Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. $$ \text { Vertex } V(3,-5), \quad \text { directrix } x=2 $$

Step-by-Step Solution

Answer

The equation is \((y + 5)^2 = 4(x-3)\).

Step 1: Understand the Components of the Parabola

Since the vertex of the parabola is given as \( V(3, -5) \) and the directrix is \( x = 2 \), the parabola opens sideways. Specifically, it opens to the right because the vertex is at \( x = 3 \), which is greater than the directrix \( x = 2 \). Similarly, if the vertex were at \( x \) less than 2, it would open to the left.

Step 2: Identify the Form of the Parabola Equation

For a parabola that opens sideways, the standard form of the equation is \((y-k)^2 = 4p(x-h)\), where \((h, k)\) is the vertex. Here, \( h = 3 \) and \( k = -5 \). The parameter \( p \) is the distance from the vertex \( V \) to the directrix, so \( p = 3 - 2 = 1 \). Since the parabola opens to the right, \( p \) is positive.

Step 3: Substitute Values into the Parabola Equation

Substitute \( h = 3 \), \( k = -5 \), and \( p = 1 \) into the equation \((y-k)^2 = 4p(x-h)\). This gives us:\((y + 5)^2 = 4(1)(x-3)\).

Step 4: Simplify the Equation

Simplify the expression to get the final equation of the parabola:\((y + 5)^2 = 4(x-3)\).

Key Concepts

Vertex

The vertex of a parabola is a significant point that represents the turning point or the highest or lowest point of the parabola. In the context of this exercise, the vertex is given at \( V(3, -5) \). This point is crucial for determining not only the position of the parabola on the coordinate plane but also the direction in which it opens.

• The vertex \( (h, k) \) represents the coordinates \( (3, -5) \) in this case.
• It determines the axis of symmetry of the parabola, which for a sideways parabola runs horizontally.
• The position of the vertex relative to the directrix (a line that helps define the parabola) also indicates the direction in which the parabola opens - left or right for sideways opening parabolas.

It is through the vertex that we can apply the standard form equation of a parabola to find its specific equation.

Directrix

The directrix is a line that, along with the vertex and focus, defines a parabola. It is positioned opposite the parabola's opening direction relative to the vertex. For this specific exercise, the directrix is given as \( x = 2 \). This information allows us to set up certain parameters in the parabola's equation.

• Its position illustrates how the parabola is balanced, maintaining a constant distance between any point on the parabola and the focus versus that same point's perpendicular distance to the directrix.
• Given that our vertex is at \( x = 3 \), the directrix \( x = 2 \) indicates that the parabola opens to the right, since the vertex's x-value is larger than the directrix.
• This line helps us determine the distance \( p \), which is the distance from the vertex to the directrix. In this case, \( p = 1 \).

The directrix acts as a foundational reference for deriving the accurate equation of a sideways-opening parabola.

Standard Form of a Parabola

The standard form of a parabola provides a generalized equation that can be adapted based on its opening direction and orientation. For a sideways opening parabola, the standard form is \[(y - k)^2 = 4p(x - h)\]where:

• \( (h, k) \) are the coordinates of the vertex.
• \( p \) is the distance between the vertex and the directrix.

In the exercise example:

• The values are \( h = 3 \), \( k = -5 \), and \( p = 1 \).
• Substituting these into the standard form gives us:\((y + 5)^2 = 4(x - 3)\)

This equation uses the relationship between the squared y-difference and the x-difference multiplied by 4 times the distance \( p \). The horizontal symmetry of the parabola is maintained by the squared y-term, showing sideways orientation.

Sideways Opening Parabola

A sideways opening parabola, unlike the more common up-down opening parabolas, extends either left or right. Identifying the direction of the opening is key to understanding and expressing its equation. In the given exercise, with the vertex at \( (3, -5) \) and the directrix at \( x = 2 \), the parabola opens to the right.

• If the vertex's x-coordinate is greater than that of the directrix, the parabola opens to the right.
• Conversely, if it is less, it opens to the left.
• The standard form is adjusted to reflect this sideways direction by squaring the y-term and adjusting parameters in the x-term.

The sideways orientation requires careful consideration of the vertex and the directrix to ensure the correct formulation of the parabola equation, reflecting this alternate direction of opening.