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 \((\mathbf{0}, \mathbf{0})\). $$x=-3(y+4)^2+2$$

Short answer

The vertex is at (-4,2), the focus is approximately at (1.91667,-4), the equation of the directrix is \(x \approx 2.08333\), and the axis of symmetry is \(y=-4\). The graph of the parabola has been reflected with respect to the y-axis, then stretched by a factor of 3, and finally translated 4 units down and 2 units to the right.

Step-by-step solution

Identify the standard form

The given equation, \(x=-3(y+4)^2+2\), is of the form \(x=a(y-h)^2+k\), where (h,k) is the vertex of the parabola.

Find the Vertex

Given this standard form, the vertex of the parabola can be identified as (h,k) or in this case (-4,2).

Identify the direction of the parabola

Since the 'y' term is squared, the parabola is horizontal. Given that the coefficient of the \((y+4)^2\) term is negative, this parabola opens left.

Calculate the Focus and Directix

The distance from the vertex to the focus (or to the directrix) is given by \(\frac{1}{4|a|}\). Here, \(a=-3\), so the distance is \(\frac{1}{4*3}\), which equals \(\frac{1}{12}\). As the parabola opens to the left, subtract this value from the x-coordinate of the vertex to get the focus, and add this value to the x-coordinate of the vertex to get the directrix. The focus is thus located at \((2-\frac{1}{12}, -4)\), or approximately (1.91667, -4), and the equation of the directrix is \(x=2+\frac{1}{12}\), or approximately x=2.08333.

Determine Axis of Symmetry

The axis of symmetry for a horizontal parabola is a horizontal line through the vertex, so in this case, the axis is \(y=-4\).

Describe the transformations of the graph

The graph of the parabola has been translated 4 units down and 2 units to the right from the origin (or the vertex of the standard parabola). Additionally, the negative in front of the 3 in the equation indicates a reflection in the y-axis, and the 3 itself is a vertical stretch by a factor of 3.

Key concepts

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. It serves as the cornerstone for understanding a parabola's shape and positioning. In our equation, the vertex is identified directly from the standard form of a parabola's equation, which for a horizontal parabola is given by \(x=a(y-h)^2+k\). Here, \(h\) and \(k\) represent the coordinates of the vertex.

From the given equation \(x=-3(y+4)^2+2\), we can see that the vertex is at the point (-4, 2). This finding is crucial as it helps us visualize the parabola and serves as a starting point to deduce other characteristics.

Direction of a Parabola

Understanding the direction in which a parabola opens is key to plotting its graph. In general, if the \(y\)-term is squared and the coefficient of that term is positive, the parabola opens to the right. Conversely, if the coefficient is negative, as in our equation, the parabola opens to the left.

This comprehension provides insight into the orientation of the parabola on the Cartesian plane. With a negative coefficient of \(-3\) in front of our \((y+4)^2\) term, we determine that the parabola in question opens leftward from its vertex.

Focus and Directrix of a Parabola

The focus of a parabola is a fixed point that, along with the directrix, helps to define the curve. Every point on the parabola is equidistant to both the focus and the directrix. The formula to find the distance from the vertex to the focus or directrix is given by \(\frac{1}{4|a|}\), where \(a\) is the coefficient from the standard form of the parabola's equation.

For our example, since \(a=-3\), this distance is \(\frac{1}{12}\). Because the parabola opens to the left, we calculate the focus by subtracting \(\frac{1}{12}\) from the x-coordinate of the vertex, and we calculate the directrix by adding \(\frac{1}{12}\) to the x-coordinate of the vertex. Consequently, we find the focus at approximately (1.91667, -4) and the directrix to be the vertical line with the equation \(x=2.08333\).

Axis of Symmetry

The axis of symmetry is an imaginary line that divides the parabola into two mirror-image halves. For a horizontal parabola, this axis is always a horizontal line. The significance of the axis of symmetry lies in its representation as the vertical line that passes through the vertex, helping us establish symmetry in the parabola's graph.

In our problem, with a horizontal parabola that has a vertex at (-4, 2), the axis of symmetry is the horizontal line \(y=-4\). This aids in graphically plotting the parabola and understanding its reflective properties.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. These transformations help describe how the graph of a parabola differs from its standard position.

In the example of \(x=-3(y+4)^2+2\), the given parabola has been translated 4 units down and 2 units to the right, relative to the vertex of the standard equation with a vertex at the origin (0, 0). Additionally, the negative sign indicates a reflection across the y-axis. Lastly, the coefficient of 3 represents a vertical stretch by a factor of 3. Understanding these transformations allows us to visualize the parabola's graph even before we start plotting it.