A linear function g is defined by \(\mathrm{g(x) = mx + c}\), where m and c are constants. In the...
GMAT Advanced Math : (Adv_Math) Questions
A linear function g is defined by \(\mathrm{g(x) = mx + c}\), where m and c are constants. In the xy-plane, the y-intercept of the graph of \(\mathrm{y = g(x)}\) is 4 more than twice the slope of the line. The graph of the function \(\mathrm{h(x) = g(x) - 12}\) has an x-intercept at \(\mathrm{(2, 0)}\). What is the value of c?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{g(x) = mx + c}\) (linear function with slope m and y-intercept c)
- "y-intercept is 4 more than twice the slope" → \(\mathrm{c = 2m + 4}\)
- \(\mathrm{h(x) = g(x) - 12}\) has x-intercept at \(\mathrm{(2, 0)}\)
2. INFER what the x-intercept condition tells us
- If h(x) has x-intercept at \(\mathrm{(2, 0)}\), then \(\mathrm{h(2) = 0}\)
- Since \(\mathrm{h(x) = g(x) - 12}\), we have: \(\mathrm{h(2) = g(2) - 12 = 0}\)
- Therefore: \(\mathrm{g(2) = 12}\)
3. TRANSLATE this into a second equation
- Using \(\mathrm{g(x) = mx + c}\): \(\mathrm{g(2) = 2m + c = 12}\)
- Now we have two equations:
- \(\mathrm{c = 2m + 4}\)
- \(\mathrm{2m + c = 12}\)
4. SIMPLIFY by solving the system
- Substitute the first equation into the second:
\(\mathrm{2m + (2m + 4) = 12}\)
\(\mathrm{4m + 4 = 12}\)
\(\mathrm{4m = 8}\)
\(\mathrm{m = 2}\) - Find c: \(\mathrm{c = 2(2) + 4 = 8}\)
Answer: C) 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students struggle to convert "4 more than twice the slope" into the equation \(\mathrm{c = 2m + 4}\), often writing it backwards as \(\mathrm{m = 2c + 4}\) or getting confused about which variable represents which quantity.
This leads to an incorrect first equation, making their entire system wrong and leading them to select an incorrect answer or get stuck and guess.
Second Most Common Error:
Poor INFER reasoning: Students don't recognize that "x-intercept at \(\mathrm{(2, 0)}\)" means \(\mathrm{h(2) = 0}\), or they forget that \(\mathrm{h(x) = g(x) - 12}\), so they can't establish the connection that gives them \(\mathrm{g(2) = 12}\).
Without this key insight, they can't set up the second equation needed to solve for both m and c, leading to confusion and guessing.
The Bottom Line:
This problem requires careful translation of verbal statements into mathematical equations, then recognizing how the x-intercept condition creates a second constraint. Success depends on systematic setup more than complex calculations.