The graph of the function f is a line in the xy-plane. If the line has slope 3/4 and \(\mathrm{f(0)...

1. TRANSLATE the problem information

• Given information:
  • The function f is a line
  • Slope = \(\frac{3}{4}\)
  • \(\mathrm{f(0) = 3}\)

2. INFER the approach

• Since f is a line, use the linear function form: \(\mathrm{f(x) = mx + b}\)
• The slope tells us \(\mathrm{m = \frac{3}{4}}\)
• The condition \(\mathrm{f(0) = 3}\) will help us find b (the y-intercept)

3. TRANSLATE f(0) = 3 into the equation

• Substitute \(\mathrm{x = 0}\) into \(\mathrm{f(x) = mx + b}\):

\(\mathrm{f(0) = m(0) + b = b}\)

• Since \(\mathrm{f(0) = 3}\), we have \(\mathrm{b = 3}\)

4. INFER the complete function

• Now we know: \(\mathrm{m = \frac{3}{4}}\) and \(\mathrm{b = 3}\)
• Therefore: \(\mathrm{f(x) = \frac{3}{4}x + 3}\)

Answer: B. \(\mathrm{f(x) = \frac{3}{4}x + 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the meaning of \(\mathrm{f(0) = 3}\) or make a sign error with the y-intercept.

Some students think \(\mathrm{f(0) = 3}\) means "subtract 3" rather than recognizing it as the y-intercept value. Others correctly identify that \(\mathrm{b = 3}\) but then write \(\mathrm{f(x) = \frac{3}{4}x - 3}\) due to careless sign handling.

This may lead them to select Choice A (\(\mathrm{f(x) = \frac{3}{4}x - 3}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse the slope fraction and use the denominator (4) as the coefficient instead of the full fraction \(\frac{3}{4}\).

When seeing slope = \(\frac{3}{4}\), some students focus on the denominator 4 and write \(\mathrm{f(x) = 4x + b}\), completely missing that the slope should be \(\frac{3}{4}\), not 4.

This may lead them to select Choice D (\(\mathrm{f(x) = 4x + 3}\)) or Choice C (\(\mathrm{f(x) = 4x - 3}\)) if they also make the sign error.

The Bottom Line:

This problem tests whether students can accurately connect the standard linear function form \(\mathrm{f(x) = mx + b}\) with given slope and point information. Success requires careful attention to both the fractional slope and the meaning of function notation.