Problem 1

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$ y^{2}=4 x $$

Step-by-Step Solution

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Answer

Focus: (1,0). Directrix: x = -1.

1Step 1: Identify the form of the equation

The given equation is $ y^2 = 4x $, which is a parabola that opens horizontally. This matches the standard form $ y^2 = 4px $, where $ p $ determines the distance from the vertex to the focus and the directrix.

2Step 2: Determine the value of p

By comparing $ y^2 = 4x $ to the standard form $ y^2 = 4px $, we see that $ 4p = 4 $. Solving for $ p $, we get $ p = 1 $.

3Step 3: Find the vertex of the parabola

For $ y^2 = 4px $, the vertex is at the origin $(0,0)$.

4Step 4: Locate the focus

Since the parabola opens to the right and $ p = 1 $, the focus is located $ p $ units to the right of the vertex. Thus, the focus is at $ (1,0) $.

5Step 5: Write the equation of the directrix

The directrix is a vertical line $ p $ units to the left of the vertex. Given $ p = 1 $, the equation of the directrix is $ x = -1 $.

6Step 6: Make a sketch

Draw a horizontal parabola opening to the right with the vertex at $(0,0)$. Plot the focus at $(1,0)$ and the directrix line at $ x = -1 $.

Key Concepts

Focus of a ParabolaDirectrix of a ParabolaVertex of a ParabolaEquation of a Parabola

Focus of a Parabola

The focus of a parabola is a specific point located in relation to the vertex. It plays an integral role in defining the shape and direction of the parabola. In the equation of a parabola that opens horizontally, like in the form $ y^2 = 4px $, the focus is found $ p $ distance from the vertex. Here, $ p $ represents the distance from the vertex to the focus along the axis of symmetry of the parabola.
For example, in the parabola described by the equation $ y^2 = 4x $:

We find $ p = 1 $ by comparing it to the standard form $ y^2 = 4px $.
The focus is then located at $(p, 0) = (1, 0)$ because the parabola opens to the right.

Therefore, understanding the location of the focus helps in sketching and analyzing the parabola's structure.

Directrix of a Parabola

The directrix of a parabola is a crucial component that, together with the focus, defines the parabola's properties. It is a straight line that is perpendicular to the axis of symmetry of the parabola.
For parabolas in the form $ y^2 = 4px $, like our example $ y^2 = 4x $, the directrix is located at a distance $ p $ units away from the vertex, opposite the focus. This means:

If the parabola opens to the right, as in $ y^2 = 4x $, the directrix will be a vertical line to the left of the vertex.
So, for $ p = 1 $, the directrix is at $ x = -1 $ since it is $ p $ units left of the vertex $(0, 0)$.

The directrix, together with the focus, ensures that any point on the parabola is equidistant from the focus and the directrix.

Vertex of a Parabola

The vertex of a parabola is the point where the parabola changes direction and is either the highest or lowest point, depending on its orientation. In equations like $ y^2 = 4px $, the vertex often serves as the reference point for locating both the focus and the directrix. For the equation $ y^2 = 4x $, the vertex is straightforward to determine:

The vertex is at the origin, $(0,0)$, because there is no constant term shifting the parabola from this position.
This vertex is the midpoint between the focus $(1,0)$ and the directrix $ x=-1 $, serving as the base from which the parabola expands.

Being able to identify the vertex is crucial for graphing the parabola and understanding its orientation in space.

Equation of a Parabola

The equation of a parabola gives us critical information about its shape and direction. Parabolas that open sideways, like in the form $ y^2 = 4px $, have an axis of symmetry that is horizontal. The distance $ p $ determines how wide or narrow the parabola is and its direction from the vertex.
In our example, the equation:

$ y^2 = 4x $ indicates a horizontal parabola opening to the right.
The coefficient 4 represents $ 4p $ where $ p $ is the distance from the vertex to the focus and directrix, calculated as $ p = 1 $.

Grasping the equation of a parabola is essential for predicting and understanding how the parabola behaves.

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