Use the drawing tool(s) to form the correct answer on the provided graph.

Question

The two tables of values represent linear relationships between x and y.

x y
15 30
25 40
35 50
x y
20 10
30 15
40 20

One of the table of values represents a proportional relationship. Graph the linear relationship represented by that table.

Answer 1

To determine which table of values represents a proportional relationship, we need to check if the ratio of \(y\) to \(x\) is constant for each pair of \((x, y)\) values.

For the first table:

• For \( (15, 30) \): \( \frac{y}{x} = \frac{30}{15} = 2 \)
• For \( (25, 40) \): \( \frac{y}{x} = \frac{40}{25} = 1.6 \)
• For \( (35, 50) \): \( \frac{y}{x} = \frac{50}{35} \approx 1.43 \)

Since the ratios are not constant, the first table does not represent a proportional relationship.

For the second table:

• For \( (20, 10) \): \( \frac{y}{x} = \frac{10}{20} = 0.5 \)
• For \( (30, 15) \): \( \frac{y}{x} = \frac{15}{30} = 0.5 \)
• For \( (40, 20) \): \( \frac{y}{x} = \frac{20}{40} = 0.5 \)

Since the ratios are the same (0.5), the second table represents a proportional relationship.

Now, let's graph the linear relationship represented by the second table of values. The points to plot from the second table are:

• \( (20, 10) \)
• \( (30, 15) \)
• \( (40, 20) \)

Steps to Graph:

1. On the x-axis, plot the points: \(20\), \(30\), and \(40\).
2. Correspondingly, on the y-axis, plot the points: \(10\), \(15\), and \(20\).
3. For each point:
   • At \(x = 20\), go up to \(y = 10\).
   • At \(x = 30\), go up to \(y = 15\).
   • At \(x = 40\), go up to \(y = 20\).
4. Draw a line through the points, extending it across the graph.

The slope of the line will indicate the constant rate of change, showing that it passes through the origin and the points lie on the line \(y = 0.5x\), confirming the proportional relationship.

If you need an actual graph illustration, please provide a tool or a medium to execute the drawing as I am currently unable to create visual content.

Answer 2

what points on the graph should I mark?

Answer 3

For the second table of values, which represents the proportional relationship between \(x\) and \(y\), you should mark the following points on the graph:

1. Point A: \( (20, 10) \)
2. Point B: \( (30, 15) \)
3. Point C: \( (40, 20) \)

Summary of Points to Plot:

• (20, 10): Start by marking the point where \(x = 20\) and \(y = 10\).
• (30, 15): Next, mark the point where \(x = 30\) and \(y = 15\).
• (40, 20): Finally, mark the point where \(x = 40\) and \(y = 20\).

Steps for Graphing:

1. Locate \(x = 20\) on the x-axis, move up to \(y = 10\) and mark that point.
2. Locate \(x = 30\) on the x-axis, move up to \(y = 15\) and mark that point.
3. Locate \(x = 40\) on the x-axis, move up to \(y = 20\) and mark that point.
4. Draw a straight line through all three points, which represents the linear relationship \(y = 0.5x\).

These points illustrate the linear relationship that is proportional, confirming that as \(x\) increases, \(y\) increases at a consistent rate.

Answer 4

do I start the line at (0, 0) or (20, 10)?

Answer 5

Since the relationship represented by the second table of values is a proportional relationship, you would start the line at the origin \((0, 0)\) and extend it through the plotted points.

Here’s the reasoning:

In a proportional relationship, the line goes through the origin because both \(x\) and \(y\) are zero when there is no value present. Therefore, for the equation of the line representing the relationship \(y = 0.5x\), it not only passes through the points \((20, 10)\), \((30, 15)\), and \((40, 20)\), but also through the point \((0, 0)\).

Steps for Drawing Your Line:

1. Start at the origin \( (0, 0) \).
2. Then draw a straight line through the points \( (20, 10) \), \( (30, 15) \), and \( (40, 20) \).
3. Extend the line in both directions to show that it represents the complete linear relationship.

By doing this, you visually indicate that all points along the line maintain the property of proportionality between \(x\) and \(y\).