A function f is represented by the following table of values.xy-11/2011224Which of the following graphs in the xy-plane could represent...

1. TRANSLATE the table information

• Given information:
  • Table shows points \((-1, \frac{1}{2})\), \((0, 1)\), \((1, 2)\), \((2, 4)\)
  • Need to identify which graph description matches this function

2. INFER the pattern in the data

• Look at how y-values change as x increases:
  • From \(\frac{1}{2}\) to 1: multiply by 2
  • From 1 to 2: multiply by 2
  • From 2 to 4: multiply by 2
• Constant ratio of 2 between consecutive terms indicates exponential growth
• This means \(\mathrm{f(x) = 2^x}\)

3. INFER the properties of \(\mathrm{y = 2^x}\)

• Since base \(\mathrm{2 \gt 1}\), the function is increasing
• Exponential functions are always concave up
• When \(\mathrm{x = 0}\), \(\mathrm{y = 2^0 = 1}\), so passes through \((0, 1)\)
• As x approaches negative infinity, \(\mathrm{2^x}\) approaches 0 (horizontal asymptote)

4. TRANSLATE each answer choice and match properties

• (A) Decreasing function - doesn't match our increasing function
• (B) Linear through origin - doesn't match exponential through \((0, 1)\)
• (C) U-shaped parabola - doesn't match exponential curve
• (D) Increasing, concave up, through \((0, 1)\), approaches x-axis as x decreases - matches all properties! ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students notice the y-values are increasing but fail to recognize the constant ratio pattern that identifies exponential growth. Instead, they might think it's linear growth because the differences between x-values are constant.

Without recognizing \(\mathrm{f(x) = 2^x}\), they can't match the exponential properties to the graph descriptions. This often leads to confusion about which characteristics are important, causing them to select Choice (B) because they see the function is increasing, or to abandon systematic analysis and guess.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students correctly identify \(\mathrm{f(x) = 2^x}\) but struggle to convert the verbal graph descriptions into mathematical properties they can verify. They might not realize that "approaches the x-axis as x decreases" means horizontal asymptote at \(\mathrm{y = 0}\).

This confusion about graph terminology may lead them to select Choice (C) if they focus only on the "increasing" property, or causes them to get stuck and guess randomly.

The Bottom Line:

Success requires recognizing exponential patterns from numerical data AND translating between mathematical properties and verbal graph descriptions. Students who master pattern recognition but struggle with graph interpretation (or vice versa) will find this problem challenging.