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 \((\mathbf{0}, \mathbf{0})\). $$x=4(y+5)^2-1$$

Short answer

The vertex of the parabola is at (-1,-5), the focus is at (0,-5), the directrix is \(x=-2\), and the axis of symmetry is \(y=-5\). The parabola is shifted 1 unit to the left and 5 units down from the origin.

Step-by-step solution

Identify the Vertex

In the standard form of a parabola that opens either to the right or left, any terms within the square represent a shift vertically and the final number a shift horizontally. In the equation \(x=4(y+5)^2-1\), it means there is a shift 5 units down and 1 unit to the left. Therefore, the vertex is at (-1, -5).

Identify the Focus and Directrix

For the basic form \(x = 4py\), the value of \(p\) determines the location of the focus and the directrix. Here, \(p\) is 1, we can determine that the focus will be one unit to the right of the vertex thus, the focus is at (0, -5). Similarly, the directrix will be one unit to the left of the vertex, thus, the directrix is at \(x=-2\).

Determine the Axis of Symmetry

The axis of symmetry for a parabola that opens to the sides is the line \(y = k\), where \(k\) is the y-coordinate of the vertex. Thus, for this equation, the axis of symmetry is \(y = -5\).

Identify the Transformations

Comparing to the basic form \(x = 4py\), which is centered at the origin, the given parabola is translated 1 unit left and 5 units down. Hence, the transformations are 'shift left 1 unit, and shiftdown 5 units'.

Key concepts

Vertex of a Parabola

The vertex of a parabola represents the highest or lowest point on the graph, depending on whether it opens upwards or downwards for vertical parabolas, or leftwards or rightwards for horizontal parabolas. In the equation x=4(y+5)^2-1, by analyzing the squared term and constant, we can determine the vertex's location. Here, the equation suggests a movement 5 units down and 1 unit to the left from the origin, placing the vertex at (-1, -5). Understanding the vertex helps in sketching the parabola's rough shape and locating its peak or trough on a graph.

Focus and Directrix

The focus of a parabola is a specific point from which distances to any point on the parabola are theoretically equal to the distances from that point to a line called the directrix. For the equation given, the focus is determined to be one unit to the right of the vertex, which results in the focus at (0, -5). Conversely, the directrix is a vertical line that lies one unit to the left of the vertex, or at x=-2. This relationship between the focus and directrix maintains the 'reflective property' of parabolas, ensuring that light or sound waves entering the parabola parallel to its axis of symmetry will reflect through the focus.

Axis of Symmetry

The axis of symmetry in a parabola is a vertical or horizontal line that divides the parabola into two mirror-image halves. For the given parabola x=4(y+5)^2-1, the axis of symmetry is horizontal and it corresponds to the y-coordinate of the vertex. Hence, for our equation, the axis of symmetry is the line y = -5. It acts like a mirror, and every point on the parabola has a mirrored counterpart across this axis. Identifying the axis of symmetry is valuable when plotting the curve and ensuring balance in the graph's representation.

Graph Transformations

Graph transformations involve moving the graph of a standard equation around the coordinate plane without changing its shape. The standard equation of a parabola with vertex at the origin is x=4py. Any adjustments in the form of additions or subtractions to the variables indicate a shift in the parabola's position. In the solution provided, the parabola described by x=4(y+5)^2-1 is a transformation of the standard parabola - it has been translated 1 unit to the left and 5 units down. By recognizing these translations, students can more accurately graph the parabola starting from the basic shape centered at the origin.