For the given function \(\mathrm{f(x) = 2x + 3}\), the graph of \(\mathrm{y = f(x)}\) in the xy-plane is parallel...

1. TRANSLATE the function into line form

• Given information:
  • \(\mathrm{f(x) = 2x + 3}\)
  • Line j is parallel to the graph of \(\mathrm{y = f(x)}\)
  • Need to find the slope of line j

• The graph of \(\mathrm{y = f(x)}\) means we replace f(x) with its definition:
  \(\mathrm{y = 2x + 3}\)

2. INFER the slope from the equation

• The equation \(\mathrm{y = 2x + 3}\) is in slope-intercept form: \(\mathrm{y = mx + b}\)
• In this form, m represents the slope and b represents the y-intercept
• Comparing: \(\mathrm{y = 2x + 3}\) with \(\mathrm{y = mx + b}\)
• Therefore: \(\mathrm{slope = 2}\), \(\mathrm{y\text{-}intercept = 3}\)

3. INFER the relationship between parallel lines

• Line j is parallel to the graph of \(\mathrm{y = f(x)}\)
• Key property: parallel lines have identical slopes
• Since the graph of \(\mathrm{y = f(x)}\) has slope 2, line j must also have slope 2

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand that "the graph of \(\mathrm{y = f(x)}\)" means the same as writing \(\mathrm{y = 2x + 3}\). They might think they need to do something special with the function notation or get confused about what "graph of \(\mathrm{y = f(x)}\)" actually represents.

This confusion leads them to not identify the slope correctly and causes them to get stuck and guess.

Second Most Common Error:

Missing conceptual knowledge about parallel lines: Students might correctly identify that the slope of \(\mathrm{y = 2x + 3}\) is 2, but forget that parallel lines have equal slopes. They might think they need to perform some calculation or transformation to find the slope of line j.

This leads to confusion and potentially guessing among answer choices.

The Bottom Line:

This problem tests whether students can connect function notation to standard linear equations and apply the fundamental property of parallel lines. Success requires recognizing that \(\mathrm{y = f(x)}\) is just another way to write the linear equation.