Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = x² + 6x + 5

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = x^2 + 6x + 5\).
2. Rewrite the quadratic function in vertex form by completing the square: Start with \(f(x) = x^2 + 6x + 5\). To complete the square, take half of the coefficient of \(x\) (which is 6), divide by 2 to get 3, then square it to get 9. Add and subtract 9 inside the function to maintain equality.
3. Express the function as \(f(x) = (x^2 + 6x + 9) + 5 - 9\), which simplifies to \(f(x) = (x + 3)^2 - 4\). This form reveals the vertex clearly.
4. From the vertex form \(f(x) = (x + 3)^2 - 4\), identify the vertex as \((-3, -4)\) and the axis of symmetry as the vertical line \(x = -3\).
5. Determine the domain and range: The domain of any quadratic function is all real numbers, so \((-\infty, \infty)\). Since the parabola opens upward (coefficient of \(x^2\) is positive), the range is \([-4, \infty)\), starting from the vertex's \(y\)-value.

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. It can be found using the formula (-b/2a, f(-b/2a)) when the function is in standard form f(x) = ax^2 + bx + c. The vertex helps in graphing and understanding the function's behavior.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is x = -b/(2a) for a quadratic function in standard form. This line is crucial for graphing and analyzing the symmetry of the parabola.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward (a > 0), the range is all values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate.