The table shows values of x and their corresponding values y for a linear function. What is the y-intercept of...
GMAT Algebra : (Alg) Questions
The table shows values of \(\mathrm{x}\) and their corresponding values \(\mathrm{y}\) for a linear function. What is the y-intercept of the graph in the xy-plane?
| \(\mathrm{x}\) | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| \(\mathrm{y}\) | 2 | 4 | 6 | 8 |
1. INFER the solution strategy
- To find the y-intercept, we need the equation of the line
- Since we have a table of values for a linear function, we can find the slope and then use point-slope form
2. SIMPLIFY to find the slope
- Using any two points from the table, let's use \((2, 2)\) and \((3, 4)\):
- Slope = \(\frac{4-2}{3-2}\) = \(\frac{2}{1}\) = \(2\)
- Verify with another pair \((3, 4)\) and \((4, 6)\): slope = \(\frac{6-4}{4-3}\) = \(\frac{2}{1}\) = \(2\) ✓
3. SIMPLIFY to find the equation
- Using point-slope form with point \((2, 2)\) and slope \(\mathrm{m = 2}\):
- \(\mathrm{y - 2 = 2(x - 2)}\)
- \(\mathrm{y - 2 = 2x - 4}\)
- \(\mathrm{y = 2x - 2}\)
4. INFER how to find the y-intercept
- The y-intercept occurs when \(\mathrm{x = 0}\)
- Substitute \(\mathrm{x = 0}\) into our equation: \(\mathrm{y = 2(0) - 2 = -2}\)
- Therefore, the y-intercept is \((0, -2)\)
Answer: A. (0, -2)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students try to find patterns in the y-values without recognizing they need to establish the linear equation first.
They might notice the y-values increase by 2 each time and incorrectly assume this means the y-intercept should be positive, or they might try to extend the pattern backwards without using proper linear function methods. This leads to confusion and guessing among the positive y-intercept choices.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly identify the need to find the equation but make algebraic errors.
Common mistakes include: expanding \(\mathrm{y - 2 = 2(x - 2)}\) incorrectly (getting \(\mathrm{y = 2x + 2}\) instead of \(\mathrm{y = 2x - 2}\)), or sign errors when distributing. This may lead them to select Choice B \((0, 2)\) if they get \(\mathrm{y = 2x + 2}\).
The Bottom Line:
This problem tests whether students can systematically work with linear functions rather than just looking for patterns. The key insight is that finding the y-intercept requires establishing the complete linear equation first.