Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. $$ (y-2)^{2}=\frac{1}{4}(x-3) $$

Step-by-Step Solution

Answer

The vertex is (3, 2), the focus is \(\left(\frac{49}{16}, 2\right)\), and the directrix is \(x = \frac{47}{16}\).

Step 1: Identify the Form of the Parabola

The given equation \((y - 2)^2 = \frac{1}{4}(x - 3)\) is in the standard form of a parabola that opens either to the right or left: \((y - k)^2 = 4p(x - h)\). This suggests that the parabola is horizontal.

Step 2: Identify the Vertex

The vertex \((h, k)\) of the parabola is directly given by the equation. Here, \(h = 3\) and \(k = 2\), so the vertex is \((3, 2)\).

Step 3: Determine the Value of p

From the standard form \((y - k)^2 = 4p(x - h)\), we compare \(4p = \frac{1}{4}\). Solving for \(p\), we have \(p = \frac{1}{16}\).

Step 4: Locate the Focus

For a horizontal parabola \((y - k)^2 = 4p(x - h)\), the focus is \((h + p, k)\). Plugging in the values: \((3 + \frac{1}{16}, 2) = (\frac{49}{16}, 2)\).

Step 5: Find the Directrix

The directrix of a horizontal parabola is a vertical line given by \(x = h - p\). Substituting the values, \(x = 3 - \frac{1}{16} = \frac{47}{16}\).

Step 6: Sketch the Graph

To sketch the graph: plot the vertex at \((3, 2)\), the focus at \((\frac{49}{16}, 2)\), and the directrix as the vertical line \(x = \frac{47}{16}\). The parabola opens to the right since \(p\) is positive.

Key Concepts

Vertex

The vertex of a parabola is a crucial point that defines its shape and position on a graph. In the equation \((y - 2)^2 = \frac{1}{4}(x - 3)\), the vertex is found directly by observing the values of \(h\) and \(k\) in the general form \((y - k)^2 = 4p(x - h)\). Here, the vertex is \((3, 2)\).

• "h" value corresponds to the horizontal position of the vertex.
• "k" value corresponds to the vertical position of the vertex.

The vertex is considered the midpoint of the parabola, a balanced point where the curve begins to change direction. It can either be the highest or lowest point for vertical parabolas, or the furthest left or right point for horizontal parabolas. In this case, since the parabola opens horizontally, it means that the vertex is neither high nor low, but rather marks the turning point left or right.
Understanding the vertex's coordinates helps in sketching the graph of the parabola and represents an essential characteristic when analyzing its equation. The vertex shows where the symmetry occurs and helps define the complete shape of the parabola on a diagram.

Focus

In the parabola world, the focus is another essential part that works hand-in-hand with the vertex and directrix to define its precise structure. The focus is found inside the parabola, on the side where it opens.For the equation \((y - 2)^2 = \frac{1}{4}(x - 3)\), the parabola opens horizontally, and the focus is calculated using \((h + p, k)\). Here, after finding \(p = \frac{1}{16}\), the focus turns out to be \((\frac{49}{16}, 2)\).

• The focus of a horizontal parabola is always on the line specified by \(k\).
• For vertical openings, the focus shifts along the \(h\) line.

The focus is a special point because every point on the parabola is equidistant from it and the directrix. This property is fundamental in understanding how parabolas are structured.In terms of the parabolic shape, the balls thrown or projectiles launched in physics often follow a path that resembles a parabola having a focus as one of its defining elements.

Directrix

The directrix of a parabola also plays a significant role. It is an imaginary line that contributes to the parabola's shape, serving as a boundary that all points of the parabola are equidistant from, along with the focus. For our horizontal parabola equation \((y - 2)^2 = \frac{1}{4}(x - 3)\), the directrix is computed as \(x = h - p\). After calculating, it becomes \(x = \frac{47}{16}\).

• The directrix is a vertical line in the case of a horizontal parabola.
• In vertical parabolas, the directrix is horizontal.

Like the focus, the directrix helps in shaping the graph of the parabola, ensuring the symmetry and balance of the entire curve. It marks a critical limit that serves the balancing role opposite the focus.When sketching parabolas, knowing the directrix assists in understanding how wide or narrow a parabola appears, offering insight into its opening angle as well. Ultimately, the relationship of the directrix with the parabola emphasizes its symmetry, helping students grasp the geometric properties of parabolas easily.