The graph of the linear function f is shown, where \(\mathrm{y = f(x)}\). What is the y-intercept of the graph...

1. TRANSLATE what the problem is asking

• The question asks: "What is the y-intercept of the graph of f?"

• Key understanding:
  • The y-intercept is the point where the graph crosses the y-axis
  • The y-axis is the vertical line where \(\mathrm{x = 0}\)
  • We need to find the coordinates of this intersection point

2. VISUALIZE to locate the y-intercept on the graph

• Look at the graph and identify the y-axis (the vertical line in the middle)

• Trace the linear function (the diagonal line) until it crosses the y-axis

• Reading from the graph:
  • The line clearly intersects the y-axis at a point above the origin
  • Counting the grid lines on the y-axis, the intersection occurs at \(\mathrm{y = 8}\)
  • Since this is on the y-axis, the x-coordinate is 0

3. TRANSLATE the visual information into coordinate form

• The y-intercept as a point: \(\mathrm{(0, 8)}\)
  • First coordinate (x): 0 (on the y-axis)
  • Second coordinate (y): 8 (read from the graph)

Answer: D. \(\mathrm{(0, 8)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about intercepts: Students confuse the y-intercept with the x-intercept.

The y-intercept is where the graph crosses the y-axis (where \(\mathrm{x = 0}\)), but the x-intercept is where the graph crosses the x-axis (where \(\mathrm{y = 0}\)). Looking at the graph, the line crosses the x-axis somewhere around \(\mathrm{x = 6}\). A student who doesn't have a solid understanding of the difference between these two concepts might look for the wrong intersection point.

This confusion, combined with misreading the graph, may lead them to select Choice A (\(\mathrm{(0, 0)}\)) if they think the intercepts should somehow be at the origin, or they might become confused and guess.

Second Most Common Error:

Weak VISUALIZE skill - Misreading the sign of the y-coordinate: Students incorrectly identify the y-coordinate as negative instead of positive.

If a student doesn't carefully observe which side of the x-axis the intercept is on, they might read the y-intercept as \(\mathrm{-8}\) instead of \(\mathrm{+8}\). This could happen if they're not paying attention to whether the point is above (positive) or below (negative) the x-axis, or if they confuse the direction of positive values on the y-axis.

This may lead them to select Choice C (\(\mathrm{(0, -8)}\)).

The Bottom Line:

This problem tests two key skills: knowing what a y-intercept actually is (conceptual understanding) and being able to accurately read coordinates from a graph (visualization). The most important step is TRANSLATING "y-intercept" into "where the graph crosses the y-axis at \(\mathrm{x = 0}\)," then carefully VISUALIZING to read that point's coordinates from the graph.