Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Former professional basketball player Shaquille O'Neal is 7 feet, 1 inch tall and weighs 325 pounds. The table lists his shoe sizes at certain ages. $$\begin{array}{|l|c|c|c|c|} \hline \text { Age } & 20 & 21 & 22 & 23 \\ \hline \text { Shoe Size } & 19 & 20 & 21 & 22 \\\ \hline \end{array}$$ (a) Write a formula that gives his shoe size \(y\) at age \(x=20,21,22,\) and 23 (b) Suppose that after age 23 his shoe size did not change. Sketch a graph of a continuous, piece wise-defined function \(f\) that models his shoe size between the ages 20 and 26, inclusive.

Step-by-Step Solution

Answer

The shoe size increases linearly as \(y = x - 1\) from age 20 to 23, and remains constant at 22 from age 24 to 26.

Step 1: Identify Data Points

From the given data table, identify the pairs of age and shoe size: \((20, 19), (21, 20), (22, 21), (23, 22)\). These will be used to find the pattern in the shoe sizes.

Step 2: Calculate Differences

Observe the shoe sizes increase by 1 for each year. Calculate the difference between consecutive shoe sizes: \(20-19=1\), \(21-20=1\), \(22-21=1\), \(22-22=0\) (no change assumed after age 23).

Step 3: Derive the Formula for Shoe Size

Since the shoe size increases by 1 each year starting from 19 at age 20, we can express the shoe size as a linear function of age: \(y = x - 1\). Verify this: At age 20, \(y=20-1=19\), at age 21, \(y=21-1=20\), and so on.

Step 4: Piecewise Function for Continuous Shoe Size

Define the function \( f(x) \) where \( f(x) = x - 1 \) for ages 20 to 23, and since there is no change after 23, \( f(x) = 22 \) for ages greater than 23 up to 26. Thus, the function can be written as: \[ f(x) = \begin{cases} x - 1 & \text{if } 20 \leq x \leq 23 \ 22 & \text{if } 23 < x \leq 26 \end{cases} \]

Step 5: Sketch the Piecewise Function

Draw a graph with age on the x-axis and shoe size on the y-axis. Plot the continuous line from (20,19) to (23,22) following the formula \( y = x - 1 \). From age 23 onwards, plot a horizontal line at \( y = 22 \) extending to age 26.

Key Concepts

Linear Function

A linear function is a type of mathematical function where the relationship between the input variable and the output variable is a straight line. In the context of shoe sizes, the linear function used is simple and directly relates age to shoe size. This can be described by the equation \( y = x - 1 \). Here, \( x \) is the age and \( y \) is the shoe size.

• The determining feature of a linear function is its constant rate of change. In this case, as age increases by one year, the shoe size increases by one as well.
• This relationship is maintained across ages 20 to 23.
• Linear functions such as this one are easy to graph because they create a straight line.

Understanding linear functions helps us see how changes in one variable equate to changes in another, making them great tools for representing real-world data like Shaquille O'Neal's shoe sizes.

Data Points

Data points are crucial elements when analyzing or modeling any function. They represent definite values of x (input) and their corresponding y (output) values. In the exercise, these data points denote ages and shoe sizes, respectively.

• The data given is: \((20, 19), (21, 20), (22, 21), (23, 22)\).
• Each pair tells us the shoe size at a particular age. Notice how the shoe size increases consistently by 1, reflecting a linear increase.
• Data points can be plotted to visualize the relationship clearly.

By pinpointing and using data points, it's possible to determine patterns and create an accurate model, which in this case is a linear function describing shoe size changes over time.

Graph Sketching

Graph sketching involves plotting data points and their corresponding functions on a graph to visually interpret the relationship between variables. This is particularly useful for understanding piecewise functions like the one described.

• The x-axis represents age, and the y-axis represents shoe size.
• The points \((20, 19)\) through \((23, 22)\) are connected by a line to display the equation \(y = x - 1\).
• After age 23, the graph shows a horizontal line at shoe size 22, indicating that the shoe size remains constant beyond age 23 up to age 26.

This method of graph sketching helps visualize the piecewise nature of functions, exhibiting changes or constancy over different intervals.

Age-dependent Function

An age-dependent function changes according to the value of age. In Shaquille O'Neal's shoe size problem, it describes how shoe size evolved from age 20 to 26. This is a piecewise function, meaning it has different expressions depending on the value of the input variable, age.

• For ages 20 through 23, the shoe size increases linearly following \( f(x) = x - 1 \).
• After age 23, up to age 26, the function remains constant, expressed as \( f(x) = 22 \).
• This reflects typical growth patterns in practical scenarios, such as when physical changes occur only over a limited timeframe.

Piecewise age-dependent functions are valuable for modeling situations where changes happen in distinct phases or segments.