Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, identify whether the statement represents an exponential function. Explain. For each training session, a personal trainer charges his clients \(\$ 5\) less than the previous training session.

Step-by-Step Solution

Answer

No, it's not an exponential function; it's linear.

Step 1: Understand the statement

The statement describes the situation where a personal trainer charges his clients less money with each training session.

Step 2: Define the rate of change

An exponential function is characterized by a constant percentage change. Here, the decrease is constant (\(\$5\) less), not a percentage.

Step 3: Determine the type of sequence

With \(\$5\) less per session, the sequence is arithmetic, not exponential, as the same amount is deducted each time.

Step 4: Conclusion

The statement does not represent an exponential function because the rate of change is linear (subtraction of $5 each time) rather than multiplicative (percentage change).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This fixed value that we add or subtract each time is called the common difference. In our example with the personal trainer, they charge $5 less every session. Here, $5 is the common difference, making it an arithmetic sequence.

Some key characteristics of arithmetic sequences include:

• A constant addition or subtraction.
• The difference, or step, is always the same between subsequent numbers.

When visualized, an arithmetic sequence forms a straight line in a graph, which points directly to the consistency of the sequence's growth or diminution.

Linear Change

A linear change involves a value that increases or decreases by a fixed amount with each step. This kind of change is common in arithmetic progressions, such as the one mentioned with the personal trainer charging less.

Linear change signifies that if you were to graph the changes, you'd see a straight line pointing upwards or downwards, depending on whether the change adds or subtracts from the value. In our example, it subtracts, showing a downward trend.
Linear changes are simple:

• The amount of increase or decrease remains the same.
• It shows a predictable pattern.

This type of change makes calculations and predictions straightforward, given its uniformity.

Percentage Change

Percentage change represents how much a quantity increases or decreases in terms of a percentage of its original value. Unlike linear change, percentage change involves multiplication instead of addition or subtraction.

For example, if a value grows by 10%, you multiply it by 1.1 to find its new value. If it decreases by 10%, you multiply by 0.9. An exponential function embodies this concept, where each change is a product of a percentage change.

With percentage change, key points include:

• Growth or decline is compounded.
• Values might increase or decrease at an accelerating rate.

Such change is more complex than linear as it forms curves rather than straight lines on a graph, representing exponential growth or decay.

Rate of Change

The rate of change refers to how quickly or slowly a quantity increases or decreases over time. This rate can be constant or variable, influencing how we model or predict behavior in different contexts.

For the personal trainer example, the rate of change is constant, represented by the $5 decrease each session. This rate leads to a straightforward, predictable pattern, typical of arithmetic sequences and linear changes.

Different rates of change include:

• Constant rate, like our example, simple and predictable.
• Variable or percentage rate, often seen in exponential functions, where the rate depends on the current value.

Understanding the rate of change helps in determining the nature of a sequence or function, guiding us in the correct classification—linear, exponential, or other types.