Problem 17 - Applied Calculus

Question

Which of the following tables could represent linear functions? $$ \begin{array}{l} \text { (a) }\\\ \begin{array}{l|c|c|c|c} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 27 & 25 & 23 & 21 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (b) }\\\ \begin{array}{l|l|l|l|l} \hline t & 15 & 20 & 25 & 30 \\ \hline s & 62 & 72 & 82 & 92 \\ \hline \end{array} \end{array} $$ $$ \begin{array}{l} \text { (c) }\\\ \begin{array}{c|c|c|c|c} \hline u & 1 & 2 & 3 & 4 \\ \hline w & 5 & 10 & 18 & 28 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

Tables (a) and (b) represent linear functions; table (c) does not.

Step 1: Check if Table (a) represents a linear function

To determine if the table represents a linear function, we need to calculate the differences between consecutive y-values. For the table with y-values 27, 25, 23, and 21, we find:\(25 - 27 = -2\)\(23 - 25 = -2\)\(21 - 23 = -2\)Since the differences between consecutive y-values are constant, table (a) represents a linear function.

Step 2: Check if Table (b) represents a linear function

For a linear function, the difference between consecutive y-values should be constant. Look at the table with s-values 62, 72, 82, and 92:\(72 - 62 = 10\)\(82 - 72 = 10\)\(92 - 82 = 10\)All differences are consistent, which indicates that table (b) represents a linear function.

Step 3: Check if Table (c) represents a linear function

With the w-values in the table as 5, 10, 18, and 28, calculate the differences:\(10 - 5 = 5\)\(18 - 10 = 8\)\(28 - 18 = 10\)The differences are not consistent for table (c), which means it does not represent a linear function.

Step 4: Final Conclusion

Tables (a) and (b) have constant differences between their consecutive y-values, so they represent linear functions. However, table (c) does not have constant differences, therefore it does not represent a linear function.

Key Concepts

Consecutive Differences

Understanding consecutive differences is key in identifying linear functions. A linear function describes a straight line when graphed. To determine if a table represents a linear function, we check the differences between consecutive y-values. If these differences remain constant, the relationship is linear. Imagine walking a set of stairs. Each step is the same height, just as each consecutive difference should be the same in a linear function. This consistency among differences ensures that the graph of the data points forms a straight line.

For instance, if we look at a sequence of numbers, like 27, 25, 23, and 21, calculating the differences gives us -2 consistently. This steady decrease confirms a linear pattern. Constant consecutive differences signal a reliable, unchanging rate of change, which is a hallmark of linear functions.

Function Representation

Linear functions can be represented in various ways, each offering a unique perspective on the relationship between variables. The most common representations are table, graph, and algebraic equation.

• Table: Just like in the original exercise, a table lists values for each variable, allowing us to observe patterns and differences easily.
• Graph: A plot of all the points from the table will yield a straight line if the function is linear. This visual representation makes it easier to see the consistency of the relationship.
• Equation: Often written as \(y = mx + b\), where \(m\) is the slope (or rate of change) and \(b\) is the y-intercept. The equation gives a concise formula to calculate y for any value of x, matching the consistent differences seen in the table.

Each representation provides a tool to understand linear functions better and to switch between visual and numerical perspectives with ease.

Constant Rate of Change

The term "constant rate of change" directly relates to the slope in linear functions. It's the 'm' in the equation \(y = mx + b\). For a function to be linear, this rate must never alter. It signifies how fast or slow y changes concerning x.

In practical terms, this is like traveling down a straight road. No matter how far you go, your speed (rate) stays the same. A constant rate of change tells us:

• The amount by which y increases or decreases as x increases by one unit.
• The direction of the slope: positive or negative.
• The uniformity and predictability of the relationship between variables.

In tables where data doesn't follow this constant change, such as with differences like 5, 8, and 10, the function isn't linear. Maintaining a constant rate is like ensuring every step on those stairs is equal. Without it, the relationship loses its straight-line integrity.