Problem 23 - Precalculus

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x=-1\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with the vertex at the origin and directrix \(x = -1\) is \(y^2 = 4x\)

Step 1: Identify vertex, focus and directrix

The vertex of the parabola is given at the origin (0,0). The directrix of the parabola is marked by the equation \(x=-1\). Since a parabola is defined by its equidistance from a point called 'focus' and a line called 'directrix', the Focus (F) of the parabola is therefore the point (1,0).

Step 2: Apply the information to the general parabola equation

For a parabola that opens to the right the standard form of its equation is \((y - k)^2 = 4p(x - h)\), where (h,k) is the vertex of the parabola and 4p is the distance from the vertex to the focus or the directrix. Our vertex is at (0,0) and the distance from the vertex to the focus (p) is 1 unit. Therefore, our equation becomes \(y^2 = 4 * 1 * x\)

Step 3: Simplify the equation

Our final equation simplifies to \(y^2 = 4x\)

Key Concepts

Vertex of Parabola

Understanding the vertex of a parabola is critical for graphing and analyzing its properties. The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards respectively. In the given exercise, the vertex of the parabola is located at the origin, point (0,0). The coordinates of the vertex also determine the axis of symmetry of the parabola, which is a vertical line that passes through the vertex. The symmetry of the parabola about this axis is an essential property and aids in sketching the graph accurately. Whenever you're given a parabola problem, be sure to first identify the vertex as it sets the stage for the rest of your calculations.

Focus and Directrix of Parabola

The focus and directrix are two foundational concepts in understanding parabolas. The focus is a fixed point located inside the curve of the parabola, while the directrix is a line outside the curve that the parabola approaches but never intersects. The key property that defines a parabola is that every point on the curve is equidistant from the focus and the directrix. This characteristic helps in determining the shape and orientation of the parabola. In our case, with the directrix at \(x=-1\) and the vertex at the origin (0,0), the focus is then logically situated at (1,0), due to the equidistance principle. These components also influence the width of the parabola; the closer the focus is to the vertex, the narrower the parabola will be, and vice versa.

General Parabola Equation

The general equation of a parabola provides a framework for all possible parabolic shapes. It can be adapted to specific parabolas by altering its parameters based on the vertex, focus, and directrix. The equation takes the form \( (y - k)^2 = 4p(x - h) \) for parabolas that open left or right, where \( (h,k) \) is the vertex and \( p \) is the distance from the vertex to the focus (positive for rightward, negative for leftward). For parabolas that open upward or downward, the equation is \( (x - h)^2 = 4p(y - k) \), with \( p \) positive for upward and negative for downward openings. Applying this information, such as in the solved exercise, allows us to express the unique equation for any parabola, specifying its precise graph on the coordinate plane. Remember to simplify the equation to its standard form to make it easier to use in further analysis or graphing.