Questions: Answer the question. Clearly label each answer. Show your work as needed. For the quadratic function, f(x)=(x+2)^2-3, find the following. (a) the vertex (b) the axis of symmetry (c) the domain (d) the range

Solution Steps

To solve the given quadratic function \( f(x) = (x+2)^2 - 3 \), we need to identify key features of the function:

(a) The vertex of a quadratic function in the form \( f(x) = (x-h)^2 + k \) is given by the point \( (h, k) \). Here, \( h = -2 \) and \( k = -3 \).

(b) The axis of symmetry for a quadratic function in vertex form is the vertical line \( x = h \). In this case, it is \( x = -2 \).

(c) The domain of any quadratic function is all real numbers, since there are no restrictions on the values that \( x \) can take.

(d) The range of the function depends on the direction of the parabola. Since the coefficient of the squared term is positive, the parabola opens upwards, and the range is \( y \geq -3 \).

The vertex of the quadratic function \( f(x) = (x + 2)^2 - 3 \) is given by the point \( (h, k) \) where \( h = -2 \) and \( k = -3 \). Therefore, the vertex is: \[ \text{Vertex} = (-2, -3) \]

The axis of symmetry for a quadratic function in vertex form is the vertical line \( x = h \). For this function, the axis of symmetry is: \[ \text{Axis of Symmetry} = x = -2 \]

The domain of any quadratic function is all real numbers, which can be expressed as: \[ \text{Domain} = \mathbb{R} \]

Since the parabola opens upwards (the coefficient of the squared term is positive), the range is determined by the vertex's \( y \)-coordinate. Thus, the range is: \[ \text{Range} = y \geq -3 \]

Final Answer