Question
The two tables of data represent relationships between x and y. Graph the relationship that is proportional.
x y
1 2.5
3 7.5
8 20
x y
1 3
2 10
4 24
x y
1 2.5
3 7.5
8 20
x y
1 3
2 10
4 24
Answers
GPT-4o mini
To determine which of the two sets of data represents a proportional relationship, we need to check whether the ratio of \( y \) to \( x \) is constant.
### First Table:
Let's examine the first table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2.5 \\
3 & 7.5 \\
8 & 20 \\
\hline
\end{array}
\]
Calculating the ratio \( \frac{y}{x} \):
- For \( x = 1 \), \( \frac{y}{x} = \frac{2.5}{1} = 2.5 \)
- For \( x = 3 \), \( \frac{y}{x} = \frac{7.5}{3} = 2.5 \)
- For \( x = 8 \), \( \frac{y}{x} = \frac{20}{8} = 2.5 \)
Since the ratio \( \frac{y}{x} = 2.5 \) is constant for all points, the first table represents a proportional relationship.
### Second Table:
Now let's look at the second table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
2 & 10 \\
4 & 24 \\
\hline
\end{array}
\]
Calculating the ratio \( \frac{y}{x} \):
- For \( x = 1 \), \( \frac{y}{x} = \frac{3}{1} = 3 \)
- For \( x = 2 \), \( \frac{y}{x} = \frac{10}{2} = 5 \)
- For \( x = 4 \), \( \frac{y}{x} = \frac{24}{4} = 6 \)
Since the ratios are different, \( 3 \), \( 5 \), and \( 6 \), this table does not represent a proportional relationship.
### Conclusion:
The first table represents a proportional relationship given that \( \frac{y}{x} \) remains constant at \( 2.5 \).
### Graph of the First Table:
To graph the relationship from the first table, you would plot the points:
- \( (1, 2.5) \)
- \( (3, 7.5) \)
- \( (8, 20) \)
The line through these points will have a slope equal to the constant ratio \( 2.5 \), indicating a proportional relationship.
When you plot it, you'll see a straight line that passes through the origin (0,0), confirming it is a proportional relationship.
### First Table:
Let's examine the first table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2.5 \\
3 & 7.5 \\
8 & 20 \\
\hline
\end{array}
\]
Calculating the ratio \( \frac{y}{x} \):
- For \( x = 1 \), \( \frac{y}{x} = \frac{2.5}{1} = 2.5 \)
- For \( x = 3 \), \( \frac{y}{x} = \frac{7.5}{3} = 2.5 \)
- For \( x = 8 \), \( \frac{y}{x} = \frac{20}{8} = 2.5 \)
Since the ratio \( \frac{y}{x} = 2.5 \) is constant for all points, the first table represents a proportional relationship.
### Second Table:
Now let's look at the second table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
2 & 10 \\
4 & 24 \\
\hline
\end{array}
\]
Calculating the ratio \( \frac{y}{x} \):
- For \( x = 1 \), \( \frac{y}{x} = \frac{3}{1} = 3 \)
- For \( x = 2 \), \( \frac{y}{x} = \frac{10}{2} = 5 \)
- For \( x = 4 \), \( \frac{y}{x} = \frac{24}{4} = 6 \)
Since the ratios are different, \( 3 \), \( 5 \), and \( 6 \), this table does not represent a proportional relationship.
### Conclusion:
The first table represents a proportional relationship given that \( \frac{y}{x} \) remains constant at \( 2.5 \).
### Graph of the First Table:
To graph the relationship from the first table, you would plot the points:
- \( (1, 2.5) \)
- \( (3, 7.5) \)
- \( (8, 20) \)
The line through these points will have a slope equal to the constant ratio \( 2.5 \), indicating a proportional relationship.
When you plot it, you'll see a straight line that passes through the origin (0,0), confirming it is a proportional relationship.
what spots should I plot
GPT-4o mini
To plot the points from the first table that represents a proportional relationship, you'll use the following coordinates:
1. \( (1, 2.5) \)
2. \( (3, 7.5) \)
3. \( (8, 20) \)
### Steps to Plot:
1. **Set Up Your Axes:**
- Draw a horizontal axis (x-axis) and a vertical axis (y-axis).
- Label your x-axis with values (consider setting up at least from \( 0 \) to \( 10 \) or higher, as needed).
- Label your y-axis with values (consider ranging from \( 0 \) to \( 25 \) or higher to accommodate the y-values).
2. **Plot the Points:**
- **For \( (1, 2.5) \):**
- Move to \( x = 1 \) on the x-axis, then go up to \( y = 2.5 \) (this is halfway between 2 and 3).
- Mark this point.
- **For \( (3, 7.5) \):**
- Move to \( x = 3 \) on the x-axis, then go up to \( y = 7.5 \) (this is halfway between 7 and 8).
- Mark this point.
- **For \( (8, 20) \):**
- Move to \( x = 8 \) on the x-axis, then go up to \( y = 20 \).
- Mark this point.
3. **Draw the Line:**
- After plotting the points, draw a straight line through them. Since this is a proportional relationship, the line should pass through the origin (0, 0) and extend through the plotted points.
### Summary of Points to Plot:
- \( (1, 2.5) \)
- \( (3, 7.5) \)
- \( (8, 20) \)
This will illustrate the linear relationship between \( x \) and \( y \) described in your first table.
1. \( (1, 2.5) \)
2. \( (3, 7.5) \)
3. \( (8, 20) \)
### Steps to Plot:
1. **Set Up Your Axes:**
- Draw a horizontal axis (x-axis) and a vertical axis (y-axis).
- Label your x-axis with values (consider setting up at least from \( 0 \) to \( 10 \) or higher, as needed).
- Label your y-axis with values (consider ranging from \( 0 \) to \( 25 \) or higher to accommodate the y-values).
2. **Plot the Points:**
- **For \( (1, 2.5) \):**
- Move to \( x = 1 \) on the x-axis, then go up to \( y = 2.5 \) (this is halfway between 2 and 3).
- Mark this point.
- **For \( (3, 7.5) \):**
- Move to \( x = 3 \) on the x-axis, then go up to \( y = 7.5 \) (this is halfway between 7 and 8).
- Mark this point.
- **For \( (8, 20) \):**
- Move to \( x = 8 \) on the x-axis, then go up to \( y = 20 \).
- Mark this point.
3. **Draw the Line:**
- After plotting the points, draw a straight line through them. Since this is a proportional relationship, the line should pass through the origin (0, 0) and extend through the plotted points.
### Summary of Points to Plot:
- \( (1, 2.5) \)
- \( (3, 7.5) \)
- \( (8, 20) \)
This will illustrate the linear relationship between \( x \) and \( y \) described in your first table.