Problem 73 - Algebra and Trigonometry

Question

Population Growth The population \(P(\text { in thousands })\) of San Jose, California, from 1988 to 2000 is shown in the table. (Midyear estimates are given.) Draw a rough graph of \(P\) as a function of time \(t\). $$ \begin{array}{|c|c|}\hline t & {P} \\ \hline 1988 & {733} \\ {1990} & {782} \\\ {1992} & {800} \\ {1994} & {817} \\ {1996} & {838} \\ {1998} & {861} \\\ {2000} & {895} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

Plot given coordinates and connect them to sketch a rough increasing graph.

Step 1: Understand the Data Table

The table provides the population (in thousands) for the city of San Jose at various points in time, from 1988 to 2000. Each row corresponds to a specific year (t) and the population (P) in that year. These data points will be used to create a graph.

Step 2: Identify the Graph Components

To create a graph, identify the components: the x-axis will represent the time in years (t) from 1988 to 2000, and the y-axis will represent the population (P) in thousands. Each row from the data table will transform into a coordinate point to be plotted, such as (1988, 733), (1990, 782), etc.

Step 3: Plot Data Points

Plot each data point on the graph using the coordinate pairs derived from the table: (1988, 733), (1990, 782), (1992, 800), (1994, 817), (1996, 838), (1998, 861), and (2000, 895). Mark these points on the respective year on the x-axis and corresponding population on the y-axis.

Step 4: Draw the Rough Graph

Connect the plotted points using straight lines to form a rough graph. This visual representation will help understand the trend or pattern of the population growth over the years. It's a rough sketch, so it's not necessary for the lines to be perfect but enough to illustrate an increasing trend.

Key Concepts

Data Table Analysis

Understanding the given data table is the first step in assessing population growth effectively. In our case, the table presents population data for San Jose from 1988 to 2000.
Here is a breakdown of what each column signifies:

• The first column indicates the year, denoted by the variable \( t \). It acts as the time span over which the population is recorded.
• The second column shows the population \( P \), measured in thousands. This indicates the number of inhabitants at each midyear.

By interpreting this table, you find individual data points that connect time with population size. For instance, in 1988, the population was 733,000.
These data entries will act as foundational blocks for plotting a graph. A clear observation of steady growth across these years can serve as an initial insight into how the graph is expected to behave.

Graph Plotting

To visualize the changes in population over time, graph plotting is essential. Start by setting up the graph axes.

• The x-axis represents the time variable \( t \), marked with years from 1988 to 2000.
• The y-axis represents the population variable \( P \), marked with population counts in thousands.

Each row from your data table, such as (1988, 733), becomes a point on this graph. You plot each point based on where its year falls on the x-axis and its population falls on the y-axis.
Once you have all points plotted, you draw lines connecting them. You may end up with a series of straight lines connecting each year to the next, presenting a rough line graph. The graph helps visually represent trends, making it clear that population grew steadily over these years.

Coordinate Geometry

In graphing, coordinate geometry helps elucidate the placement and relationship between each point. Consider each data entry from the table as a coordinate pair \( (t, P) \).
This pair is plotted on your graph, where:

• \( t \) (the year) represents the x-coordinate.
• \( P \) (the population) represents the y-coordinate.

Understanding coordinate geometry ensures each point is placed correctly in relation to others. For example, the point (1988, 733) means the year 1988 on the x-axis lines up with 733 on the y-axis.
By utilizing these coordinates, a clear pattern emerges, showcasing how each year's population relates to the next. This method anchors the abstract data into a visual form that is easier to interpret and analyze for growth trends.