The graph of a quadratic function is shown in the xy-plane above. The function reaches a maximum value at the...

1. TRANSLATE the problem information

• Given information:
  • A quadratic function \(\mathrm{y = -x^2 + 6x - 5}\) is graphed
  • The function reaches a maximum at point (h, k)
  • We need to find the value of k

• What this tells us:
  • We're looking for the y-coordinate of the maximum point
  • The notation (h, k) means h is the x-coordinate and k is the y-coordinate

2. INFER what the maximum represents

• Key insight: For any quadratic function, the maximum (or minimum) occurs at the vertex of the parabola
  • Since this parabola opens downward (we can see from the graph, and the coefficient of x² is negative), the vertex is at the top—the maximum point

• Strategic approach: We need to:
  1. Locate the vertex (highest point) on the graph
  2. Read the y-coordinate of that point
  3. That y-coordinate is k

3. VISUALIZE the vertex location on the graph

• Locate the highest point:
  • Scan the parabola to find where it peaks
  • The curve reaches its highest point at approximately the middle of the graph
  • The peak occurs at x = 3 (where the parabola is symmetric)

• Read the coordinates:
  • At x = 3, trace vertically to the curve
  • From that point, trace horizontally to the y-axis
  • The y-value at the vertex is 4
  • Therefore, the vertex is at (3, 4)

4. Identify the answer

• The maximum point is (h, k) = (3, 4)
• The question asks for k, which is the y-coordinate
• k = 4

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill—confusing coordinates: Students understand they need to find the vertex but confuse which coordinate the question is asking for. They correctly identify the vertex at (3, 4) but mistakenly report h = 3 as the answer instead of k = 4.

The confusion stems from not fully processing that k specifically represents the y-coordinate (the height/maximum value), while h represents the x-coordinate (the horizontal position where the maximum occurs). Since the problem asks "What is the value of k?" they might quickly look at the vertex and grab the first number they see, which is 3.

This leads to an incorrect answer of 3 instead of the correct answer 4.

Second Most Common Error:

Weak VISUALIZE skill—misreading the graph: Students may struggle to accurately locate the vertex on the graph. They might:

• Miscount the grid lines and think the vertex is at a different height
• Read the vertex as (3, 3) or (4, 4) due to hasty graph reading
• Not recognize that the vertex is precisely where the parabola is perfectly symmetric

This causes them to report an incorrect y-coordinate value instead of 4.

The Bottom Line:

This problem requires careful attention to two distinct pieces of information: (1) where the vertex is located on the graph, and (2) which coordinate—h or k—the question is asking for. Students who rush through either the graph reading or the interpretation of what k represents will likely answer incorrectly, even though the problem itself is conceptually straightforward.