In which of the following tables is the relationship between the values of x and their corresponding y-values nonlinear?

1. INFER the strategy needed

• A linear relationship has a constant rate of change (slope) between any two points
• A nonlinear relationship has a changing rate of change
• Strategy: Calculate the rate of change between consecutive points in each table and find which one has non-constant rates

2. SIMPLIFY by calculating rates of change for each table

For Choice A (x: 1,2,3,4; y: 8,11,14,17):

• Rate from \((1,8)\) to \((2,11)\): \(\frac{11-8}{2-1} = \frac{3}{1} = 3\)
• Rate from \((2,11)\) to \((3,14)\): \(\frac{14-11}{3-2} = \frac{3}{1} = 3\)
• Rate from \((3,14)\) to \((4,17)\): \(\frac{17-14}{4-3} = \frac{3}{1} = 3\)
• All rates equal 3 → Linear

For Choice B (x: 1,2,3,4; y: 4,8,12,16):

• Rate from \((1,4)\) to \((2,8)\): \(\frac{8-4}{2-1} = \frac{4}{1} = 4\)
• Rate from \((2,8)\) to \((3,12)\): \(\frac{12-8}{3-2} = \frac{4}{1} = 4\)
• Rate from \((3,12)\) to \((4,16)\): \(\frac{16-12}{4-3} = \frac{4}{1} = 4\)
• All rates equal 4 → Linear

For Choice C (x: 1,2,3,4; y: 8,13,18,23):

• Rate from \((1,8)\) to \((2,13)\): \(\frac{13-8}{2-1} = \frac{5}{1} = 5\)
• Rate from \((2,13)\) to \((3,18)\): \(\frac{18-13}{3-2} = \frac{5}{1} = 5\)
• Rate from \((3,18)\) to \((4,23)\): \(\frac{23-18}{4-3} = \frac{5}{1} = 5\)
• All rates equal 5 → Linear

For Choice D (x: 1,2,3,4; y: 6,12,24,48):

• Rate from \((1,6)\) to \((2,12)\): \(\frac{12-6}{2-1} = \frac{6}{1} = 6\)
• Rate from \((2,12)\) to \((3,24)\): \(\frac{24-12}{3-2} = \frac{12}{1} = 12\)
• Rate from \((3,24)\) to \((4,48)\): \(\frac{48-24}{4-3} = \frac{24}{1} = 24\)
• Rates are 6, 12, 24 (not constant) → Nonlinear

3. INFER the final answer

• Only Choice D shows changing rates of change
• The rates double each time (\(6 \rightarrow 12 \rightarrow 24\)), indicating exponential growth

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to check ALL consecutive point pairs to determine linearity. They might calculate only one rate of change per table, or they might look for patterns in the y-values without calculating rates of change at all.

For example, they might notice that in Choice D, the y-values (6, 12, 24, 48) are doubling, but not connect this to calculating actual rates of change. Or they might calculate just one rate and assume the whole table follows that pattern. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when computing the rates of change, particularly with the subtraction in the numerator or forgetting that all x-differences equal 1.

For instance, they might calculate \(\frac{24-12}{3-2}\) as \(24-12 = 12\), but then forget to divide by 1, or they might incorrectly compute 48-24 as 22 instead of 24. These calculation mistakes could lead them to incorrectly identify a linear table as nonlinear, causing them to select Choice A, B, or C instead of the correct answer.

The Bottom Line:

This problem requires systematic checking - students must calculate multiple rates of change and compare them, rather than looking for shortcuts or patterns in just the y-values alone.