Problem 25 - Precalculus with Limits

Question

In Exercises 19-32, 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 equation of the parabola with directrix \(y = 1\) and vertex at the origin is \(y = -\frac {1}{4}x^2\).

Step 1: Recall the Formula

Recall that the formula for the standard form of a parabola is \(y=ax^2\), where \(a\neq0\). The vertex of the parabola is at the origin, so no adjustments for a moved vertex need to be made.

Step 2: Identify the values

The directrix \(y=1\) is above the vertex (at the origin). Parabolas always open away from the directrix, so this parabola opens downward.

Step 3: Calculate the value of 'a'

Since the directrix is \(y = 1\) and the vertex is at the origin, the distance from the vertex to the directrix is 1. In the equation \(y = ax^2\), the value of 'a' is \(-\frac {1}{4p}\) where 'p' is the distance from the vertex to the directrix. Therefore, \(a = -\frac {1}{4*1} = -\frac {1}{4}\).

Step 4: Write the Equation

Replace 'a' in the standard equation \(y = ax^2\) with the calculated value. The equation of the parabola is thus \(y = -\frac {1}{4}x^2\).

Key Concepts

Vertex at Origin

When a parabola has its vertex at the origin, it simplifies how we can write its equation in standard form. The origin in the coordinate plane is the point

• (0, 0), which means the vertex is located at this point.
• With the vertex at the origin, we automatically define the center of the parabola, from which it either opens upwards, downwards, or sideways.

This makes describing the parabola's orientation simpler, as we don't have any added horizontal or vertical shifts.In the context of the standard form of a parabola, this is often written as:

• For a vertical orientation:
  \(y = ax^2\)
• For a horizontal orientation:
  \(x = ay^2\)

But in our problem, since we are focused on a parabola with the equation \(y = ax^2\), all terms related to shifting are zeroed out since the vertex is at (0, 0). This makes analyzing other components like the directrix and the parabola's opening much more direct.

Directrix

The directrix in the context of a parabola is a line that helps define its shape and orientation. Opposite the directrix line from the vertex, the parabola opens away.For a parabola with a vertex at the origin

• (0, 0), the directrix determines if the parabola opens upwards or downwards when the directrix is horizontal.
• Our exercise specifies the directrix as \(y = 1\).

Since the directrix is above the vertex, it implies that our parabola will open downwards. This type of parabolic orientation creates a mirror effect, focusing on a point that is equal distance from both the directrix and the vertex. This ensures a perfect, symmetric curve. With the vertex at the origin and the directrix at \(y = 1\), the vertex form automatically starts with the vertex parabola equation noted as \(y = ax^2\) since the parabola opens vertically.

Value of 'a' in Parabolas

The value of 'a' in a parabola's equation, \(y = ax^2\), significantly defines how the parabola is shaped and oriented:

• The sign of 'a' denotes direction:
  Positive values indicate an upward opening, while negative values indicate a downwards opening.
• The absolute value of 'a' affects the width:
  Smaller absolute values (closer to 0) make the parabola wider, while larger absolute values create a narrower shape.

To calculate 'a', we use the distance from the vertex to the directrix

• 'labeled as 'p'.

In our specific problem, the vertex is at zero, and the directrix is at \(y = 1\), making this distance \(p = 1\). Therefore, with the formula \(a = -\frac{1}{4p}\), we plug in our value of 'p' to find

• \(a = -\frac{1}{4 * 1} = -\frac{1}{4}\).

This means our parabola equation becomes \(y = -\frac{1}{4}x^2\), indicating it is a downward-opening parabola.