The table shown includes some values of x and their corresponding values of y.xy001244168Which of the following graphs in the...

1. TRANSLATE the problem information

• Given information:
  • Data points: (0, 0), (1, 2), (4, 4), (16, 8)
  • Need to identify which graph represents this relationship

2. INFER the mathematical relationship

• Look for patterns in the x-values: 0, 1, 4, 16
• These are perfect squares: \(0^2, 1^2, 2^2, 4^2\)
• Test the relationship \(\mathrm{y = 2\sqrt{x}}\):
  • At \(\mathrm{x = 0}\): \(\mathrm{y = 2\sqrt{0} = 0}\) ✓
  • At \(\mathrm{x = 1}\): \(\mathrm{y = 2\sqrt{1} = 2}\) ✓
  • At \(\mathrm{x = 4}\): \(\mathrm{y = 2\sqrt{4} = 4}\) ✓
  • At \(\mathrm{x = 16}\): \(\mathrm{y = 2\sqrt{16} = 8}\) ✓
• The relationship is \(\mathrm{y = 2\sqrt{x}}\)

3. INFER the function's behavior

• Increasing or decreasing? As x increases, \(\sqrt{\mathrm{x}}\) increases, so \(2\sqrt{\mathrm{x}}\) increases
• Concavity analysis: Examine rate of change between consecutive points:
  • From \(\mathrm{x = 0}\) to \(\mathrm{x = 1}\): y increases 2 units over 1 unit of x
  • From \(\mathrm{x = 1}\) to \(\mathrm{x = 4}\): y increases 2 units over 3 units of x
  • From \(\mathrm{x = 4}\) to \(\mathrm{x = 16}\): y increases 4 units over 12 units of x
• The rate of increase is getting smaller as x gets larger - this means concave down

4. APPLY CONSTRAINTS to match graph characteristics

• Need: increasing function that is concave down
• Choice A: Straight line (constant rate) - doesn't match
• Choice B: Increasing and concave up - wrong concavity
• Choice C: Increasing and concave down - perfect match
• Choice D: Decreasing - wrong direction

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize the pattern in x-values or don't test the square root relationship systematically.

Without identifying \(\mathrm{y = 2\sqrt{x}}\), students might try to guess based on the general increasing trend alone, leading them to select Choice A (straight line) because it's the simplest increasing function they can think of.

Second Most Common Error:

Conceptual confusion about concavity: Students correctly identify \(\mathrm{y = 2\sqrt{x}}\) and that it's increasing, but confuse concave up with concave down.

They might think "the function is growing faster" means concave up, when actually the square root function grows more slowly as x increases (concave down). This leads them to select Choice B (increasing and concave up).

The Bottom Line:

This problem requires recognizing a non-linear pattern from discrete data points and then correctly analyzing the geometric properties of that function. The key insight is that decreasing rate of change corresponds to concave down behavior.