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. f(x)=4−(x−1)²

Verified step by step guidance

1. Identify the given quadratic function: \(f(x) = 4 - (x - 1)^2\). Notice it is in vertex form \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
2. Determine the vertex by comparing: here, \(h = 1\) and \(k = 4\), so the vertex is at the point \((1, 4)\).
3. Find the axis of symmetry, which is the vertical line passing through the vertex's x-coordinate: \(x = 1\).
4. Calculate the y-intercept by evaluating \(f(0)\): substitute \(x = 0\) into the function to find the point where the graph crosses the y-axis.
5. Determine the domain and range: the domain of any quadratic function is all real numbers, \((-\infty, \infty)\). Since the parabola opens downward (because of the negative sign before the squared term), the range is all \(y\) values less than or equal to the vertex's y-value, so \((-\infty, 4]\).

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, represented by a parabola. For functions in the form f(x) = a(x - h)^2 + k, the vertex is at (h, k). It helps determine the shape and position of the parabola and is essential for sketching the graph.

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 = h, where h is the x-coordinate of the vertex. This line helps in graphing and understanding the parabola'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 parabola's direction; if it opens downward, the range is all values less than or equal to the vertex's y-coordinate, and if upward, all values greater than or equal to it.