In Exercises 45–48, give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

Verified step by step guidance

1. Identify the vertex of the quadratic function, which is given as a point \((h, k)\). This point represents the minimum point of the parabola since it opens upward.
2. Recall that the general form of a quadratic function with vertex \((h, k)\) is \(f(x) = a(x - h)^2 + k\), where \(a > 0\) because the parabola opens up.
3. Determine the domain of the function. Since quadratic functions are defined for all real numbers, the domain is \(\{x \mid x \in \mathbb{R}\}\) or \((-\infty, \infty)\).
4. Determine the range of the function. Because the parabola opens upward and the vertex is the minimum point, the range is all \(y\)-values greater than or equal to \(k\), expressed as \([k, \infty)\).
5. Summarize: The domain is all real numbers, and the range is \(y \geq k\), where \(k\) is the \(y\)-coordinate of the vertex.

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, which can open upward or downward depending on the sign of the coefficient a. Understanding the shape and orientation of the parabola is essential for analyzing its properties.

Vertex of a Parabola

The vertex is the highest or lowest point on the parabola, representing either a maximum or minimum value of the quadratic function. It is given by the coordinates (h, k) in vertex form f(x) = a(x - h)² + k. Knowing the vertex helps determine the function’s range and the axis of symmetry.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since you can input any x-value. The range depends on the vertex and the direction the parabola opens: if it opens upward, the range is all y-values greater than or equal to the vertex’s y-coordinate; if downward, all y-values less than or equal to it.