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Regression analysis is a powerful tool used to uncover relationships between variables. It helps us understand how one variable affects another. For example, in our exercise, we used regression to determine how the hour of the day affects the number of vehicles parked.
There are different types of regression, such as linear and non-linear, but here we're focusing on quadratic regression. This is where the relationship isn't just a straight line, but instead follows a curve. This type of regression is perfect for our data as vehicle parking doesn't linearly increase or decrease with time, it changes in a curved pattern.
By using a quadratic equation of the form \(y = ax^2 + bx + c\), we find the best-fit curve that represents our data points scattered on a graph. Software or a calculator can quickly compute the values of \(a\), \(b\), and \(c\) to best fit our data.

A quadratic model is a type of mathematical model that takes the form of a quadratic equation. The generic form is \(y = ax^2 + bx + c\), where:

• \(y\) is the dependent variable (e.g., vehicles parked).
• \(x\) is the independent variable (here, time in hours).
• \(a\), \(b\), and \(c\) are constants that determine the shape and position of the parabola.

In our problem, this model helps us see how parking changes over time. We found it using regression analysis. For our dataset, the quadratic model obtained, \( y = -0.135x^2 + 1.092x + 6.200 \), fits the trend of our data points.
This model shows that as time progresses from morning to evening, the number of parked vehicles first increases to a peak and then decreases, forming a parabolic curve.

Functional notation is a way of writing expressions that efficiently describe relationships in a compact form. It typically uses \(f(x)\) to imply that \(y\) is a function of \(x\).
For instance, when the task required us to calculate the number of vehicles at 2 P.M., we translated that into a function: \(y = f(x)\). In mathematical terms, this could be shown as \(f(x) = -0.135x^2 + 1.092x + 6.200\) where \(x\) represents the time of day.
Using functional notation makes it straightforward to substitute specific \(x\) values to find corresponding \(y\) values, like plugging in the time 2 P.M. (or \(x = 5\)) to estimate the number of vehicles.

Data modeling involves constructing a model that represents data and its relationships. In our exercise, we used a quadratic model to represent how the number of parked vehicles varied with time throughout the day.
Data modeling is crucial because it transforms raw data into a meaningful understanding of real-world phenomena. We began by collecting data at various times, interpreted as vehicle counts, and then applied a suitable model.
This approach helps in making informed predictions and identifying patterns. By determining the vertex of the quadratic and evaluating the model at specific times, like estimating the parking at 2 P.M. or determining the peak time, data modeling aids in planning and decision-making processes.