The graph above shows the distance traveled d, in feet, by a product on a conveyor belt m minutes after...

1. TRANSLATE the graph information

Looking at the graph carefully:

• The line is straight (linear relationship)
• The line passes through the origin: point \((0, 0)\)
• The line goes upward from left to right (positive slope)

TRANSLATE specific points from the graph:

• When m = 1 minute, d = 2 feet → point \((1, 2)\)
• When m = 2 minutes, d = 4 feet → point \((2, 4)\)
• When m = 3 minutes, d = 6 feet → point \((3, 6)\)

2. INFER the equation form

Since the line passes through the origin \((0, 0)\), this means:

• There is NO y-intercept (\(\mathrm{b = 0}\))
• The equation must be in the form: \(\mathrm{d = km}\)
• We just need to find the value of \(\mathrm{k}\) (the slope)

This insight immediately rules out Choices C and D, which both have \(\mathrm{+ 2}\) (meaning a y-intercept of 2).

3. SIMPLIFY to find the slope k

Use any point on the line (other than the origin) and substitute into \(\mathrm{d = km}\).

Using point \((2, 4)\):

• \(\mathrm{4 = k(2)}\)
• \(\mathrm{k = 4/2 = 2}\)

We can verify with another point \((1, 2)\):

• \(\mathrm{2 = k(1)}\)
• \(\mathrm{k = 2}\) ✓

4. Write the final equation

Substituting \(\mathrm{k = 2}\) into \(\mathrm{d = km}\):
\(\mathrm{d = 2m}\)

This matches Choice A.

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misreading the scale or coordinates on the graph

Students sometimes misread where the line crosses grid lines. For example:

• They might think at m = 2, the distance is \(\mathrm{d = 1}\) instead of \(\mathrm{d = 4}\)
• Or they confuse the x and y values, reading horizontally instead of vertically

If they incorrectly read point \((2, 1)\) instead of \((2, 4)\):

• They calculate: \(\mathrm{1 = k(2)}\), so \(\mathrm{k = 1/2}\)
• This leads them to select Choice B (\(\mathrm{d = 1/2\, m}\))

Second Most Common Error:

Weak INFER skill: Not recognizing that a line through the origin has no constant term

Some students see the number "2" in the problem (either from the slope or from coordinates) and think it must appear as a \(\mathrm{+ 2}\) in the equation. They don't recognize that:

• A line through \((0, 0)\) means the y-intercept \(\mathrm{b = 0}\)
• The equation cannot have a constant term added

This confusion might lead them to select Choice D (\(\mathrm{d = 2m + 2}\)), which has the correct slope but an incorrect y-intercept.

The Bottom Line:

This problem tests whether students can accurately extract information from a visual representation (the graph) and connect it to the algebraic form of a linear equation. The key insight is recognizing that passing through the origin means the equation is simply \(\mathrm{d = km}\) with no additional constant.