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 quadratic function, the domain is all real numbers. In interval notation, this is expressed as \((-\infty, \infty)\).
2. Locate the vertex of the parabola from the graph or the function. The vertex form of the function is \(f(x) = -7(x+5)^2 + 7\), so the vertex is at \((-5, 7)\).
3. Determine the direction the parabola opens. Because the coefficient of the squared term is negative (\(-7\)), the parabola opens downward.
4. Find the range of the function. Since the parabola opens downward and the vertex is the highest point, the range includes all \(y\)-values less than or equal to the vertex's \(y\)-coordinate. So, the range is \((-\infty, 7]\).
5. Summarize the domain and range: Domain is all real numbers \((-\infty, \infty)\), and range is all real numbers less than or equal to 7, written as \((-\infty, 7]\).

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 x can take any value from negative to positive infinity.

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 downward, like f(x) = -7(x+5)^2 + 7, the range is all values less than or equal to the vertex's y-coordinate, since the vertex represents the maximum point.

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 based on the sign of 'a'. Here, the vertex is (-5, 7), and the parabola opens downward because a = -7.