x\(\mathrm{h(x)}\)01.2321.5441.94The table shows the exponential relationship between the number of years, x, since Hana started training in pole vaul...

1. INFER the function structure

• Given information:
  • Table shows exponential relationship
  • Three data points: \(\mathrm{(0, 1.23), (2, 1.54), (4, 1.94)}\)
• What this tells us: We need \(\mathrm{h(x) = Ca^x}\) where C is initial value and a is growth factor

2. INFER which point gives us the easiest start

• The point \(\mathrm{(0, 1.23)}\) directly gives us C since any number to the 0 power equals 1
• Strategy: Use \(\mathrm{x = 0}\) first, then use another point to find a

3. SIMPLIFY to find the initial value C

• When \(\mathrm{x = 0}\): \(\mathrm{h(0) = Ca^0 = C(1) = C}\)
• From table: \(\mathrm{h(0) = 1.23}\)
• Therefore: \(\mathrm{C = 1.23}\)
• Our function is now \(\mathrm{h(x) = 1.23a^x}\)

4. SIMPLIFY to find the base a

• When \(\mathrm{x = 2}\): \(\mathrm{h(2) = 1.23a^2 = 1.54}\)
• Divide both sides by 1.23: \(\mathrm{a^2 = 1.54/1.23 ≈ 1.252}\) (use calculator)
• Take square root: \(\mathrm{a = \sqrt{1.252} ≈ 1.12}\) (use calculator)

5. INFER the final function

• Substituting back: \(\mathrm{h(x) = 1.23(1.12)^x}\)
• This matches choice D

Answer: D. \(\mathrm{h(x) = 1.23(1.12)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{x = 0}\) gives the easiest path to finding C, or they don't understand that \(\mathrm{Ca^0 = C}\) because \(\mathrm{a^0 = 1}\).

Instead, they might try to use the other data points first, leading to more complex systems of equations. This confusion about strategy often causes them to abandon systematic solution and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when computing \(\mathrm{1.54/1.23}\) or finding \(\mathrm{\sqrt{1.252}}\), leading to incorrect values for a.

For example, if they incorrectly calculate \(\mathrm{a = 0.12}\) instead of \(\mathrm{1.12}\), this may lead them to select Choice C (\(\mathrm{h(x) = 1.23(0.12)^x}\)).

The Bottom Line:

This problem tests whether students can systematically extract parameters from data points using the structure of exponential functions, rather than just recognizing exponential patterns.