The function f is defined by \(\mathrm{f(x) = 7x^3}\). In the xy-plane, the graph of \(\mathrm{y = g(x)}\) is the...

1. TRANSLATE the problem information

• Given information:
  • Original function: \(\mathrm{f(x) = 7x^3}\)
  • New function \(\mathrm{g(x)}\) is created by shifting \(\mathrm{f(x)}\) down 2 units
• What we need to find: The equation that defines \(\mathrm{g(x)}\)

2. INFER the transformation rule

• Vertical shifts follow a specific pattern:
  • Shifting UP k units: add k → \(\mathrm{g(x) = f(x) + k}\)
  • Shifting DOWN k units: subtract k → \(\mathrm{g(x) = f(x) - k}\)
• Since we're shifting DOWN 2 units, we need to subtract 2

3. Apply the transformation

• Start with the shift formula: \(\mathrm{g(x) = f(x) - 2}\)
• Substitute the given function: \(\mathrm{g(x) = 7x^3 - 2}\)

Answer: D. \(\mathrm{g(x) = 7x^3 - 2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the direction of vertical shifts and think "down 2 units" means adding 2 to the function.

They might reason: "Moving the graph down means making the y-values bigger, so I need to add 2." This backwards thinking about how shifts work leads them to write \(\mathrm{g(x) = 7x^3 + 2}\).

This may lead them to select Choice C (\(\mathrm{g(x) = 7x^3 + 2}\)).

Second Most Common Error:

Missing conceptual knowledge: Students don't understand how vertical transformations work and think shifts might affect the coefficient or the variable somehow.

They might think "shifting changes the steepness" or get confused about which part of the function to modify, leading to random guessing between the available choices.

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

The Bottom Line:

The key insight is that vertical shifts are surprisingly simple - they only affect the constant term. Moving a graph up or down doesn't change the shape or steepness, just the vertical position, which translates to adding or subtracting a constant from the entire function.