For the linear function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane has a slope of 2 and...

1. TRANSLATE the problem information

• Given information:
  • The graph has a slope of 2
  • The y-intercept is at the point (0, -5)
• What this tells us: We need \(\mathrm{slope = 2}\) and \(\mathrm{y\text{-}intercept\ value = -5}\)

2. INFER the appropriate approach

• Since we have slope and y-intercept, we should use slope-intercept form
• The slope-intercept form is \(\mathrm{f(x) = mx + b}\) where:
  • \(\mathrm{m = slope}\)
  • \(\mathrm{b = y\text{-}intercept\ value}\) (the y-coordinate where the line crosses the y-axis)

3. Substitute the values

• \(\mathrm{m = 2}\) (given slope)
• \(\mathrm{b = -5}\) (y-coordinate of y-intercept point (0, -5))
• Therefore: \(\mathrm{f(x) = 2x + (-5) = 2x - 5}\)

4. Verify by checking answer choices

• Choice D: \(\mathrm{f(x) = 2x - 5}\) has \(\mathrm{slope = 2}\) ✓ and \(\mathrm{y\text{-}intercept = -5}\) ✓

Answer: D. f(x) = 2x - 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the y-intercept point (0, -5) with just needing the number -5. Some students might think the y-intercept means the slope should involve -5, or they might incorrectly use +5 instead of -5.

This leads to selecting wrong answers like Choice A or Choice B where the y-intercept value is correct but the slope is wrong.

Second Most Common Error:

Missing conceptual knowledge about slope-intercept form: Students who don't clearly remember that \(\mathrm{f(x) = mx + b}\) requires \(\mathrm{m = slope}\) and \(\mathrm{b = y\text{-}intercept\ value}\) might mix up which number goes where.

This causes confusion about whether the slope or y-intercept should be 2, potentially leading them to select Choice C (\(\mathrm{f(x) = -2x - 5}\)) or abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can cleanly TRANSLATE verbal descriptions into the mathematical components of slope-intercept form. The key is recognizing that "slope of 2" means \(\mathrm{m = 2}\) and "y-intercept at (0, -5)" means \(\mathrm{b = -5}\).