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)=2(x+2)²−1

Verified step by step guidance

1. Identify the given quadratic function: \(f(x) = 2(x+2)^2 - 1\). 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, \(h = -2\) and \(k = -1\), so the vertex is at \((-2, -1)\).
3. Find the axis of symmetry, which is the vertical line passing through the vertex: \(x = h\), so the axis of symmetry is \(x = -2\).
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 x-intercepts by setting \(f(x) = 0\) and solving for \(x\): solve the equation \$2(x+2)^2 - 1 = 0$ to find the points where the graph crosses the x-axis.

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is f(x) = 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 sign of a. For f(x) = 2(x+2)^2 - 1, the vertex is at (-2, -1).

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. Its equation is x = h, where h is the x-coordinate of the vertex. For the given function, the axis of symmetry is x = -2.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since the parabola extends infinitely left and right. The range depends on the vertex and the direction the parabola opens. Since a = 2 > 0, the parabola opens upward, so the range is all y-values greater than or equal to the vertex's y-coordinate, i.e., y ≥ -1.