Question

which coordinate grid shown points whose coordinates have a proportional relationship between x and y? (four coordinate grids with points and circles for selection are shown, but the images are omitted here as per ocr text processing instructions)

Explanation:

Step1: Recall Proportional Relationship

For a proportional relationship between \(x\) and \(y\), the ratio \(\frac{y}{x}\) (or slope) should be constant for all points, and the line should pass through the origin \((0,0)\). The formula for a proportional relationship is \(y = kx\), where \(k=\frac{y}{x}\) is the constant of proportionality.

Step2: Analyze Top - Left Graph

• For point \((4,4)\): \(\frac{y}{x}=\frac{4}{4} = 1\)
• For point \((8,10)\): \(\frac{y}{x}=\frac{10}{8}=\frac{5}{4}=1.25\)
• The ratios are not equal, so this graph does not show a proportional relationship.

Step3: Analyze Top - Right Graph

• For point \((2,4)\): \(\frac{y}{x}=\frac{4}{2}=2\)
• For point \((6,6)\): \(\frac{y}{x}=\frac{6}{6} = 1\)
• The ratios are not equal, so this graph does not show a proportional relationship.

Step4: Analyze Bottom - Left Graph

• For point \((4,6)\): \(\frac{y}{x}=\frac{6}{4}=\frac{3}{2}=1.5\)
• For point \((8,12)\): \(\frac{y}{x}=\frac{12}{8}=\frac{3}{2}=1.5\)
• For point \((12,18)\): \(\frac{y}{x}=\frac{18}{12}=\frac{3}{2}=1.5\)
• All ratios are equal (\(k = 1.5\)) and the line passes through the origin (implicitly, since when \(x = 0\), \(y=0\) in the proportional relationship), so this graph shows a proportional relationship.

Step5: Analyze Bottom - Right Graph

• For point \((6,4)\): \(\frac{y}{x}=\frac{4}{6}=\frac{2}{3}\approx0.67\)
• For point \((10,6)\): \(\frac{y}{x}=\frac{6}{10}=\frac{3}{5}=0.6\)
• The ratios are not equal, so this graph does not show a proportional relationship.

Answer:

The bottom - left coordinate grid (with points \((4,6)\), \((8,12)\), \((12,18)\))