Scatter Plot
A scatter plot is a graphical representation of two variables that shows their relationship. Each point on the scatter plot represents an individual data point from the dataset.
In the context of the presented exercise, creating a scatter plot would involve plotting revenue per share on the y-axis against time on the x-axis, with the year 1996 represented as 6 and incrementing by one for each subsequent year. Such a plot helps visualize how revenue per share has changed over time and can reveal patterns in the data that may not be immediately obvious from the table alone.
Linear Model
A linear model attempts to summarize the relationship between two variables with a straight line. In algebra, this line is represented by an equation of the form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
Within the exercise, finding the linear model involves using regression analysis to determine the line that best fits the scatter plot data for revenue per share. This linear approximation is useful for making predictions and understanding the general trend of the data.
Quadratic Model
On the other hand, a quadratic model includes an additional squared term, resulting in an equation of the form \(y = ax^2 + bx + c\). This model is capable of capturing non-linear, parabolic trends in data.
For the Amazon revenue data, a quadratic model may provide a better fit than a linear model if the revenue growth is accelerating or decelerating over time rather than increasing at a consistent rate.
Cubic Model
A cubic model incorporates a cubic term to the equation: \(y = ax^3 + bx^2 + cx + d\). Such a model is even more flexible than a quadratic one, allowing for the representation of S-shaped curves, which can reflect more complex relationships between variables.
Applying a cubic model to the revenue data might reveal intricate dynamics in the growth pattern that simpler models cannot capture, such as inflection points where the rate of revenue increase changes direction.
Quartic Model
The quartic model, represented by \(y = ax^4 + bx^3 + cx^2 + dx + e\), introduces a fourth-degree term, making it very versatile for approximating a variety of data patterns.
Although a quartic model has the potential to fit the scatter plot for Amazon's revenue data very closely, it runs the risk of 'overfitting'—where the model is too closely tailored to the sample data, reducing its ability to predict future, unseen trends accurately.
Graphing Utility
A graphing utility is a software tool that assists in plotting data and performing various types of regression analyses to create models. It simplifies the creation of scatter plots and the computation of best-fit lines or curves.
When tackling the exercise, you would use this utility to not only plot the initial data but also to derive the linear, quadratic, cubic, and quartic models, and then to overlay these models onto the scatter plot for comparison and analysis.
Data Fitting
Data fitting is a statistical process used to create a model that approximates the relationship between the independent and dependent variables in a dataset.
Throughout the exercise, data fitting involves using regression methods to construct models (linear, quadratic, cubic, quartic) that best summarize Amazon's revenue data. The goal is to determine which model is most appropriate for the given data by examining how closely the model's predictions align with the actual data points.
Prediction in Algebra
Prediction in algebra involves using mathematical models to estimate future values based on past or present data. By plugging in different values into the model's equation, predictions about the variable of interest can be made.
In the context of the Amazon revenue example, once models are created, algebra is used to predict in which year the revenue per share will reach $37 by solving for \(t\) in each of the regression equations. Differences in the predictions from each model can help to understand the models' strengths and weaknesses.
