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

1. TRANSLATE the question

The problem asks: "What is the y-intercept of the graph of \(\mathrm{y = f(x)}\)?"

TRANSLATE this into what you need to find:

• The y-intercept is the point where the graph crosses the y-axis
• The y-axis is the vertical line in the middle of the coordinate plane
• At any point on the y-axis, the x-coordinate is always 0
• So you're looking for a point with coordinates \(\mathrm{(0, y)}\) for some value of y

2. VISUALIZE the y-axis on the graph

Look at the graph provided:

• Locate the y-axis (the vertical line labeled with "y" at the top)
• This is where \(\mathrm{x = 0}\)
• Trace the linear function and see where it crosses this vertical line

3. VISUALIZE the exact intersection point

Following the line from left to right:

• The line passes through the y-axis
• At the intersection point, \(\mathrm{x = 0}\) and \(\mathrm{y = 2}\)
• The point of intersection is \(\mathrm{(0, 2)}\)

Answer: C. \(\mathrm{(0, 2)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill - Confusing x-intercept with y-intercept:

Students sometimes confuse the definition of x-intercept and y-intercept. They might think "y-intercept" means finding where \(\mathrm{y = 0}\) (which is actually the x-intercept definition). Looking at the graph, they see the line crosses the x-axis at approximately \(\mathrm{(-5, 0)}\) and select that point.

This may lead them to select Choice A (\(\mathrm{(-5, 0)}\))

Second Most Common Error:

Conceptual confusion about ordered pair notation:

Some students correctly identify that they need to find where the graph crosses the y-axis, and they correctly identify that \(\mathrm{y = 2}\) at that point. However, they get confused about ordered pair notation and write \(\mathrm{(2, 0)}\) instead of \(\mathrm{(0, 2)}\), mixing up which value goes first.

This may lead them to select Choice B (\(\mathrm{(2, 0)}\))

The Bottom Line:

This problem tests your understanding of coordinate plane terminology. The key insight is remembering that:

• Y-intercept → crosses the y-axis → \(\mathrm{x = 0}\) → point looks like \(\mathrm{(0, something)}\)
• X-intercept → crosses the x-axis → \(\mathrm{y = 0}\) → point looks like \(\mathrm{(something, 0)}\)

Once you have this straight, reading the correct value from the graph is straightforward.