For the function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane has a slope of 3 and passes...

1. TRANSLATE the problem information

• Given information:
  • Slope of the graph = 3
  • Graph passes through point \(\mathrm{(0, -8)}\)
• What this tells us: We need to find a linear equation using these two pieces of information.

2. INFER the connection to linear function form

• Linear functions have the form \(\mathrm{f(x) = mx + b}\)
• The slope tells us \(\mathrm{m = 3}\)
• The point \(\mathrm{(0, -8)}\) is special because it's on the y-axis - this is the y-intercept!

3. INFER the y-intercept value

• When a line passes through \(\mathrm{(0, -8)}\), the y-intercept is -8
• This means \(\mathrm{b = -8}\) in our equation \(\mathrm{f(x) = mx + b}\)

4. Substitute the values

• \(\mathrm{f(x) = mx + b}\) becomes \(\mathrm{f(x) = 3x + (-8)}\)
• Simplifying: \(\mathrm{f(x) = 3x - 8}\)

Answer: B. \(\mathrm{f(x) = 3x - 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the point \(\mathrm{(0, -8)}\) directly gives them the y-intercept value.

Instead, they might think they need to use point-slope form or somehow "solve for" the y-intercept. They may get confused about what to do with the point \(\mathrm{(0, -8)}\) and either ignore it or use it incorrectly. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that \(\mathrm{b = -8}\) but then get confused about the negative sign when writing the equation.

They might write \(\mathrm{f(x) = 3x + 8}\) instead of \(\mathrm{f(x) = 3x - 8}\), thinking that since the y-intercept is -8, they should "add" it to get \(\mathrm{f(x) = 3x + (-8)}\), but then incorrectly simplify this. This may lead them to select Choice C (\(\mathrm{f(x) = 3x + 5}\)) or Choice D (\(\mathrm{f(x) = 3x + 11}\)) if they make additional errors.

The Bottom Line:

This problem tests whether students understand that a point \(\mathrm{(0, k)}\) directly gives the y-intercept value k, and whether they can correctly handle negative values in the slope-intercept form.