Give the domain and the range of each quadratic function whose graph is described. Maximum = -6 at x = 10

Verified step by step guidance

1. Identify the general form of a quadratic function: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
2. Since the problem states there is a maximum value of \(-6\) at \(x = 10\), the vertex is at \((10, -6)\) and the parabola opens downward (which means \(a < 0\)).
3. The domain of any quadratic function is all real numbers, so the domain is \((-\infty, \infty)\).
4. Because the parabola opens downward and the maximum value is \(-6\), the range includes all \(y\)-values less than or equal to \(-6\). So, the range is \((-\infty, -6]\).
5. Summarize: Domain is \((-\infty, \infty)\) and range is \((-\infty, -6]\) based on the vertex and the direction the parabola opens.

Key Concepts

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and vertex of the parabola is essential for analyzing its properties.

Vertex and Maximum/Minimum Values

The vertex of a parabola is the point where the function reaches its maximum or minimum value. If the parabola opens downward, the vertex represents the maximum point; if it opens upward, the vertex is the minimum. The vertex form f(x) = a(x-h)² + k helps identify this point as (h, k).

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens downward, the range is all y-values less than or equal to the maximum y-value at the vertex; if it opens upward, the range is all y-values greater than or equal to the minimum y-value.