The graph of the function f is shown in the xy-plane. What is the maximum value of the function f?

1. TRANSLATE what the question is asking

When the problem asks for "the maximum value of the function f," we need to understand:

• Maximum value = the greatest y-coordinate on the graph
• This is the highest output value the function produces
• We're looking for a y-value, not an x-value

2. VISUALIZE the graph structure

Looking at the graph:

• The curve is a parabola (U-shaped curve)
• It opens downward (like an upside-down U)
• The highest point on the curve is at the top of the "hill"

3. INFER where the maximum occurs

For a downward-opening parabola:

• The vertex (the turning point) is the highest point
• This is where the function reaches its maximum
• Before the vertex, the function increases; after the vertex, it decreases

4. VISUALIZE by reading the vertex coordinates

Locate the highest point on the graph:

• The vertex appears at the point \((-3, 8)\)
• The x-coordinate is \(-3\) (this tells us WHERE the maximum occurs)
• The y-coordinate is \(8\) (this tells us WHAT the maximum value is)

5. INFER the final answer

The question asks for the maximum value (not the location):

• The value of the function = the y-coordinate
• At the vertex, \(\mathrm{y = 8}\)
• Therefore, the maximum value is 8

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

TRANSLATE weakness + INFER confusion: Students confuse "maximum value" with "where the maximum occurs" and report the x-coordinate instead of the y-coordinate.

When they see the vertex at \((-3, 8)\), they might think:

• "The highest point is at \(\mathrm{x = -3}\), so the answer is \(-3\)"

This confusion between the location of the maximum (\(\mathrm{x = -3}\)) and the value of the maximum (\(\mathrm{y = 8}\)) is very common. The problem asks "what is the maximum value," which specifically means the y-coordinate, but students sometimes answer with the x-coordinate where that maximum happens.

This would lead them to incorrectly answer -3 instead of the correct answer 8.

Second Most Common Error:

VISUALIZE weakness: Students misread the graph and don't accurately identify the coordinates of the vertex.

They might:

• Read the y-coordinate as 7 or 9 instead of 8 (not reading the gridlines carefully)
• Estimate the peak incorrectly if they're not precise with the graph

This leads to selecting an incorrect value close to 8, or causes confusion and guessing.

The Bottom Line:

This problem tests whether students understand the difference between a function's value (output/y-coordinate) and the input (x-coordinate) where that value occurs. The word "value" in "maximum value" is the key—it specifically refers to the y-coordinate. Students must also be able to accurately read coordinates from a graph, which requires careful attention to the gridlines and the precise location of the vertex.