Consider the graph of each quadratic function.
a) Give the domain and range.

Verified step by step guidance

1. Identify the domain of the quadratic function. Since it is a parabola that opens upwards and extends infinitely left and right, the domain includes all real numbers. In interval notation, this is expressed as \((-\infty, \infty)\).
2. Locate the vertex of the parabola on the graph. The vertex is given as \((-13, -15)\), which represents the minimum point of the parabola because it opens upwards.
3. Determine the range of the function. Since the parabola opens upwards, the function values (y-values) start at the vertex's y-coordinate and go to positive infinity. Therefore, the range is all real numbers greater than or equal to \(-15\).
4. Express the range in interval notation as \([-15, \infty)\), indicating that the function's output values start at \(-15\) and increase without bound.
5. Summarize the domain and range: Domain is all real numbers \((-\infty, \infty)\), and range is \([-15, \infty)\).

Key Concepts

Domain of a Quadratic Function

The domain of a quadratic function includes all possible input values (x-values) for which the function is defined. Since quadratic functions are polynomials, their domain is all real numbers, meaning any real number can be substituted for x.

Range of a Quadratic Function

The range of a quadratic function is the set of all possible output values (y-values). For a parabola opening upwards, the range starts at the vertex's y-coordinate and extends to positive infinity. For the given function, the minimum value is -15, so the range is y ≥ -15.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = a(x-h)^2 + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and determine the direction the parabola opens. Here, the vertex is (-13, -15), and since a = 1 > 0, the parabola opens upwards.