The function g is defined by \(\mathrm{g(x) = 5x + 45}\). At what point does the graph of \(\mathrm{y =...

1. TRANSLATE the intersection requirement

• Given information:
  • Function \(\mathrm{g(x) = 5x + 45}\)
  • Need intersection with line \(\mathrm{y = 15}\)
• What this means: At the intersection point, both graphs have the same y-value, so g(x) must equal 15

2. TRANSLATE this into an equation

• Set the function equal to 15:
  \(\mathrm{5x + 45 = 15}\)

3. SIMPLIFY to solve for x

• Subtract 45 from both sides:
  \(\mathrm{5x = 15 - 45 = -30}\)
• Divide both sides by 5:
  \(\mathrm{x = -6}\)

4. INFER the complete intersection point

• Since we found \(\mathrm{x = -6}\) and we know the intersection occurs where \(\mathrm{y = 15}\)
• The intersection point is \(\mathrm{(-6, 15)}\)

Answer: B (-6, 15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete INFER reasoning: Students solve correctly for \(\mathrm{x = -6}\) but then report the intersection point as \(\mathrm{(-6, 0)}\) instead of \(\mathrm{(-6, 15)}\).

This happens because they forget that the intersection is with the horizontal line \(\mathrm{y = 15}\), not with the x-axis. They correctly find where \(\mathrm{x = -6}\), but then assume the y-coordinate is 0 because they're thinking of x-intercepts rather than the specific line \(\mathrm{y = 15}\).

This leads them to select Choice C (-6, 0).

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{5x + 45 = 15}\), particularly with the subtraction step.

A common mistake is: \(\mathrm{5x = 15 + 45 = 60}\), leading to \(\mathrm{x = 12}\). However, since 12 isn't among the x-coordinates in the choices, this typically leads to confusion and guessing rather than a specific wrong answer.

The Bottom Line:

This problem requires students to understand that "intersection with \(\mathrm{y = 15}\)" means finding where the function output equals 15, and then remembering that the intersection point includes both the x-value they calculated AND the y-value from the given horizontal line.