In Exercises 45–48, 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: $f\left(x\right)=ax{²}^{}+bx+c$. Since the graph has a maximum, the parabola opens downward, so .
2. Use the vertex form of a quadratic function: $f(x)\; =\; a(x\; -\; h)^2\; +\; k$, where $(h,\; k)$ is the vertex. Here, the vertex is given as $(10,\; -6)$ because the maximum value is -6 at $x\; =\; 10$.
3. Determine the domain of the function. Since quadratic functions are defined for all real numbers, the domain is $(-infty,\; infty)$.
4. Determine the range of the function. Because the parabola opens downward and the maximum value is -6, the range includes all values less than or equal to -6. So, the range is $(-infty,\; -6]$.
5. Summarize: Domain is all real numbers, and range is all real numbers less than or equal to -6, reflecting the maximum point at $x\; =\; 10$.

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 maximum or minimum values.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value. For a quadratic function, the vertex coordinates (h, k) can be found using the formula h = -b/(2a) and k = f(h). In this question, the vertex is given as (10, -6), indicating the maximum value.

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 and the parabola's direction: if it opens downward, the range is all values less than or equal to the maximum y-value; if upward, all values greater than or equal to the minimum y-value.