Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: Determine the direction of the Parabola

Since the directrix is \(y = -1\) and the vertex is the origin, it is clear that the parabola opens downwards. A downward opening parabola has the standard form of \(x^2 = -4ay\). The sign in front of the 4a tells us the direction of the opening.

Step 2: Identify the focus and the directrix

Next, identify the focus and the directrix. Since the vertex is at the origin, and we know that the parabola is opening downwards, the focus will be at \(y = -a\). The directrix will be \(y = a\). Because the directrix is given as \(y = -1\), it can be concluded that \(a = 1\).

Step 3: Create the Standard Form of The Equation

Now, substitute the value of \(a\) into the standard form of our equation \(x^2 = -4ay\). You get, \(x^2 = -4*1*y\), which simplifies to \(x^2 = -4y\).

Key Concepts

Standard Form

The standard form of a parabola is essential to understanding its equation and orientation. It is given by the equation \[y = ax^2 + bx + c \] for parabolas opening up or down, or \[x = ay^2 + by + c \] for parabolas opening left or right.
In our exercise, since the parabola opens downwards, the applicable standard form becomes \[x^2 = -4ay \].
This specific form indicates:

• The parabola’s axis of symmetry is vertical.
• If the coefficient of \(x^2\) is negative, it opens downwards.

Understanding the standard form allows us to deduce other elements such as the focus and directrix. It provides a clear framework to analyze and graph parabolas successfully.

Directrix

The directrix of a parabola plays a pivotal role in defining its shape. It is a line parallel to the axis of symmetry of the parabola. The relationship between the directrix and the parabola dictates the proximity and orientation of the curve.
In our exercise, the directrix is given as \(y = -1\), which tells us that the parabola opens downwards.
This information:

• Assists in determining the position of the vertex relative to the focus.
• Is essential in identifying the correct standard form of the equation.

The directrix is not part of the parabola but rather a guideline that helps in constructing the parabola's shape accurately.

Vertex

The vertex of a parabola is a point where the curve changes direction. It is considered the turning point and is either the highest or lowest point on the graph depending on the parabola's orientation.
In the given exercise, the vertex is at the origin (0, 0).
Knowing the vertex:

• Helps in plotting the parabola precisely.
• Acts as a starting point for further calculations regarding the focus and directrix.

The position of the vertex is crucial as it anchors the graph of the parabola, influencing its orientation and curvature.

Focus

The focus of a parabola is a fixed point that, along with the directrix, helps in forming the shape of the curve. It is located inside the parabola and is equidistant from any point on the curve, when combined with the directrix.
From our standard equation \(x^2 = -4ay\), we can find the focus at point (0, -a). This exercise reveals that the focus lies at \(y = -1\), which makes \(a = 1\).
Knowing the focus:

• It helps in understanding how the parabola "wraps" itself around this point.
• Works together with the directrix to define the shape of the parabola.

In summary, the focus is instrumental in determining the unique characteristics and behavior of the parabola graph.