Problem 27 - Precalculus: Mathematical for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(-8,0)\)

Step-by-Step Solution

Answer

The equation is \(y^2 = -32x\).

Step 1: Identify the Parabola Type

Since the vertex of the parabola is at the origin, \(0,0\), and the focus, \((-8,0)\), lies on the x-axis, the parabola opens horizontally (left or right).

Step 2: Equation of the Parabola

The standard form for a parabola that opens horizontally with its vertex at the origin is \(y^2 = 4px\). Here, \(p\) is the distance from the vertex to the focus.

Step 3: Determine the Value of p

Since the focus is at \((-8,0)\), and the vertex at \(0,0\), \(p = -8\) (the negative sign indicates the parabola opens left).

Step 4: Formulate the Equation

Substituting \(p = -8\) into the equation \(y^2 = 4px\), gives \[y^2 = 4(-8)x = -32x.\Therefore, the equation of the parabola is \space y^2 = -32x.\]

Key Concepts

Vertex Form

The vertex form of a parabola is a particular way of expressing the parabola's equation, which directly provides information about its vertex. The vertex is a crucial point that represents the "tip" or turning point of the parabola. For parabolas that open horizontally, the general vertex form looks like this:

• For horizontal parabolas: \[ y^2 = 4p(x - h) + k\]
• For vertical parabolas: \[ x^2 = 4p(y - k) + h\]

Here, \(h, k\) is the vertex of the parabola. When the vertex is at the origin, like in our given problem, \(h = 0\) and \(k = 0\), the equation simplifies to \(y^2 = 4px\) or \(x^2 = 4py\) depending on the direction the parabola is opening. Understanding the vertex form is essential as it makes it easier to determine the geometry of the parabola by simply looking at the equation.

Focus of a Parabola

The focus of a parabola is a special point that, together with the directrix (a line), defines the parabola. Each point on the parabola is equidistant from the focus and the directrix. This characteristic forms the basis of the parabola's geometric definition.
In the problem at hand, the focus is given as \(F(-8, 0)\). With the vertex at the origin \(0,0\), the focus helps us not only to understand the shape of the parabola but also its orientation. The distance between the vertex and the focus is denoted by \(p\), which in this case is -8. The sign of \(p\) indicates the direction in which the parabola opens. Recognizing the focus and its placement relative to the vertex is crucial in graphing the parabola and understanding its physical shape.

Opening Direction of Parabola

The direction in which a parabola opens is vital in understanding its behavior. A parabola can open upwards, downwards, left, or right. When given the vertex at the origin and a specific focus point, determining this direction becomes straightforward.In our example, since the focus \(F(-8, 0)\) is located on the negative x-axis, and the vertex is at \(0,0\), the parabola opens to the left. If the parabola opens to the left or right, we know that it is a horizontal parabola, and its equation takes the form \(y^2 = 4px\).

• If \(p > 0\), the parabola opens to the right.
• If \(p < 0\), it opens to the left.

Understanding the opening direction aids in graphing the parabola accurately and helps in predicting its path and spread.

Distance Calculation in Parabolas

Calculating distances in parabolas is fundamental for constructing their equations and understanding their configurations. The parameter \(p\) represents the distance from the vertex of the parabola to its focus.In this specific problem, the given focus is \(-8,0\), while the vertex is at the origin \(0,0\). The distance \(p\) is effectively the x-coordinate of the focus, meaning \(p = -8\). The negative sign shows direction on the x-axis.The distance calculation between these two points forms the crucial link in determining \(p\) for the parabola's equation \(y^2 = 4px\). Once you know \(p\), you can substitute it directly into the parabola's equation form to find the complete equation. Understanding this distance calculation enables accurate formulation of the parabola's key characteristics.