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)=(x−1)²+2

Verified step by step guidance

1. Identify the given quadratic function: \(f(x) = (x - 1)^2 + 2\). This is in vertex form, \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
2. Find the vertex by comparing: here, \(h = 1\) and \(k = 2\), so the vertex is at the point \((1, 2)\).
3. Determine the axis of symmetry, which is the vertical line passing through the vertex. The equation is \(x = h\), so here it is \(x = 1\).
4. Find the y-intercept by evaluating \(f(0)\): substitute \(x = 0\) into the function to get \(f(0) = (0 - 1)^2 + 2\). This gives the point where the graph crosses the y-axis.
5. Find the x-intercepts by setting \(f(x) = 0\) and solving for \(x\): solve \((x - 1)^2 + 2 = 0\). Since the square term plus 2 cannot be zero for real \(x\), check if there are real x-intercepts or not.
6. Determine the domain and range: the domain of any quadratic function is all real numbers, \((-\infty, \infty)\). The range depends on the vertex and the direction of the parabola (opens upward since the coefficient of the squared term is positive). The minimum value is \(k = 2\), so the range is \([2, \infty)\).

Key Concepts

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, represented by the point (h, k) in vertex form f(x) = (x - h)^2 + k. It indicates the parabola's maximum or minimum value and is essential for sketching the graph accurately.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing 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 helps in understanding the parabola's symmetry and graphing.

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, the range is all y-values greater than or equal to the vertex's y-coordinate; if downward, all y-values less than or equal to it. This helps describe the function's output values.