Math Problem Statement

Consider the graph of the quadratic function f(x) = −3(x − 2)2 − 4 to complete the statement.

The graph of the function f(x) = −3(x − 2)2 − 4 has a ____ value of .

Solution

The given quadratic function is:

f(x)=3(x2)24f(x) = -3(x - 2)^2 - 4

This is in vertex form, f(x)=a(xh)2+kf(x) = a(x - h)^2 + k, where:

  • a=3a = -3 (indicating the parabola opens downwards because aa is negative)
  • h=2h = 2 (the x-coordinate of the vertex)
  • k=4k = -4 (the y-coordinate of the vertex)

Since the parabola opens downward, the vertex represents the maximum point of the graph.

Thus, the graph of the function f(x)=3(x2)24f(x) = -3(x - 2)^2 - 4 has a maximum value of -4.


Would you like more details or have any other questions?

Here are some related questions for further practice:

  1. What is the vertex of the quadratic function f(x)=2(x+3)2+5f(x) = 2(x + 3)^2 + 5?
  2. How does the value of aa affect the shape and direction of a parabola?
  3. What is the axis of symmetry for the function f(x)=3(x2)24f(x) = -3(x - 2)^2 - 4?
  4. How would you determine the x-intercepts of a quadratic function?
  5. How do you convert a quadratic function from standard form to vertex form?

Tip: The vertex form of a quadratic function makes it easy to identify the maximum or minimum value directly.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form of a Quadratic
Graphing Parabolas

Formulas

Vertex Form: f(x) = a(x - h)² + k
Maximum or Minimum Value of a Parabola

Theorems

A parabola opens downward if the coefficient a < 0, and the vertex represents the maximum value.

Suitable Grade Level

Grades 9-11