In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

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1. Rewrite the given quadratic function in standard vertex form: \( f(x) = (x + 4)^2 - 2 \). Here, the vertex form is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the equation, the vertex is \( (-4, -2) \).
2. Identify the axis of symmetry. The axis of symmetry is a vertical line that passes through the x-coordinate of the vertex. Therefore, the equation for the axis of symmetry is \( x = -4 \).
3. Find the x-intercepts by setting \( f(x) = 0 \) and solving for \( x \): \( 0 = (x + 4)^2 - 2 \). Add 2 to both sides to get \( (x + 4)^2 = 2 \). Take the square root of both sides, remembering to include both the positive and negative roots, to get \( x + 4 = \pm \sqrt{2} \). Finally, solve for \( x \) by subtracting 4 from both sides: \( x = -4 \pm \sqrt{2} \).
4. Find the y-intercept by setting \( x = 0 \) in the function: \( f(0) = (0 + 4)^2 - 2 \). Simplify to find the y-intercept as a point \( (0, y) \).
5. Determine the domain and range of the function. Since this is a quadratic function, the domain is all real numbers, \( (-\infty, \infty) \). The range is determined by the vertex and the direction of the parabola. Since the parabola opens upwards (the coefficient of \( (x + 4)^2 \) is positive), the range is \( [-2, \infty) \), starting at the y-coordinate of the vertex.

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function in the form f(x) = a(x - h)^2 + k is the point (h, k) where the parabola changes direction. It represents either the maximum or minimum value of the function, depending on the sign of 'a'. For the given function f(x) = (x + 4)^2 - 2, the vertex is at (-4, -2).

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It can be expressed as x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry for f(x) = (x + 4)^2 - 2 is x = -4.

Domain and Range of Quadratic Functions

The domain of a quadratic function is all real numbers, as there are no restrictions on the input values. The range, however, depends on the vertex; if the parabola opens upwards, the range starts from the y-coordinate of the vertex to positive infinity. For f(x) = (x + 4)^2 - 2, the range is [-2, ∞) since the vertex is the minimum point.