Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. y−1=(x−3)²

Verified step by step guidance

1. Rewrite the given equation in standard vertex form: \(y - 1 = (x - 3)^2\). This shows the vertex is at the point \((3, 1)\).
2. Identify the axis of symmetry, which is the vertical line passing through the vertex's x-coordinate. So, the axis of symmetry is \(x = 3\).
3. Find the y-intercept by substituting \(x = 0\) into the equation: \(y - 1 = (0 - 3)^2\), then solve for \(y\).
4. Find the x-intercepts by setting \(y = 0\) and solving the equation \(0 - 1 = (x - 3)^2\) for \(x\).
5. Determine the domain and range: The domain of any quadratic function is all real numbers, \((-\infty, \infty)\). Since the parabola opens upward and the vertex is the minimum point at \(y=1\), the range is \([1, \infty)\).

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, which is the highest or lowest point on the graph, depending on the parabola's orientation.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. For a quadratic in vertex form y = a(x - h)^2 + k, the axis of symmetry is the line x = h.

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: if it opens upward, the range is y ≥ k; if downward, y ≤ k, where k is the y-coordinate of the vertex.