Question

1. table analysis

determine whether the relationship in the table is proportional.

\\(\

$$\begin{array}{cc} x & y \\\\ 3 & 12 \\\\ 6 & 24 \\\\ 9 & 36 \\\\ 12 & 48 \\end{array}$$

\\)

is the relationship proportional? yes

explain your reasoning:

Explanation:

Identify the given data points

We are given a table of values representing a relationship between \(x\) and \(y\):

• \((3, 12)\)
• \((6, 24)\)
• \((9, 36)\)
• \((12, 48)\)

Define proportional relationships

Using the Linear Modeling knowledge point, a relationship is proportional if the ratio of \(y\) to \(x\) is constant for all pairs:

$$ \frac{y}{x} = k $$

where \(k\) is the constant of proportionality.

Calculate the ratios for each pair

We compute the ratio \(\frac{y}{x}\) for each data point:

• For \((3, 12)\): \(\frac{12}{3} = 4\)
• For \((6, 24)\): \(\frac{24}{6} = 4\)
• For \((9, 36)\): \(\frac{36}{9} = 4\)
• For \((12, 48)\): \(\frac{48}{12} = 4\)

Determine proportionality

Since the ratio \(\frac{y}{x}\) is constantly \(4\) for every pair, the relationship is proportional.

Answer:

Determine whether the relationship in the table is proportional.

Is the relationship proportional? <blank>Yes</blank>

Explain your reasoning: <blank>The ratio of \(y\) to \(x\) is constant for all pairs in the table, where \(\frac{y}{x} = 4\).</blank>