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Chapter 3: Problem 8

Given \(f(x)=a(x-h)^{2}+k,\) if \(a<0,\) then the maximum of \(f\) is ______.

Short Answer

Expert verified
The maximum of f is k.

Step by step solution

01

- Identify the Vertex Form of the Parabola

The given function is in the form of the vertex formula for a parabola: f(x) = a(x-h)^2+kwhere - a is the coefficient that determines the direction and width of the parabola,- (h, k) is the vertex of the parabola.
02

- Determine the Coefficient Sign

The problem states that a < 0. This means that the parabola opens downwards.
03

- Identify the Maximum Point

For a downward opening parabola, the vertex represents the maximum point. Therefore, the maximum value of f(x) is simply the y-coordinate of the vertex, which is k.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parabola maximum point
A parabola is a curved graph that comes from a quadratic function. When we have the equation in the vertex form: \[f(x) = a(x-h)^2 + k\] the maximum or minimum value of this function is located at the 'vertex.' If the coefficient \(a\) is less than 0, the parabola opens downwards, leading to a maximum point at the vertex. The vertex of the parabola yields the highest point on the graph, and this value is crucial for students to understand when analyzing quadratic functions.
vertex coordinates
The vertex of a parabola provides key information. In vertex form, \[f(x) = a(x-h)^2 + k\], the vertex is given by the coordinates guided by 鈥� \( (h, k) \) where: < ul >
  • \( h \): Represents the x-coordinate of the vertex
  • \( k \): Represents the y-coordinate of the vertex
    • < / ul > These coordinates pinpoint the location of the parabola's maximum or minimum point on a coordinate plane. Understanding the vertex coordinates is crucial as it determines where the graph changes direction, either reaching a peak (maximum) or a trough (minimum).
    downward opening parabola
    When the coefficient \(a\) in the vertex form \[f(x) = a(x-h)^2 + k\] is negative (\(a < 0\)), the parabola opens downwards. This means the graph forms a 'U' shape facing downwards, indicating that the vertex represents the highest (maximum) point on the graph. Recognizing this downward opening is vital in determining where the maximum value of the function is located. Keep in mind: < ul >
  • The vertex still represents the point (\(h, k\))
  • Since `a` is negative, the 'U' shape faces downwards. < /ul>

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