Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=(y-2)^{2}-1$$

Step-by-Step Solution

Answer

A horizontal parabola centered at (-1, 2) opens to the right.

Step 1: Identify the Conic Section

The given equation is in the form of \(x = (y-k)^2 + h\), which is the equation of a parabola with a horizontal axis. Here, \(h = -1\) and \(k = 2\). This is a parabola opening to the right.

Step 2: Determine Vertex and Axis of Symmetry

The vertex of the parabola is at \( (h, k) \) which is \((-1, 2)\). The axis of symmetry is horizontal and given by the line \(y = 2\).

Step 3: Identify Direction and Width of Opening

Since the coefficient of \((y-2)^2\) is positive, the parabola opens to the right. The standard parabola \(x = y^2\) has a vertex at the origin and opens to the right as well, indicating the shape is generally the same, just shifted.

Step 4: Plot the Vertex

Begin by plotting the vertex on the coordinate plane at the point \((-1, 2)\). This point is crucial because it defines where the parabola will originate.

Step 5: Sketch the Parabola

Since the parabola opens to the right, plot a few additional points on each side of the vertex to guide your drawing. For example, when \(y = 0\), then \(x = 3\) (since \(x = (0 - 2)^2 - 1 = 3\)), and when \(y = 4\), \(x = 3\) as well.

Step 6: Draw the Parabola

Connect the points smoothly to form a 'U' shaped curve opening to the right. Make sure the curve passes through the additional points and keep it symmetric about the line \(y = 2\).

Key Concepts

Parabola

A parabola is a type of conic section that looks like a U-shaped curve on a graph. While a circle is round and closed, a parabola is open and it can face up, down, left, or right. In this particular exercise, the equation \(x = (y-2)^2 - 1\) describes a parabola that opens to the right. We can tell this because the variable \(y\) is squared rather than \(x\). This indicates the parabola is oriented horizontally. In general, the equation of a parabola can vary, but the key to identifying it is the presence of a squared term.

• The leading variable indicates the direction of the opening.
• A positive coefficient means the parabola opens in a 'positive' direction - right for \(x\) and up for \(y\).

When graphing, it is essential to know how the parabola fits into the entire coordinate plane. Understanding its structure helps in predicting its shape, size, and the location of its key components.

Vertex

The vertex of a parabola is a crucial point on its graph. It is perceived as the peak or the lowest point of the parabola, depending on its orientation. For the equation \(x = (y-2)^2 - 1\), the vertex can be identified at the point \((-1, 2)\).

• The vertex is calculated using \(h\) and \(k\) from the equation \(x = (y - k)^2 + h\).
• Here, \(h = -1\) and \(k = 2\), making the vertex located at \((-1, 2)\).

This is the point where the parabola starts, and the symmetry of the curve revolves around this particular spot. It's important for graphing because all other points on the parabola are related in distance to the vertex.

Axis of Symmetry

The axis of symmetry in a parabola is a line that vertically divides the parabola into two mirror-image halves. In simpler terms, it's like drawing an invisible line through the parabola, ensuring both sides are evenly balanced. For our equation, the axis of symmetry is horizontal and it is represented by the line \(y = 2\).

• This axis is determined by the value of \(k\) in the equation \((y-k)^2\).
• In this case, \(y = 2\) means that the parabola is symmetrically balanced on this horizontal line.

The axis of symmetry is important as it provides a reference line to plot additional points. It assures that the distances on either side of the axis are mirrored, aiding in accurately sketching and understanding the parabola's structure.