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Difficulty: EasyQuadratic Functions and Graphs

The function ff is defined by f(x)=−(x−5)2+4f(x) = -(x - 5)^2 + 4. What is the maximum value of the function ff?

  1. A
    −4-4
  2. 44Answer
  3. C
    55
  4. D
    −5-5

Answer

The maximum value of the function is 4.
The quadratic function f(x)=−(x−5)2+4f(x) = -(x - 5)^2 + 4 is in vertex form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where the vertex of the parabola is (h,k)=(5,4)(h, k) = (5, 4). Because the leading coefficient a=−1a = -1 is negative, the parabola opens downward, which means the vertex is the highest point on the graph. The maximum value of the function is the y-coordinate of this vertex, which is 4.

Step-by-Step Solution

1
Identify the form of the quadratic function.
The function f(x)=−(x−5)2+4f(x) = -(x - 5)^2 + 4 is written in vertex form, f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.
Recognizing the vertex form allows us to directly identify the vertex coordinates of the parabola.
2
Identify the vertex (h,k)(h, k) and the direction the parabola opens.
The vertex is (5,4)(5, 4) and the leading coefficient is a=−1a = -1.
Since the leading coefficient aa is negative, the parabola opens downward, meaning the vertex is the highest point on the graph.
3
Determine the maximum value of the function.
The maximum value of the function is the y-coordinate of the vertex, which is 4.
The maximum value of a function refers to its largest output (y-value), which corresponds to the y-coordinate of its highest point.

Key Concept

Quadratic Functions and Graphs
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Topics

Quadratic Functions and Graphs

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