Question

Identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex \((\mathbf{0}, \mathbf{0})\). $$y=-\frac{1}{4}(x+2)^2+1$$

Short answer

Vertex: (-2,1), Focus: (-2,0), Directrix: y=0, Axis of Symmetry: x=-2. Transformations: Shift 2 units to the left, 1 unit upward and reflection over the x-axis.

Step-by-step solution

Identify the vertex

The vertex is the point where the graph of the parabola intersects its axis of symmetry. In this equation, the standard form is \(y=a(x-h)^2 + k\), where \((h,k)\) are the coordinates of the vertex. Compared with its standard form, we see that \(h = -2\) and \(k = 1\). Therefore, the vertex is at \((-2,1)\).

Identify the focus

The focus lies within the parabola, directly across from the directrix. Distance from the vertex to the focus and is \(p = -1/4a\), where \(a\) is the coefficient of the \(x^2\) term which is -1/4 in the given equation. Thus, \(p = 1\). Here, the negative sign shows the parabola opens downwards. Since, the parabola is vertical, then the focus will be at \((-2,1-p)\), which is \((-2,0)\).

Identify the directrix

The directrix is a line that lies opposite to the focus. For a vertical parabola, directrix is a horizontal line \(y = k-p\), which is \(y = 1 - 1\). So, the directrix is \(y=0\).

Identify the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex, and it has the equation \(x = h\), thus here it is \(x = -2\).

Describe the transformations of the graph

From the standard equation \(y = -\frac{1}{4}x^2\) to \(y = -\frac{1}{4}(x+2)^2 +1\), there's a horizontal shift two units to the left and a vertical shift one unit upward. The negative sign indicates a reflection over the x-axis.

Key concepts

Vertex

The vertex of a parabola is a crucial point as it represents the peak or the bottommost part of the curve. For parabolas expressed in the form \(y = a(x-h)^2 + k\), the vertex can be quickly identified as \((h, k)\).
In the given equation \(y=-\frac{1}{4}(x+2)^2+1\), comparing with the standard form reveals that \(h = -2\) and \(k = 1\). Hence, the vertex of this parabola is located at the point \((-2, 1)\).
This point is significant as it also indicates the point from which the parabola opens, either upwards or downwards, depending on the sign of \(a\). Here, since \(a\) is negative, the parabola opens downwards.

Focus

The focus of a parabola lies inside the curve itself and serves as a fixed point that helps define the entire curve. It is known that the distance from the vertex to the focus is described by \(p = \frac{1}{4|a|}\), where \(a\) is the coefficient of the \(x^2\) term.
For the equation \(y = -\frac{1}{4}(x+2)^2+1\), \(a = -\frac{1}{4}\). Consequently, \(p = 1\). Since the parabola opens downward, the focus will be below the vertex.
This means the focus is situated at \((-2, 1-p)\) or \((-2, 0)\). This point serves as an anchor that ensures the unique reflective properties of parabolas.

Directrix

The directrix is a line that stands in opposition to the focus and plays a significant role in defining the parabola. For a vertical parabola, the directrix will be a horizontal line. Its equation can be determined as \(y = k - p\).
From the parabola equation \(y=-\frac{1}{4}(x+2)^2+1\), we have already found that \(p = 1\) and \(k = 1\). Thus, the equation of the directrix is \(y = 1 - 1\) or simply \(y = 0\).
This line helps to maintain the geometric integrity of the parabola and serves as a boundary opposite to the focus.

Axis of Symmetry

The axis of symmetry of a parabola is an invisible line that divides the parabola into two perfect mirror-image halves. This line is essential for understanding the parabola's shape and orientation.
For parabolas represented by \(y = a(x-h)^2 + k\), the axis of symmetry can be derived as the vertical line \(x = h\).
Considering the given equation \(y=-\frac{1}{4}(x+2)^2+1\), we know \(h = -2\). Therefore, the axis of symmetry is given by the equation \(x = -2\). This line goes through the vertex and ensures the parabola is symmetric relative to it.
Understanding this symmetry is key for graphing and analyzing the properties and behavior of the parabola.