For the linear function f, the graph of \(\mathrm{y = f(x)}\) in the xy-plane has a slope of 1/4 and...

1. TRANSLATE the problem information

• Given information:
  • Slope of the graph = \(\frac{1}{4}\)
  • Graph passes through point \((0,5)\)
• We need to find the equation that defines function f

2. INFER the approach

• For any linear function, we use slope-intercept form: \(\mathrm{f(x) = mx + b}\)
• We need to identify values for m (slope) and b (y-intercept)
• The point \((0,5)\) is special because \(\mathrm{x = 0}\) means this point is on the y-axis

3. TRANSLATE the slope value

• Given slope = \(\frac{1}{4}\)
• Therefore: \(\mathrm{m = \frac{1}{4}}\)

4. INFER the y-intercept from the given point

• The graph passes through \((0,5)\)
• When \(\mathrm{x = 0}\), \(\mathrm{y = 5}\)
• This means the y-intercept is 5
• Therefore: \(\mathrm{b = 5}\)

5. Substitute into the slope-intercept form

• \(\mathrm{f(x) = mx + b}\)
• \(\mathrm{f(x) = \frac{1}{4}x + 5}\)

Answer: A. f(x) = \(\frac{1}{4}\)x + 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing that the point \((0,5)\) directly gives the y-intercept value.

Students might see "passes through \((0,5)\)" and think they need to do something more complex with this information, rather than recognizing that any point with x-coordinate 0 is automatically the y-intercept. They might try to use point-slope form unnecessarily or become confused about how to use this point.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can connect the geometric concept of "passing through a point on the y-axis" with the algebraic concept of "y-intercept value" in the slope-intercept form equation.