x\(\mathrm{f(x)}\)029132235For the linear function f, the table shows three values of x and their corresponding values of \(\mathrm{f(x)}\). Which equ...

1. TRANSLATE the table information

• Given information from the table:
  • When \(\mathrm{x = 0}\), \(\mathrm{f(x) = 29}\)
  • When \(\mathrm{x = 1}\), \(\mathrm{f(x) = 32}\)
  • When \(\mathrm{x = 2}\), \(\mathrm{f(x) = 35}\)

• What this gives us: Three coordinate points \(\mathrm{(0, 29)}\), \(\mathrm{(1, 32)}\), and \(\mathrm{(2, 35)}\)

2. INFER the most efficient approach

• Since we need \(\mathrm{f(x) = mx + b}\), we should find b first using the \(\mathrm{x = 0}\) point
• Why? When \(\mathrm{x = 0}\), the equation becomes \(\mathrm{f(0) = m(0) + b = b}\)
• This immediately gives us the y-intercept without any algebra

3. SIMPLIFY to find the y-intercept

• Using point \(\mathrm{(0, 29)}\): \(\mathrm{f(0) = m(0) + b}\)
• \(\mathrm{29 = 0 + b}\)
• Therefore: \(\mathrm{b = 29}\)

4. SIMPLIFY to find the slope

• Now we know \(\mathrm{f(x) = mx + 29}\)
• Using point \(\mathrm{(1, 32)}\): \(\mathrm{32 = m(1) + 29}\)
• \(\mathrm{32 = m + 29}\)
• \(\mathrm{m = 32 - 29 = 3}\)

5. Write the final equation

• \(\mathrm{f(x) = 3x + 29}\)

Answer: A. \(\mathrm{f(x) = 3x + 29}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which column represents the input (x) and which represents the output f(x), or they try to find slope using the wrong values.

For example, they might think the slope is \(\mathrm{29/0}\) or try to use f(x) values as x-coordinates. This leads to completely incorrect equations and may cause them to select Choice B (\(\mathrm{f(x) = 29x + 32}\)) or get stuck and guess.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the strategic advantage of using \(\mathrm{x = 0}\) first to find the y-intercept, instead attempting to calculate slope first using the slope formula with two points.

While this approach can work, it's more prone to arithmetic errors and may lead to calculation mistakes that result in selecting Choice C (\(\mathrm{f(x) = 35x + 29}\)) or Choice D (\(\mathrm{f(x) = 32x + 35}\)).

The Bottom Line:

This problem tests whether students can systematically extract information from a table and use the structure of linear equations strategically. The key insight is recognizing that \(\mathrm{x = 0}\) immediately gives you the y-intercept, making the problem much simpler than it first appears.