Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = -(1/2)(x + 1)² - 3

Verified step by step guidance

1. Identify the given quadratic function: \(f(x) = -\frac{1}{2} (x + 1)^2 - 3\). Notice it is in vertex form, which is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
2. Find the vertex by comparing the given function to the vertex form. Here, \(h = -1\) (note the sign change inside the parentheses) and \(k = -3\). So, the vertex is \((-1, -3)\).
3. Determine the axis of symmetry, which is the vertical line that passes through the vertex. The axis is given by \(x = h\), so here it is \(x = -1\).
4. State the domain of the quadratic function. Since it is a parabola that opens either up or down, the domain is all real numbers, expressed as \((-\infty, \infty)\).
5. Find the range by considering the direction the parabola opens. Since \(a = -\frac{1}{2}\) is negative, the parabola opens downward, so the range is all \(y\) values less than or equal to the vertex's \(y\)-coordinate, expressed as \((-\infty, -3]\).

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, given in vertex form as ƒ(x) = a(x - h)^2 + k, where (h, k) is the vertex. It represents the point where the parabola changes direction, and identifying it 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. For a quadratic in vertex form, the axis of symmetry is x = h, where h is the x-coordinate of the vertex. It is essential for graphing and analyzing the function's symmetry.

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 and the direction the parabola opens: if it opens upward (a > 0), the range is all y-values greater than or equal to the vertex's y-coordinate; if downward (a < 0), the range is all y-values less than or equal to the vertex's y-coordinate.