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

Verified step by step guidance

1. Identify the given quadratic function: \(f(x) = (x - 2)^2\). This 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 \(f(x) = (x - 2)^2\) to the vertex form. Here, \(h = 2\) and \(k = 0\), so the vertex is at \((2, 0)\).
3. Determine the axis of symmetry, which is the vertical line that passes through the vertex. The axis is \(x = 2\).
4. State the domain of the function. Since this is a quadratic function, the domain is all real numbers, written as \((-\infty, \infty)\).
5. Find the range by considering the vertex and the direction the parabola opens. Since \(a = 1 > 0\), the parabola opens upward, so the range is \([0, \infty)\).

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. For functions in the form f(x) = (x - h)^2 + k, the vertex is at (h, k). In this question, the vertex is at (2, 0), indicating the parabola opens upward from this point.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and has the equation x = h for a quadratic in vertex form. Here, the axis of symmetry is x = 2, reflecting the parabola's symmetry about this line.

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 parabola's direction; for f(x) = (x - 2)^2, the parabola opens upward with a minimum at y = 0, so the range is y ≥ 0.