What is the y-intercept of the graph of y = 34x + 81 in the xy-plane?

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = 34x + 81}\)
  • Need to find: y-intercept of this line

• What this tells us: We need to find the point where this line crosses the y-axis

2. INFER the approach using equation form

• The equation \(\mathrm{y = 34x + 81}\) is in slope-intercept form: \(\mathrm{y = mx + b}\)
• In this form: \(\mathrm{m = 34}\) (slope) and \(\mathrm{b = 81}\) (y-intercept value)
• Key insight: The y-intercept is the point \(\mathrm{(0, b)}\), not just the number b

3. Apply the y-intercept rule

• Since the equation is \(\mathrm{y = 34x + 81}\), we have \(\mathrm{b = 81}\)
• Therefore, the y-intercept point is \(\mathrm{(0, 81)}\)

Answer: A. \(\mathrm{(0, 81)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse slope with y-intercept in the equation \(\mathrm{y = mx + b}\)

They see \(\mathrm{y = 34x + 81}\) and think "the y-intercept must be related to the 34 since it's attached to the variable." They don't properly apply the slope-intercept form rule that identifies b (the constant term) as the y-intercept value.

This may lead them to select Choice B (\(\mathrm{(0, 34)}\))

Second Most Common Error:

Conceptual confusion about y-intercept: Students remember that y-intercept involves the constant term but make a sign error

They might think the equation has a negative y-intercept for some reason, possibly confusing the setup with other problems or misreading the equation.

This may lead them to select Choice D (\(\mathrm{(0, -81)}\))

The Bottom Line:

This problem tests whether students can correctly identify the components of slope-intercept form. The key is remembering that in \(\mathrm{y = mx + b}\), the y-intercept is always \(\mathrm{(0, b)}\) - the point where x equals zero and y equals the constant term.