Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = x^2 + 6x + 5

Verified step by step guidance

1. Rewrite the quadratic function in vertex form by completing the square: \( f(x) = x^2 + 6x + 5 \).
2. To complete the square, take the coefficient of \( x \), which is 6, divide it by 2 to get 3, and then square it to get 9.
3. Add and subtract 9 inside the function: \( f(x) = (x^2 + 6x + 9) - 9 + 5 \).
4. Rewrite the expression as a perfect square trinomial: \( f(x) = (x + 3)^2 - 4 \).
5. Identify the vertex from the vertex form \( f(x) = (x + 3)^2 - 4 \), which is \((-3, -4)\), and determine the axis of symmetry, domain, and range.

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general shape and properties of parabolas is essential for graphing and analyzing quadratic functions.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens downwards or upwards. For a quadratic function in standard form, the vertex can be found using the formula (-b/2a, f(-b/2a)). The vertex is crucial for determining the maximum or minimum value of the function and plays a key role in graphing the function accurately.

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, while the range depends on the vertex; if the parabola opens upwards, the range starts from the vertex's y-coordinate to infinity, and if it opens downwards, it extends from negative infinity to the vertex's y-coordinate.