The function g is defined by \(\mathrm{g(x) = 4x + 16}\). The graph of \(\mathrm{y = g(x)}\) in the xy-plane...

1. TRANSLATE the intercept information

• Given: \(\mathrm{g(x) = 4x + 16}\), need intercepts first
• x-intercept occurs where the graph crosses the x-axis (\(\mathrm{y = 0}\))
• y-intercept occurs where the graph crosses the y-axis (\(\mathrm{x = 0}\))

2. TRANSLATE and solve for the x-intercept

• Set \(\mathrm{g(x) = 0}\): \(\mathrm{4x + 16 = 0}\)
• SIMPLIFY: \(\mathrm{4x = -16}\), so \(\mathrm{x = -4}\)
• x-intercept is \(\mathrm{(-4, 0)}\)

3. TRANSLATE and solve for the y-intercept

• Evaluate \(\mathrm{g(0)}\): \(\mathrm{g(0) = 4(0) + 16 = 16}\)
• y-intercept is \(\mathrm{(0, 16)}\)

4. INFER what's needed next

• We have two points: \(\mathrm{(-4, 0)}\) and \(\mathrm{(0, 16)}\)
• Need the x-coordinate of their midpoint
• This requires the midpoint formula

5. SIMPLIFY using the midpoint formula

• Midpoint x-coordinate = \(\mathrm{(x_1 + x_2)/2 = (-4 + 0)/2 = -2}\)

Answer: B (-2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't properly convert "x-intercept" and "y-intercept" into the correct mathematical procedures. They might try to find intercepts by plugging in random values or by looking at the coefficients directly without understanding that x-intercept means "set \(\mathrm{y = 0}\)" and y-intercept means "set \(\mathrm{x = 0}\)."

This leads to incorrect intercept coordinates and subsequently wrong midpoint calculations, causing them to select any of the other answer choices through guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct intercept points but make arithmetic errors when applying the midpoint formula. For example, they might calculate \(\mathrm{(-4 + 0)/2}\) as \(\mathrm{-1}\) instead of \(\mathrm{-2}\), or they might give the y-coordinate of the midpoint (8) instead of the x-coordinate.

This may lead them to select Choice A (-4) if they mistakenly think one of the intercept x-values is the midpoint, or cause confusion leading to guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically translate intercept language into mathematical procedures and then accurately apply the midpoint formula. The key insight is recognizing that finding intercepts is always about setting one coordinate to zero.