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Chapter 3: Problem 6

Fill in the blank. If \(a<0\) and \(f(x)=a(x-h)^{2}+k,\) then \(k\) is the _________ value of the function.

Short Answer

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maximum

Step by step solution

01

- Understand the Function Form

The function given is in vertex form: \[ f(x) = a(x - h)^{2} + k \].
02

- Analyzing the Coefficient of the Quadratic Term

Identify the coefficient of the quadratic term, which is \(a\). Since \(a < 0\), note that the parabola opens downward.
03

- Determine Characteristics of Parabolas Opening Downward

For a parabola that opens downward, the vertex represents the maximum point on the graph.
04

- Vertex Form and Maximum Value

In the vertex form \(f(x) = a(x - h)^{2} + k\), the point \((h, k)\) is the vertex. Since the parabola opens downward, \(k\) represents the maximum value of the function.

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Key Concepts

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

parabola
A parabola is a U-shaped curve that can open either upwards or downwards. It鈥檚 the graph of a quadratic function, and its shape and direction are determined by the coefficients in the equation.

When we talk about a parabola, there are a few key terms to remember:
  • **Vertex**: The highest or lowest point on the graph.
  • **Axis of Symmetry**: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
  • **Direction**: Whether the parabola opens upward or downward, which depends on the sign of the leading coefficient, usually denoted as 鈥榓鈥�.


In vertex form, a quadratic function is written as [f(x) = a(x - h)^{2} + k] where (h, k) is the vertex. Understanding the structure of this equation helps in analyzing and graphing the parabola.
maximum value
The maximum value of a quadratic function occurs at the vertex when the parabola opens downward. This happens when the coefficient 'a' in the equation [a(x - h)^{2} + k] is less than zero ([a < 0]).

For example, if you have [a = -3], the parabola opens downward, and the vertex (h, k) is the highest point on the graph.

This means that 鈥榢鈥� represents the maximum value of the function. To visualize:
  • Imagine the parabola as a hill. The top of the hill is the maximum point.
  • Every point on the parabola below this top represents lower values of the function.
This concept is crucial for solving optimization problems and finding the highest or lowest points in various contexts.
coefficient analysis
Understanding the role of coefficients in a quadratic function is essential for mastering the vertex form. Here's a breakdown:

  • The coefficient 'a' affects the width and direction of the parabola. If [a > 0], the parabola opens upwards, and if [a < 0], it opens downwards.
  • The absolute value of 'a' determines the 'stretch' or 'compression' of the parabola. Larger values of |a| make the parabola narrower, while smaller values make it wider.
For example, in [f(x) = -2(x - 3)^{2} + 4], 'a' is -2, so the parabola opens downward and is moderately narrow. Knowing how 'a' affects the function aids in accurately graphing and understanding the behavior of parabolas.
vertex
The vertex of a parabola is a critical point where the function reaches its maximum or minimum value. In vertex form, the quadratic function [f(x) = a(x - h)^{2} + k] has its vertex at the point (h, k).

A few more things to note about the vertex:
  • It鈥檚 the point where the axis of symmetry intersects the parabola.
  • If 'a' is positive ([a > 0]), the vertex is the minimum point.
  • If 'a' is negative ([a < 0]), the vertex is the maximum point.
To find the vertex from the standard quadratic form [ax^2 + bx + c], you can use the formula [h = -b/2a] for the x-coordinate and then plug this back into the function to find the y-coordinate. Understanding the vertex helps pinpoint the extreme value of the function and is essential for solving various practical and theoretical problems.

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