Problem 10 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin and the given focus. focus at \((-1,0)\)

Step-by-Step Solution

Answer

The equation of the parabola with vertex at the origin and focus at (-1,0) is \(y=-1/4*x\).

Step 1: Determine the Opening Direction and the Distance from the Vertex to the Focus

Here, the vertex of a parabola is at the origin (0,0) and focus is given as (-1,0). Since the y-coordinate of the focus is 0, the parabola opens sideways. The x-coordinate of the focus denotes the distance from the vertex to the focus which is -1. So, the p=-1.

Step 2: Substitute p value into the Standard Equation for Parabola

The standard equation for a parabola that opens sideways is \(x=4py\). Substituting \(p=-1\) into the equation, we get \(x=4*(-1)*y\), or \(x=-4y\).

Step 3: Rearrange the Equation

Finally, to express the parabola equation with y as the subject, rearrange the equation to get \(y= -1/4*x\).

Key Concepts

Vertex

The vertex of a parabola is a key point that signifies where the curvature changes direction. Think of it as the 'tip' or the 'turning point' of the parabola. For this specific exercise, the vertex is placed at the origin, which means the coordinates are

• (0,0)

Placing the vertex at the origin simplifies our equation and calculations. It's a central reference point for determining the shape and position of the parabola. Understanding the vertex helps us grasp the symmetry and properties of the parabola.

Focus

The focus of a parabola plays a vital role in shaping it. It's a single point inside the parabola from which all points equidistant to the directrix reflect and form the curve. In this case, the focus of the parabola is at

• (-1,0)

The position of the focus determines how the parabola opens and its width. The distance from the vertex to this point, noted as "p," is crucial for forming the parabola's equation. Here,

• p = -1

This means that the focus is one unit to the left of the vertex, influencing the orientation and direction of the parabola.

Parabola Equation

The parabola equation is what allows us to mathematically represent the curve. It is derived using the vertex and the focus. In many cases, the equation can be modeled in forms like

• Vertical opening: \[ y = ax^2 \]
• Horizontal opening: \[ x = 4py \]

For this exercise, the parabola opens sideways, so we use the equation \[ x = 4py \]Substituting the value of

• p = -1

into the standard equation, we get\[ x = -4y \]Rearranging this, we have \[ y = -\frac{1}{4}x \]This final form represents our parabola concisely, showing its relation to the vertex and focus.

Opening Direction

The opening direction of a parabola shows which way the curve faces. It can open upwards, downwards, or sideways (left or right). In this scenario, since the focus is located at

• (-1,0)

with the vertex at

• (0,0)

the parabola opens to the left. The opening direction affects how you set up the equation of the parabola. For sideways opening parabolas, like this one, the equation takes the form \[ x = 4py \]instead of the typical quadratics used for parabolas opening up or down.