The graph shows the modeled number y of customers in a help-desk queue x hours after the center opens, for...

1. TRANSLATE the problem information

The key question asks: "How many customers are in the queue at the moment the center opens?"

• TRANSLATE this timing phrase:
  • "At the moment the center opens" = \(\mathrm{x = 0}\) hours
  • We're looking for the number of customers (\(\mathrm{y}\)) when no time has passed yet (\(\mathrm{x = 0}\))

2. VISUALIZE what we need from the graph

We need to find the y-value where the curve intersects the y-axis.

• The y-axis is where \(\mathrm{x = 0}\)
• This intersection point is called the y-intercept
• Look at the leftmost point of the curve, where it touches the vertical axis

3. VISUALIZE and read the y-intercept

Looking at the graph at \(\mathrm{x = 0}\):

• The curve starts at \(\mathrm{y = 10}\) on the vertical axis
• The point is \(\mathrm{(0, 10)}\)
• This means 10 customers are in the queue at opening time

Answer: C (10)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "at the moment the center opens" and think it refers to a different time point, such as:

• The maximum point on the curve (around \(\mathrm{x = 3, y = 12}\))
• The endpoint when the center closes (\(\mathrm{x = 7, y = 8}\))
• Some other notable feature of the graph

If they confuse "opening" with "peak business time," they might identify the maximum value and select Choice D (12).

Second Most Common Error:

Poor VISUALIZE execution: Students may misread the y-axis scale or estimate the y-intercept incorrectly:

• Reading the grid lines incorrectly
• Estimating the starting point as closer to 9 or 8
• Confusing which axis represents which variable

This could lead them to select Choice B (9) if they underestimate the y-intercept, or Choice A (7) if they severely misread the graph.

The Bottom Line:

This problem tests whether students can connect verbal descriptions of time ("at the moment the center opens") to mathematical representations (\(\mathrm{x = 0}\)) and then accurately read values from a graph. The challenge isn't in complex calculations but in careful TRANSLATION of language to mathematics and precise VISUALIZATION of graph features.