xy001128327The table shown includes some values of x and their corresponding values of y. Which of the following graphs in...

1. TRANSLATE the table data into coordinate points

• Given information:
  • Table shows x-values: 0, 1, 2, 3
  • Corresponding y-values: 0, 1, 8, 27
  • These create coordinate points: \(\mathrm{(0, 0)}\), \(\mathrm{(1, 1)}\), \(\mathrm{(2, 8)}\), \(\mathrm{(3, 27)}\)

2. INFER the mathematical relationship

• Looking for a pattern in the y-values:
  • \(\mathrm{y = 0}\) when \(\mathrm{x = 0}\) → This could be \(\mathrm{0^3}\)
  • \(\mathrm{y = 1}\) when \(\mathrm{x = 1}\) → This could be \(\mathrm{1^3}\)
  • \(\mathrm{y = 8}\) when \(\mathrm{x = 2}\) → This could be \(\mathrm{2^3}\)
  • \(\mathrm{y = 27}\) when \(\mathrm{x = 3}\) → This could be \(\mathrm{3^3}\)

• The relationship is \(\mathrm{y = x^3}\) (a cubic function)

3. VISUALIZE which graph matches this function

• A cubic function \(\mathrm{y = x^3}\) should:
  • Pass through the origin \(\mathrm{(0, 0)}\)
  • Pass through \(\mathrm{(1, 1)}\)
  • Pass through \(\mathrm{(2, 8)}\)
  • Pass through \(\mathrm{(3, 27)}\)
  • Be increasing for all positive x-values
  • Be concave up for positive x-values

4. APPLY CONSTRAINTS to eliminate incorrect choices

• The solution states that only Graph B passes through the point \(\mathrm{(1, 1)}\)
• Graphs A, C, and D do not pass through \(\mathrm{(1, 1)}\)
• Since \(\mathrm{(1, 1)}\) is a required point for \(\mathrm{y = x^3}\), only Graph B can be correct

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the cubic pattern \(\mathrm{y = x^3}\) from the data points. They might see that y increases rapidly but fail to identify the specific \(\mathrm{x^3}\) relationship. Without recognizing this pattern, they cannot effectively eliminate incorrect graph choices and may end up guessing between graphs that "look increasing."

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor VISUALIZE execution: Even if students recognize \(\mathrm{y = x^3}\), they may not carefully check whether each graph passes through the critical point \(\mathrm{(1, 1)}\). Students might focus on general shape rather than specific coordinate points, leading them to select a graph that appears cubic but doesn't actually contain the required points.

This may lead them to select Choice A, C, or D based on general appearance rather than point verification.

The Bottom Line:

Success requires both pattern recognition skills (seeing \(\mathrm{y = x^3}\) in the data) and careful graph analysis (checking that chosen graphs pass through specific coordinate points). Many students struggle with one or both of these requirements.