Problem 12 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin and the given focus. focus at \((0,-5)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y = 20x\).

Step 1: Determine the Value of p

The given focus is at (0,-5). Since the vertex is at the origin (0,0), the value of p is the distance from the vertex to the focus, which is -5.

Step 2: Substitute p into the Equation

Now that we have the value for p, we can substitute it into the standard form of the equation for a parabola that opens downwards, which is \(y = -4px\). By substituting p = -5, we get \(y = -4*(-5)x\)

Step 3: Simplify the Equation

Finally, we simplify the equation to get the final form. The equation simplifies to \(y = 20x\)

Key Concepts

Vertex Form

The vertex form of a parabola is a way to express the equation of a quadratic function in a way that makes identifying the vertex easier. The standard vertex form of a parabola is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For a parabola with a vertex at the origin, \( h \) and \( k \) are both zero, simplifying the equation to \( y = ax^2 \).When writing the equation in vertex form, determining the value of \( a \) depends on the parabola's direction and width. If the parabola opens upwards or downwards, the \( a \) value will be positive or negative respectively. Using vertex form makes graphing and understanding the properties of the parabola straightforward, as the vertex is clearly defined and it highlights symmetry with respect to that vertex.

Focus and Directrix

In the context of parabolas, both the focus and the directrix play crucial roles in defining their shape. The focus of a parabola is a fixed point used in the parabola's geometric definition, and every point on the parabola is equidistant from the directrix, a fixed line, and the focus.

• Focus: In this exercise, the focus is given at \((0, -5)\). The focus is always located inside the parabola.
• Directrix: The corresponding directrix for this focus would be a horizontal line. If the vertex of the parabola is \((0, 0)\) and the focus is \((0, -5)\), then the directrix will be a line at \( y = 5 \), equidistant from the vertex but on the opposite side of the focus.

Understanding these components allows one to derive the parabola's equation and study its reflective properties. These properties are fundamental in applications of parabolas, such as satellite dishes and car headlights, where all rays parallel to the parabola's axis reflect through the focus.

Distance from Vertex to Focus

The distance from the vertex of a parabola to its focus, often denoted as \( p \), is a vital part of defining the parabola's equation. This distance helps determine the direction and width of the parabola.

• The parabola's equation includes the term \( 4p \), and this is used when putting the parabola in its standard form, \( y = \pm \frac{1}{4p}x^2 \). For a parabola with vertex at the origin that opens vertically, \( y = \pm 4px \) becomes relevant.
• For the given exercise, where the focus is \((0, -5)\), \( p = -5 \). Thus, the equation becomes \( y = -4(-5)x \), simplifying to \( y = 20x \).
• Understanding the distance \( p \) helps in visualizing the parabola's orientation: negative \( p \) indicates that the parabola opens downwards.

Grasping this concept of distance helps in quickly sketching the direction and location of a parabola based on its focus and vertex.