The graph of \(\mathrm{y = g(x)}\) is shown on a coordinate plane.The graph is a straight line that decreases as...

1. TRANSLATE the problem requirements

The problem asks for the x-intercept, which is defined as 'a point where the graph crosses the x-axis.'

• TRANSLATE this definition into mathematical terms:
  • The x-axis is where \(\mathrm{y = 0}\)
  • So I need to find the point \(\mathrm{(x, 0)}\) where the line crosses
  • The answer will be a coordinate point in the form \(\mathrm{(x, 0)}\)

2. VISUALIZE by examining the graph

• Look at where the line intersects the x-axis (the horizontal axis)
• Follow the line from where it's visible in the upper left
• The line passes through several clear grid points:
  • Around \(\mathrm{x = -2}\), the line is above the x-axis
  • At \(\mathrm{x = 0}\), the line crosses at \(\mathrm{y = 3}\) (this is the y-intercept, not what we want)
  • At \(\mathrm{x = 1}\), the line is at \(\mathrm{y = 2}\)
  • At \(\mathrm{x = 2}\), the line is at \(\mathrm{y = 1}\)
  • At \(\mathrm{x = 3}\), the line crosses the x-axis \(\mathrm{(y = 0)}\) ← This is what we need!

3. TRANSLATE the visual information into coordinate notation

• At \(\mathrm{x = 3}\), the line crosses the x-axis
• At this point, \(\mathrm{y = 0}\)
• Therefore, the x-intercept is the point \(\mathrm{(3, 0)}\)

4. Check the answer choices

Looking at the options:

• (A) \(\mathrm{(-3, 0)}\) - Wrong x-value
• (B) \(\mathrm{(0, 3)}\) - This is the y-intercept, not the x-intercept
• (C) \(\mathrm{(3, 0)}\) - This matches our answer ✓
• (D) \(\mathrm{(0, -3)}\) - Wrong intercept type and wrong values

Answer: C \(\mathrm{(3, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

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

Students sometimes mix up the definitions:

• They read 'x-intercept' but mentally think about where the graph crosses the y-axis
• Or they identify the correct crossing point but write the coordinates in the wrong order
• The y-intercept is \(\mathrm{(0, 3)}\), which appears prominently on the graph

This may lead them to select Choice B: \(\mathrm{(0, 3)}\)

Second Most Common Error:

Poor VISUALIZE skill: Misreading the graph or counting grid lines incorrectly

Students may:

• Count grid squares incorrectly when tracing the line
• Confuse which axis is which
• Look at the negative x-direction and think the intercept is at \(\mathrm{x = -3}\) instead of \(\mathrm{x = 3}\)

This may lead them to select Choice A: \(\mathrm{(-3, 0)}\)

The Bottom Line:

This problem tests whether you can correctly TRANSLATE the term 'x-intercept' into 'where the graph crosses the x-axis (where \(\mathrm{y = 0}\))' and then accurately VISUALIZE that location on the graph. The most critical moment is not confusing x-intercept (crossing the x-axis) with y-intercept (crossing the y-axis) - they are different concepts that students often mix up when reading quickly.