In the xy-plane, the graph of y = x^2 - 9 intersects line p at \(\mathrm{(1, a)}\) and \(\mathrm{(5, b)}\),...

1. TRANSLATE the problem information

• Given information:
  • Parabola: \(\mathrm{y = x^2 - 9}\)
  • Line p intersects the parabola at \(\mathrm{(1, a)}\) and \(\mathrm{(5, b)}\)
  • Need to find the slope of line p
• What this tells us: Since these points lie on both the parabola and the line, we can find their exact coordinates by substituting into the parabola equation.

2. SIMPLIFY to find the first intersection point

• Substitute \(\mathrm{x = 1}\) into \(\mathrm{y = x^2 - 9}\):
  • \(\mathrm{y = (1)^2 - 9}\)
    \(\mathrm{y = 1 - 9}\)
    \(\mathrm{y = -8}\)
  • So point \(\mathrm{(1, a) = (1, -8)}\)

3. SIMPLIFY to find the second intersection point

• Substitute \(\mathrm{x = 5}\) into \(\mathrm{y = x^2 - 9}\):
  • \(\mathrm{y = (5)^2 - 9}\)
    \(\mathrm{y = 25 - 9}\)
    \(\mathrm{y = 16}\)
  • So point \(\mathrm{(5, b) = (5, 16)}\)

4. SIMPLIFY using the slope formula

• With points \(\mathrm{(1, -8)}\) and \(\mathrm{(5, 16)}\):
  • \(\mathrm{Slope = \frac{y_2 - y_1}{x_2 - x_1}}\)
  • \(\mathrm{Slope = \frac{16 - (-8)}{5 - 1}}\)
  • \(\mathrm{Slope = \frac{16 + 8}{4}}\)
    \(\mathrm{Slope = \frac{24}{4}}\)
    \(\mathrm{Slope = 6}\)

Answer: A. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when handling the negative number in the slope calculation.

When computing \(\mathrm{(16 - (-8))}\), they might incorrectly calculate this as \(\mathrm{(16 - 8) = 8}\) instead of \(\mathrm{(16 + 8) = 24}\). This gives them a slope of \(\mathrm{\frac{8}{4} = 2}\), leading them to select Choice B (2).

Second Most Common Error:

Inadequate TRANSLATE reasoning: Students misunderstand what the intersection points represent and fail to substitute the x-values into the parabola equation to find the actual coordinates.

They might try to work directly with the variables a and b without finding their numerical values, or they might confuse which equation to use for finding the intersection points. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand that intersection points must satisfy both equations simultaneously, and whether they can handle negative numbers correctly in slope calculations. The key insight is that knowing the x-coordinates of intersection points allows you to find the y-coordinates using the parabola equation.