Problem 42 - Precalculus

Question

Find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}-4 y-4 x=0$$

Step-by-Step Solution

Answer

The vertex of the given parabola is at (-1,2), the focus is at (0,2) and the equation of the directrix is x=-2. The parabola opens to the right.

Step 1 : 'Rearrange the given equation'

Let's rewrite the given equation, \(y^{2}-4 y-4 x = 0\), in the form, \( (y - h)^2 = 4a(x - k) \), where y is the square term. We take '4' common from y terms to get, \(y^{2}-4 y = 4x, which will become, (y - 2)^2 = 4x + 4. Therefore, the standard form is (y - 2)^2 = 4(x + 1).

Step 2: 'Identify the vertex'

The vertex is given by (h, k), and in our standard form, this becomes (-1, 2). Therefore, the vertex of the parabola is (-1, 2).

Step 3 : 'Calculate the focus'

In the standard form of a parabola, 'a' is related to the distance from the vertex to the focus which is 1/4a away from the vertex along the axis of symmetry (in x direction as it's horizontally oriented). In our equation, 4a=4, therefore, a=1. Since the parabola opens to the right, to get the focus we have to add 'a' to the x-coordinate of the vertex. Therefore, the focus is (-1+1, 2) = (0, 2).

Step 4: 'Find the equation of the directrix'

The directrix is a vertical line 'a' units left from the vertex as the parabola opens to the right. The equation of the vertical line is of form x = k. So, the directrix is x = -1 - a = -2.

Step 5: 'Sketch the parabola'

Plot the points for the vertex, focus and the directrix line on a graph. The vertex is at (-1, 2), the focus is at (0, 2) and the directrix is the line x = -2. From these, we can sketch the parabola which opens to the right. It's vertex is at (-1,2), it contains the focus point and is reflected away from the directrix line.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is one of the key features that describes its shape and position. It is the point where the parabola turns or curves. Knowing the coordinates of the vertex helps in understanding the symmetry and orientation of the parabola.

For a given quadratic equation in the standard form y = ax^2 + bx + c the vertex can be found using the formula:

• The x-coordinate is \(x = -\frac{b}{2a}\).
• The y-coordinate is \(f(x) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c\).

In the exercise, after rearranging the equation to its standard form, the vertex was identified as (-1, 2). This means the parabola is centered around this point. Understanding where the vertex falls in relation to other features like the focus and directrix is crucial for graphing and interpreting the parabola's behavior.

Focus of a Parabola

The focus of a parabola is an internal point that, together with the directrix, defines the parabola's unique distance properties. Every point on the parabola is equidistant from the focus and the directrix.

For parabolas oriented along the x-axis, as shown in the exercise, the focus can be calculated using the vertex form equation y = (x-h)^2 = 4a(y-k)

• Here, \(a\) is the distance from the vertex to the focus.
• The focus for a horizontally opening parabola is at \((h+a, k)\).

In our example, with \(a=1\), the focus was calculated as (0, 2). This focus helps determine the direction and shape of the parabola, indicating how it opens and its depth. Knowing the focus is vital for constructing the parabola’s path on a graph.

Directrix of a Parabola

The directrix of a parabola is a line used in defining the curve along with the focus. It serves as a reference line that, along with the focus, maintains the parabola’s property of equidistance.

For a parabola oriented along the x-axis, the directrix is a vertical line, and its equation is usually of the formx=k-a

• Where \(k\) is the x-coordinate of the vertex, and \(a\) is the distance from the vertex to the focus.

In the context of the given problem, since \(a=1\), the directrix was determined to be the line x = -2. It serves as a guideline for sketching the parabola and is essential in verifying the positioning and orientation of the parabola on a coordinate plane.