Problem 48 - College Algebra with Modeling and Visualization

Question

Sketch a graph of the parabola. $$ (y+2)^{2}=2 x $$

Step-by-Step Solution

Answer

The parabola opens to the right with vertex at (0, -2) and focus at (0.5, -2).

Step 1: Identify the Vertex

The given equation \((y+2)^2 = 2x\) is not in the standard parabola form. However, it resembles \((x-h)^2 = 4p(y-k)\), which is a parabola that opens horizontally. The term \((y+2)^2\) indicates the form \((y-k)^2\), suggesting the vertex is at \((h, k)\). Rewrite the equation as \((y - (-2))^2 = 2(x - 0)\), so the vertex is \((0, -2)\).

Step 2: Determine the Orientation

Given the setup \((y+2)^2 = 2x\), the square is on \(y\), which means the parabola opens horizontally. Since the coefficient of \(x\) (i.e., 2) is positive, the parabola opens to the right.

Step 3: Find the Directrix and Focus

The equation \((y+2)^2 = 2(x)\) suggests \(4p = 2\). Solving gives \(p = \frac{1}{2}\). Since the parabola opens to the right, the focus is \(\left(0 + \frac{1}{2}, -2\right) = \left(\frac{1}{2}, -2\right)\) and the directrix is \(x = 0 - \frac{1}{2} = -\frac{1}{2}\).

Step 4: Sketch the Parabola

Plot the vertex at \((0, -2)\) on a coordinate grid. Mark the focus at \(\left(\frac{1}{2}, -2\right)\) and draw the vertical line \(x = -\frac{1}{2}\) as the directrix. Sketch the parabola opening to the right, ensuring it is symmetric about the horizontal line \(y = -2\).

Key Concepts

Vertex

The vertex of a parabola is a very crucial point as it represents the peak or the bottom of the curve, depending on its orientation. In our equation, \((y+2)^2 = 2x\), it closely resembles the standard form of a parabola that opens horizontally, \((y-k)^2 = 4p(x-h)\). This helps us identify the vertex position. By rewriting the given equation as \((y - (-2))^2 = 2(x - 0)\), we can deduce that the vertex is located at \((0, -2)\).

• The vertex \((h, k)\) serves as a pivotal point that divides the parabola into two symmetrical halves.
• In horizontal parabolas like ours, the vertex gives the axis of symmetry, which in this case is the line \(y = -2\).

Focus

To understand the focus, imagine the parabola as a mirror that directs lines to a single point, known as the focus. The parabola we have is in a form that suggests it opens to the right, where the focus lies along the axis of symmetry specified by \(k = -2\). In our particular case, from the equation \((y+2)^2 = 2(x)\), we find out that:

• The value of \(4p = 2\), which simplifies down to \(p = \frac{1}{2}\).
• Since the parabola opens to the right, you add \(p\) to the \(x\)-coordinate of the vertex to find the focus. Thus, the focus is situated at \(\left(\frac{1}{2}, -2\right)\).

The focus is essential for understanding the direction in which the parabola opens and how narrow or wide it is. All points on the parabola are equidistant from the focus and the directrix.

Directrix

The directrix of a parabola is a straight line that balances with the focus to define the shape and orientation of the parabola. For our horizontal parabola (which opens to the right), the directrix is a vertical line. According to our equation, again by knowing that \(p = \frac{1}{2}\):

• The directrix is \(x = h - p\). With our values, it calculates to \(x = 0 - \frac{1}{2} = -\frac{1}{2}\).
• This line provides a geometric relationship, where each point on the parabola is equidistant to the directrix and the focus. Therefore, it is important in ensuring the curvature follows consistent geometric properties.

It is noteworthy that the directrix influences how "open" or "closed" the parabola is by its distance from the vertex.

Graph Sketching

Sketching a parabola involves plotting its main components: the vertex, focus, and directrix. It is crucial because it visually conveys the orientation and the span of a parabola.Let's break the process down:

• First, you'll want to plot the vertex at \((0, -2)\) on your coordinate plane. This point acts as your starting guideline.
• Next, locate the focus at \(\left(\frac{1}{2}, -2\right)\) and draw this point. It demonstrates the parabola's center of symmetry along this axis.
• Then, draw the directrix at the vertical line \(x = -\frac{1}{2}\). This helps in visually understanding the parabola's width and direction.

By accurately marking such points and lines, your sketch will help illustrate how the parabola opens to the right, giving a clear image of its shape and direction.