Question

Phishing Phishing is a criminal activity used by an individual or group to fraudulently acquire information by masquerading as a trustworthy person or business in an electronic communication. Criminals create spoof sites on the Internet to trick victims into giving them information. The sites are designed to copy the exact look and feel of a "real" site. A model for the number of reported spoof sites from November 2005 through October 2006 is \(f(t)=88.253 t^{3}-1116.16 t^{2}+4541.4 t+4161,0 \leq t \leq 11\) where \(t\) represents the number of months since November 2005.

Short answer

To use the model function for the number of reported spoof sites, simply substitute the number of months since November 2005 into the function. Examining the results for various values of \(t\) can reveal trends in the number of reported spoof sites.

Step-by-step solution

Understanding the Function

The provided function \(f(t)=88.253 t^{3}-1116.16 t^{2}+4541.4 t+4161\), where \(t\) is the number of months since November 2005, is used to model the number of reported spoof sites across time. Since it's a cubic function, it can represent more complex trends compared to linear or quadratic functions. This might indicate that the number of reported spoof sites has a non-linear trend, possibly increasing or decreasing at varying rates as time passes.

Evaluating the function

To find out the number of reported spoof sites at any given month since November 2005, simply substitute the value of \(t\) (number of months) in the function. For example, to find the number of sites in December 2005 (one month since November 2005), substitute \(t = 1\) into the function. This works for any values of \(t\) within the given domain \(0 \leq t \leq 11\)

Analyzing the trend

By evaluating the function at several values of \(t\), it is possible to identify a trend in the number of reported spoof sites. If the function yields increasing values with increasing \(t\), this indicates that the number of reports is rising over time. If the function yields decreasing values with increasing \(t\), this indicates that the number of reports is falling over time. Other patterns can also be observed by computing several values and examining the results.

Key concepts

Algebraic Modeling

Understanding algebraic modeling is essential for representing real-world problems using mathematical expressions. In the case of phishing activities, the function

\(f(t)=88.253 t^{3}-1116.16 t^{2}+4541.4 t+4161\)

is an example of algebraic modeling, where a cubic polynomial is used to fit the pattern of reported spoof sites over a period of time. A cubic function is chosen because it can capture the nuances and fluctuations of real-life situations better than simpler functions like linear or quadratic equations. The coefficients (88.253, -1116.16, 4541.4, 4161) are specific to this scenario and were likely obtained through statistical methods to best fit the data available for the model.

This model allows researchers and law enforcement to forecast future trends and prepare for possible outcomes.

Function Evaluation

Evaluating functions is a core component of understanding how algebraic models work in various scenarios. In our phishing example, function evaluation involves substituting a particular value of

\(t\)

into the equation to find the number of spoof sites reported at that time. Evaluating functions allows one to get concrete numbers from abstract models. For instance, to know how many sites were reported 6 months after November 2005, one would plug in

\(t = 6\)

and calculate

\(f(6)\).

It is crucial to properly substitute values and perform the arithmetic carefully to ensure accurate results. Every aspect of the function—including its domain and range—must be understood to correctly interpret these results.

Trend Analysis

Trend analysis involves examining patterns within data to predict future movements. In the context of our phishing example, analyzing the trend of reported spoof sites can give us insight into whether phishing activities are increasing or decreasing over time, and at what rate.

By calculating the value of the function

\(f(t)\)

at different points, we look for patterns: is the output growing as

\(t\)

increases, indicating a rise in reports, or is it decreasing? Identifying these trends helps allocate resources, inform policy decisions, and heighten public awareness. Importantly, trend analysis not only concerns the direction but also the rate of change. The cubic nature of the function means we can expect complex behavior, such as the rate of increase or decrease changing over time, requiring a deeper analysis than simply assessing a positive or negative trajectory.