Problem 33 - Algebra 2

Question

Identify the vertex, the focus, and the directrix of each graph. Then sketch the graph. $$ y^{2}-6 x=18 $$

Step-by-Step Solution

Answer

Vertex of the graph is at (3, 0), Focus is at (4.5, 0) and the directrix is the line x = 1.5. The parabola opens towards the right.

Step 1: Rewriting the given equation in standard form

The given equation is \(y^2 - 6x = 18\). We can rewrite it in the standard form \(y^2 = 4ax\), as follows: \[y^2 = 6x - 18\] which is equivalent to \[y^2 = 6(x - 3)\]. Now it's clear that the equation is in the form \(y^2 = 4ax\) with \(4a = 6\], thus \(a = 1.5\).

Step 2: Finding Vertex

The Vertex (V) of the parabola is given by (h, k), where the standard form is \( (y - k)^2 = 4a(x - h)\). From the equation \(y^2 = 6(x - 3)\), we can say that h = 3 and k = 0. Hence, the Vertex (V) = (3, 0).

Step 3: Finding the focus

The focus (F) lies on the line x=h+a. We know that h = 3 and a = 1.5. Hence, F = (3 + 1.5, 0) = (4.5, 0)

Step 4: Finding the directrix

The directrix is given by the line x = h - a. We know that h = 3 and a = 1.5. Hence, the equation of the directrix is x = 3 - 1.5 = 1.5.

Step 5: Sketch the graph

Plot the Vertex, Focus and the directrix line on the graph. As 'a' is positive, the parabola opens towards right. Draw the parabola such that it passes through the vertex and Focus, and remains equidistant from the Focus and directrix at any point on the parabola.

Key Concepts

Vertex

The vertex of a parabola is a key point where the parabola changes direction. In the context of a horizontal parabola, the vertex is the point where the parabola makes its sharpest turn. For any parabola expressed in the standard form \[(y - k)^2 = 4a(x - h)\]the vertex is at the point \((h, k)\).In the equation \(y^2 = 6(x - 3)\),the form matches \[(y - 0)^2 = 6(x - 3)\].From this, we can clearly see that \(h = 3\) and \(k = 0\),which tells us that the vertex is located at \((3, 0)\). The vertex is important because it is used as a starting point to plot the parabola. It also serves as a symmetry point, as parabolas are symmetric around a vertical line through the vertex (in this case for a horizontally opening parabola). Knowing the vertex helps us quickly understand the parabola's position relative to the coordinate axes.

Focus

The focus of a parabola is a special point used to construct the curve. Every point on a parabola is equidistant to the focus and the directrix, which is a line outside the parabola.For the horizontal parabola in the format \((y - k)^2 = 4a(x - h)\),the focus can be found at the point \((h + a, k)\),where \(a\) is the distance from the vertex to the focus. Given our equation \(y^2 = 6(x - 3)\),we can determine \(a = 1.5\)since \(4a = 6\).Thus, the focus for this specific parabola is \((3 + 1.5, 0) = (4.5, 0)\).The focus helps define the width and the direction the parabola opens. This point dictates the "direction" of the parabola relative to its vertex indicating whether it opens to the right or to the left based on its horizontal positioning.

Directrix

The directrix of a parabola is a crucial line that, along with the focus, helps in defining the parabola itself. All points on the parabola are equidistant from the focus and the directrix. While the focus is a point inside the parabola, the directrix is a line outside.In our example, using the format \((y - k)^2 = 4a(x - h)\),we determine the directrix using the equation \(x = h - a\).For our given parabola equation \(y^2 = 6(x - 3)\),we know that \(h = 3\) and \(a = 1.5\),which makes the directrix \(x = 3 - 1.5 = 1.5\).A directrix is imperative for plotting because it sets one of the foundational guidelines for creating the parabola. Knowing the position of the directrix allows us to visualize the parabola's extent to one side and helps distinguish between the space inside and outside of the parabola.