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.

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insert step 1: Understand that a quadratic function is generally of the form f(x) = ax^2 + bx + c, where the graph is a parabola.

insert step 2: Recognize that the vertex form of a quadratic function is f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.

insert step 3: Since the parabola opens up, the coefficient 'a' in the vertex form is positive.

insert step 4: The domain of any quadratic function is all real numbers, which can be expressed as (-∞, ∞).

insert step 5: The range of the function, since the parabola opens upwards, starts from the y-coordinate of the vertex, k, and goes to infinity, expressed as [k, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open either upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of quadratic functions is essential for determining their domain and range.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards, respectively. For a parabola that opens upwards, the vertex represents the minimum value of the function. The coordinates of the vertex can be found using the formula (-b/2a, f(-b/2a)), which is crucial for identifying the range of the quadratic function.

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, but the range is determined by the vertex and the direction the parabola opens. Understanding these concepts is vital for accurately describing the behavior of quadratic functions.

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