The graph of \(\mathrm{y = h(x)}\) is shown in the xy-plane. What is the value of \(\mathrm{h(2)}\)?

1. TRANSLATE the function notation

When you see \(\mathrm{h(2)}\), this is asking:

• What is the \(\mathrm{y}\)-value (output) when \(\mathrm{x = 2}\) (input)?

This is the key translation: \(\mathrm{h(2)}\) doesn't mean "multiply h by 2" - it means "evaluate the function h at \(\mathrm{x = 2}\)."

2. VISUALIZE the location on the graph

• Locate \(\mathrm{x = 2}\) on the horizontal axis
  • Find the number 2 on the \(\mathrm{x}\)-axis
• Draw an imaginary vertical line at \(\mathrm{x = 2}\)
  • This line goes straight up and down through \(\mathrm{x = 2}\)
• Find where this vertical line intersects the curve
  • The curve crosses this vertical line at exactly one point
  • Notice the point is marked on the graph

3. VISUALIZE and read the \(\mathrm{y}\)-coordinate

• Look horizontally from the intersection point to the \(\mathrm{y}\)-axis
  • The point sits at height -1 on the vertical axis
  • The graph confirms this is the point \(\mathrm{(2, -1)}\)
• The \(\mathrm{y}\)-coordinate is -1

4. State the answer

Since the point \(\mathrm{(2, -1)}\) is on the curve, this means:

• When \(\mathrm{x = 2}\), \(\mathrm{y = -1}\)
• Therefore, \(\mathrm{h(2) = -1}\)

Answer: B) -1

Why Students Usually Falter on This Problem

Most Common Error Path:

TRANSLATE failure: Confusion about function notation

Some students see \(\mathrm{h(2)}\) and don't understand what it's asking for. They might:

• Think it means "2h" or some multiplication
• Not realize they need to find a \(\mathrm{y}\)-value
• Look for "h" somewhere on the graph instead of understanding h is the function itself

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

VISUALIZE error: Reading the wrong coordinate

Students correctly locate the point \(\mathrm{(2, -1)}\) but then:

• Report the \(\mathrm{x}\)-coordinate instead of the \(\mathrm{y}\)-coordinate, thinking "the answer must be 2"
  • This may lead them to select Choice D (2)

Alternatively, students might misread the grid:

• Miscount the gridlines or read from the wrong scale
• Confuse positive and negative, potentially reading +1 instead of -1
  • This may lead them to select Choice C (1)

The Bottom Line:

This problem tests two fundamental skills: understanding what function notation asks you to find, and accurately reading coordinates from a graph. The notation \(\mathrm{h(2)}\) is pure translation - it tells you to find the output (\(\mathrm{y}\)-value) for input \(\mathrm{x = 2}\). Once you know what you're looking for, it's just careful graph reading with attention to which axis measures what.