Let's break the problem into smaller steps and solve it accordingly.
### Step 1: Understanding the Data
We are given pairs [tex]\((x, y)\)[/tex]:
- [tex]\( (7, 41) \)[/tex]
- [tex]\( (8, 45) \)[/tex]
- [tex]\( (9, 49) \)[/tex]
- [tex]\( (10, 52) \)[/tex]
- [tex]\( (10, 53) \)[/tex]
- [tex]\( (11, 55) \)[/tex]
- [tex]\( (12, 59) \)[/tex]
In these pairs:
- [tex]\(x\)[/tex] represents the age of the boys (in years).
- [tex]\(y\)[/tex] represents their corresponding height (in inches).
### Step 2: Filling the Table
Based on the given data, we fill the table as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
7 & 41 \\
\hline
8 & 45 \\
\hline
9 & 49 \\
\hline
10 & 52 \\
\hline
10 & 53 \\
\hline
11 & 55 \\
\hline
12 & 59 \\
\hline
\end{array}
\][/tex]
### Step 3: Plotting the Points
To create an accurate and informative graph:
1. Axes: Define the axes. The horizontal axis will represent the age ([tex]\(x\)[/tex]), and the vertical axis will represent the height ([tex]\(y\)[/tex]).
2. Labeling: Clearly label both axes (e.g., "Age (years)" for the x-axis and "Height (inches)" for the y-axis).
3. Plot Points: Plot each pair as a coordinate point on the graph.
4. Title: Provide a title for the graph, such as "Age vs. Height of Boys".
Here’s how you can plot them manually:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
7 & 41 \\
8 & 45 \\
9 & 49 \\
10 & 52 \\
10 & 53 \\
11 & 55 \\
12 & 59 \\
\hline
\end{array}
\][/tex]
Using the data points:
- (7, 41)
- (8, 45)
- (9, 49)
- (10, 52)
- (10, 53)
- (11, 55)
- (12, 59)
These points are plotted on a Cartesian plane where the x-axis ranges from 7 to 12 (or a bit more for better spacing), and the y-axis ranges from 40 to 60 (with appropriate scaling).
### Step 4: Interpretation of the Table
What does [tex]\(x\)[/tex] represent?
- [tex]\(x\)[/tex] represents the age of boys in years.
What does [tex]\(y\)[/tex] represent?
- [tex]\(y\)[/tex] represents the height of boys in inches.
### Step 5: Final Output Plot
Manually plotting these points and drawing a graph provides a visual understanding of how the height changes with age in the given dataset. Each point on the graph corresponds to a pair from the data set, clearly showing trends and any points that might be outliers or indicate unusual growth patterns.
Remember, in practice, tools like graph paper, or software (e.g., spreadsheet or graphing tools), would be used to make the plotting precise and easy.
