Problem 34 - College Algebra with Modeling and Visualization

Question

Find an equation of the parabola with vertex \((0,0)\) that satisfies the given conditions. Focus \((-1,0)\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the parabola's orientation

Since the focus of the parabola is located at (-1,0), the parabola opens horizontally. A horizontal parabola with a vertex at (0,0) and focus at (-1,0) will open towards the left.

Step 2: Identifying the standard form of a horizontal parabola

The standard form of a horizontal parabola is given by \( (y-k)^2 = 4p(x-h) \). Since the vertex is at (0,0), \(h = 0\) and \(k = 0\), making the equation \( y^2 = 4px \).

Step 3: Finding the value of p

The distance p is the distance from the vertex to the focus. Since the focus is at (-1,0) and the vertex is at (0,0), the distance \(p = -1\).

Step 4: Substitute p into the standard equation

Substitute \(p = -1\) into the equation \(y^2 = 4px\): \[ y^2 = 4(-1)x \]. This simplifies to \[ y^2 = -4x \].

Step 5: Final equation of the parabola

The equation of the parabola with vertex at (0,0) and focus at (-1,0) is \[ y^2 = -4x \].

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that represents either the highest or lowest point on its curve. In our specific exercise, the vertex of the parabola is located at the origin, which is the point \((0,0)\).

This point can be seen as the "turning point" of the parabola. It is also the point from which the parabola is symmetrically balanced.

When imagining the vertex, picture it as the "hinge" or "pivot" from where the parabola extends. For the equation in standard form, this point is denoted in the parameters as \( (h, k) \). Here, \(h = 0\) and \(k = 0\), making it simpler to substitute these into any equation involving the parabola.

Focus and Its Role

The focus of a parabola is a crucial point that helps determine its shape and direction. In this exercise, the focus is located at \((-1,0)\).

The focus lies on the interior of the parabola, and all the points on the curve are equidistant from both the focus and the directrix (a line that is part of the parabola's defining properties).

For a horizontal parabola like ours, with its vertex at \( (0, 0) \), the location of the focus helps confirm that the parabola opens towards the left.

• The distance from the vertex to the focus is described as \(p\).
• For this parabola, \(p = -1\), highlighting that it stretches or compresses horizontally by that unit length.

This understanding is essential when formulating the parabola equation.

Standard Form of a Parabola

The standard form of a parabola is an equation that provides a clear framework for describing its shape and direction. It simplifies our understanding of how the parabola behaves in a coordinate system.

For a horizontally opening parabola, the standard form is written as \((y-k)^2 = 4p(x-h)\). Substituting the vertex \((0, 0)\), this simplifies to \(y^2 = 4px\).

In our example, the focus is at \((-1, 0)\), and this shifts \(p\) to \(-1\). Thus, after substituting, the standard equation becomes \(y^2 = -4x\).

• "4p" indicates the width of the parabola.
• In our case, "-4" reveals it is opening to the left.

This form is useful for graphing the curve efficiently and verifying its key attributes.

Understanding Distance in Parabolas

Distance within the context of a parabola defines relationships between points on the curve, vertex, focus, and directrix.

To solve the given exercise, understanding the distance \(p\), between the vertex \((0,0)\) and the focus \((-1,0)\) is crucial.

• This distance \(p\), is \(-1\), positioning the focus one unit to the left of the vertex.
• "\(-1\)" further implies the direction of opening and the concavity of the parabola.

Such metrics are fundamental as they offer insights into the parabola's proportionality and symmetry. Every point on the parabola is governed by these distances, ensuring it adheres to its mathematical principles.