Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is \(\left(0,-\frac{1}{9}\right)\)

Step-by-Step Solution

Answer

The equation of the parabola is \( y = -\frac{9}{4}x^2 \).

Step 1: Identify the Direction

Since the focus has a negative y-coordinate, the parabola opens downwards. This information is crucial for determining the structure of the equation.

Step 2: Recall the Standard Parabola Equation

For a parabola with its vertex at the origin and opening vertically, the standard form of the equation is \[ y = ax^2 \] where \( a \) is a constant that affects the width and direction of the parabola.

Step 3: Use the Focus to Find 'a'

The directrix of the parabola is at \( y = -p \), where the distance \( p \) is equal to the distance from the vertex to the focus. Given the focus \( (0, -\frac{1}{9}) \), the distance \( p \) is \( \frac{1}{9} \). Since the parabola opens downward, \( a = -\frac{1}{4p} \). Thus, \( a = -\frac{1}{4 \times \frac{1}{9}} = -\frac{9}{4} \).

Step 4: Write the Equation

Substitute \( a = -\frac{9}{4} \) into the standard form to get the equation \[ y = -\frac{9}{4}x^2 \] This is the standard equation of the parabola with the given focus and vertex at the origin.

Key Concepts

Focus of a Parabola

The focus of a parabola is a fixed point inside the curve that helps define its shape. In the given exercise, the focus is at the point \( (0, -\frac{1}{9}) \). This particular point, combined with the vertex, determines the parabola's direction.

The focus is crucial because it is used to calculate the value of \( p \), which is the distance from the vertex to the focus. This distance helps in defining both the directrix and the equation of the parabola.

Specifically, the value of \( p \) directly influences the formula to find \( a \) in the standard parabola equation configuration \( y = ax^2 \). Knowing \( p = \frac{1}{9} \), we determine \( a \) by using the formula \( a = -\frac{1}{4p} \) for a downward opening parabola.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. In the context of standard equations, the vertex is often set at the origin \( (0, 0) \) for simplicity.

By placing the vertex at the origin, we simplify the equation of the parabola, making it easier to understand how each parameter affects the parabola's shape.

• The vertex serves as a symmetry center for the parabola.
• In equations involving the vertex at the origin, terms involving \( h \) and \( k \) disappear, resulting in a simpler form like \( y = ax^2 \).

Overall, understanding the location of the vertex allows us to see the primary influence in shaping the parabola, especially alongside the focus.

Parabola Opening Direction

The direction in which a parabola opens is significant. It is determined by the sign of \( a \) in the equation \( y = ax^2 \).

In this exercise, the focus located at a negative y-coordinate \( (0, -\frac{1}{9}) \) means the parabola opens downward. This is because the focus lies below the vertex on the coordinate plane.

• A positive \( a \) indicates the parabola opens upwards.
• A negative \( a \) means the parabola opens downwards.

The opening direction is crucial because it affects how the graph of the parabola will appear.
Additionally, understanding this helps when constructing the correct equation based on given characteristics, like the position of the focus.