In the xy-plane, the graph of y = 8/x intersects line ℓ at the points where x = 1 and...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{y = \frac{8}{x}}\) (a rational function)
  • Line ℓ intersects this function at \(\mathrm{x = 1}\) and \(\mathrm{x = 2}\)
  • Need to find: slope of line ℓ
• What this tells us: We need two points on the line to calculate its slope

2. TRANSLATE what "intersects at x = 1 and x = 2" means

• This means we know the x-coordinates of two points on line ℓ
• We need to find the corresponding y-coordinates using the function \(\mathrm{y = \frac{8}{x}}\)
• These y-values will give us two complete coordinate pairs

3. SIMPLIFY by evaluating the function at each x-value

• At \(\mathrm{x = 1}\): \(\mathrm{y = \frac{8}{1} = 8}\) → Point \(\mathrm{(1, 8)}\)
• At \(\mathrm{x = 2}\): \(\mathrm{y = \frac{8}{2} = 4}\) → Point \(\mathrm{(2, 4)}\)

4. SIMPLIFY using the slope formula

• Apply: \(\mathrm{slope = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Substitute: \(\mathrm{slope = \frac{4 - 8}{2 - 1}}\)
  \(\mathrm{= \frac{-4}{1}}\)
  \(\mathrm{= -4}\)

Answer: \(\mathrm{-4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that "intersects at \(\mathrm{x = 1}\) and \(\mathrm{x = 2}\)" means they need to find the y-coordinates by evaluating the given function.

Instead, they might try to work directly with the function \(\mathrm{y = \frac{8}{x}}\) to find its slope (which doesn't make sense for a curve) or become confused about what information they actually have. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the points \(\mathrm{(1, 8)}\) and \(\mathrm{(2, 4)}\) but make sign errors when calculating the slope.

Common mistake: \(\mathrm{slope = \frac{8 - 4}{2 - 1}}\)
\(\mathrm{= \frac{4}{1}}\)
\(\mathrm{= 4}\), reversing the order of subtraction in the numerator. This may lead them to select Choice D \(\mathrm{(4)}\).

The Bottom Line:

This problem tests whether students can connect the abstract idea of "intersection points" to the concrete task of evaluating a function, then execute a straightforward slope calculation without sign errors.