphing a polynomial function, we observe its shape to identify features like increasing and decreasing intervals, local maxima and minima, and the end behavior—how the function behaves as \(x\) becomes very large or very small. The coefficients of the terms influence the steepness and direction of the curve. In this exercise, graphing helps visualize the trend, which is crucial for interpreting real-world models, like the one illustrating the frequency of physician visits as a function of age.

Relationship Between Age and Physician Visits

The polynomial function provided models the relationship between a person's age and the number of times they visit a physician annually. Understanding this relationship is essential for planning healthcare services and forecasting needs for different age groups. After graphing the function, we see a curve representing this relationship; in our case, visits are high for young ages, decrease through early adulthood to a low point, and then increase again.

The model suggests that infants and young children require frequent visits for vaccinations and development check-ups. There's a decline afterward, likely due to the general robust health of adolescents and young adults. Later in life, the trend reverses as aging brings more health issues, hence the increase in doctor visits. These insights, gained through graph analysis, are invaluable for healthcare policy and individual health planning.

Minimum Point of a Function

The minimum point of a function is the point where the function's value is the lowest on a given interval. In real-world models, finding this point can represent optimizing for the least cost, least time, or, as in our case, the least number of physician visits. The minimum point is often found using calculus or graphing tools. In the context of our exercise, the minimum represents the age at which a person is least likely to visit a physician. Calculated using either a TABLE feature or a minimum function capability on a graphing calculator, it's a practical way to identify a specific value that could be critical in decision-making or in understanding the behavior of a demographic model. This concept empowers students to analyze and interpret data, emphasizing its importance in fields such as economics, health sciences, and engineering.