The function h gives the sum of a number z and the square of the quantity 7 less than z....
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{h}\) gives the sum of a number \(\mathrm{z}\) and the square of the quantity 7 less than \(\mathrm{z}\). Which of the following equations defines the function \(\mathrm{h}\)?
1. TRANSLATE the problem information
- TRANSLATE each phrase carefully:
- "a number z" becomes \(\mathrm{z}\)
- "7 less than z" becomes \(\mathrm{z - 7}\) (not \(\mathrm{7 - z}\)!)
- "the square of the quantity 7 less than z" becomes \(\mathrm{(z - 7)^2}\)
- "the sum of..." means we add: \(\mathrm{z + (z - 7)^2}\)
2. SIMPLIFY by expanding the binomial
- Expand \(\mathrm{(z - 7)^2}\) using the formula \(\mathrm{(a - b)^2 = a^2 - 2ab + b^2}\):
\(\mathrm{(z - 7)^2 = z^2 - 2(z)(7) + 7^2 = z^2 - 14z + 49}\)
- So our function becomes: \(\mathrm{h(z) = z + (z^2 - 14z + 49)}\)
3. SIMPLIFY by combining like terms
- Distribute and collect terms:
\(\mathrm{h(z) = z^2 + z - 14z + 49}\)
\(\mathrm{h(z) = z^2 - 13z + 49}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Misinterpreting "7 less than z" as \(\mathrm{7 - z}\) instead of \(\mathrm{z - 7}\)
Students often think "7 less than z" means "7 minus z" because they focus on the word "less" meaning subtraction. However, "7 less than z" actually means "z decreased by 7" or \(\mathrm{z - 7}\). This fundamental translation error leads to \(\mathrm{(7 - z)^2}\) instead of \(\mathrm{(z - 7)^2}\).
This may lead them to select Choice A (\(\mathrm{h(z) = z^2 - 14z + 49}\)) after incorrectly expanding and combining terms.
Second Most Common Error:
Inadequate SIMPLIFY execution: Correctly translating but making errors when expanding \(\mathrm{(z - 7)^2}\)
Students might expand \(\mathrm{(z - 7)^2}\) incorrectly, perhaps forgetting the middle term or getting the signs wrong. Some might get \(\mathrm{z^2 + 14z + 49}\) instead of \(\mathrm{z^2 - 14z + 49}\).
This leads to confusion and incorrect final expressions that don't match any of the given choices, causing them to guess.
The Bottom Line:
The phrase "less than" in mathematics requires careful attention to order - "7 less than z" means \(\mathrm{z - 7}\), not \(\mathrm{7 - z}\). This translation skill is fundamental to converting word problems into algebraic expressions.