Question

SAT or ACT? The number of participants in SAT and ACT testing from 1995 to 2005 can be approximated by \(\left\\{\begin{array}{ll}y=0.68 t^{2}+28.1 t+903 & \text { SAT } \\ y=-0.485 t^{3}+14.88 t^{2}-115.1 t+1201 & \text { ACT }\end{array}\right.\) where \(y\) is the number of participants (in thousands) and \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use a graphing utility to determine whether the number of participants in ACT testing will exceed the number of participants in SAT testing. Do you think these models will continue to be accurate? Explain your reasoning. (Sounce: College Board; ACT, Inc.)

Short answer

The graphical analysis would help in telling whether the testers of ACT exceeded those of SAT within the given years. But, since the functions are quadratic and cubic which have different long term behaviors, without knowing real world constraints or the context, it cannot be definitively said that this model will remain accurate in the future. More complex statistical modelling and data might be needed to predict future trends accurately.

Step-by-step solution

Represent the Models Graphically

Plot both the given equations using a graphing utility. The equation for SAT is \(y=0.68 t^{2}+28.1 t+903\) and the equation for ACT is \(y=-0.485 t^{3}+14.88 t^{2}-115.1 t+1201\). Make sure to choose an appropriate range for \(t\), representing the years 1995 to 2005.

Determine Intersection Point

Examine the generated graph to see if the two curves intersect within the time frame considered. If it does intersect, then this intersection point indicates where the number of ACT and SAT takers becomes equal.

Analyze and Compare

If the ACT curve exceeds the SAT curve after a certain point, it means that more people took the ACT than the SAT after that year. As for the continuation of the model's accuracy beyond 2005, this requires an understanding of the cubic and quadratic models and their long-term behavior.

Understand the Long-Term Behavior of the Models

Bear in mind that the quadratic function(SAT) will continue to grow indefinitely as \(t\) increases, albeit at a slower pace. The cubic function(ACT), however, can increase or decrease depending on the sign of \(t\). So, the ACT numbers could decrease after a certain point.

Reasoning

Use the gathered evidence and understanding to reason if this model will still be valid in the future. The inferred accuracy of the models will be based on understanding of graphical representation and the nature of the mathematical models.

Key concepts

Graphing Utility Analysis

Graphing utility analysis is a powerful tool for visualizing the behaviors of different functions, particularly when comparing data like SAT and ACT participation over time. Using a graphing utility, students can easily plot the quadratic and cubic functions provided for SAT and ACT participation rates respectively. By doing so, they observe how the number of participants changes as years progress (with the year 1995 being denoted by t=5).

For an effective analysis, it's crucial to choose the correct scale for t. This way, the graph accurately represents the period from 1995 to 2005. Through graphing, one can identify key features such as intercepts, turning points, and intersections, providing insights at a glance that numerical data alone cannot. This visual examination can answer the question of whether ACT participation will surpass SAT participation within the given timeframe. Moreover, graphical analysis can also highlight limitations or anomalies in the data that might not be apparent through equations alone.

Quadratic and Cubic Functions

The mathematical expressions given for SAT and ACT participation rates are prime examples of quadratic and cubic functions. A quadratic function, like the one modeling SAT participation, is characterized by the highest power of t being two, represented as t^2. It creates a parabola on a graph and generally indicates a consistent rate of change that will continue to grow as t increases.

On the other hand, the cubic function used to model ACT participation involves t^3, the highest power of t being three. Cubic functions can experience more complex behavior with the potential for multiple peaks and valleys on a graph. They have an inflection point—where the curve changes from being concave up to concave down or vice versa—and their long-term behavior can exhibit both increases and decreases. Understanding these function types is essential when predicting future data since the shapes of their graphs give a clear visual representation of how participation numbers are expected to trend over time.

Mathematical Model Prediction

Mathematical model prediction involves using algebraic expressions to forecast future data points. These models can be derived from historical data, as seen with the SAT and ACT participation rates. With these predictive models, students can extend the graph beyond the given data range to forecast future participation. However, caution is needed, as these predictions assume that the trend will continue without interruption, which is not always the case in real-life scenarios.

The validity of predictions hinges on the models’ accuracy and the likelihood that the underlying factors influencing the data will remain constant. Changing educational policies, test accessibility, or cultural shifts could all affect the relevance of these models over time. Therefore, while a prediction made with a mathematical model provides an informed estimate, it should be treated as a hypothesis to be validated with actual data as time progresses.

Long-Term Behavior of Algebraic Models

To discuss the long-term behavior of algebraic models, particularly those representing real-world data like SAT and ACT participation, it's important to extend our analysis beyond the given time frame. For a quadratic function, the long-term behavior indicates indefinite growth as t increases, which in the context of SAT participation, would suggest an ever-increasing number of test-takers. This conclusion comes with the assumption that all other factors influencing participation rates remain constant.

The cubic function that models ACT participation presents a more nuanced long-term prediction. Its growth depends on the coefficients of the t terms. If the cubic coefficient (associated with t^3) is negative, for instance, the function might increase up to a certain point and then decrease. Consequently, while logic might suggest that the participation rates could keep growing, both functions will eventually diverge from real-world scenarios as they do not account for market saturation or changing educational trends. Therefore, understanding these algebraic behaviors is essential for interpreting mathematical models, but it is equally crucial to acknowledge their limitations.