Problem 32 - Precalculus Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus: \(F(5,0)\)

Step-by-Step Solution

Answer

The equation is \( y^2 = 20x \).

Step 1: Understand the Problem

We need to find the equation of a parabola with its vertex at the origin (0,0) and a focus at (5, 0). The presence of the focus tells us about the direction and properties of the parabola.

Step 2: Identify the Parabola Type

Since the focus is located on the x-axis (horizontal line), the parabola opens either to the left or right. Given that the x-coordinate of the focus is positive, the parabola opens to the right.

Step 3: Recall the Standard Equation

For a horizontally opening parabola with vertex at the origin, the standard equation is \[ y^2 = 4px \]where \( p \) is the distance from the vertex to the focus.

Step 4: Calculate the Parameter p

The distance \( p \) is the x-coordinate of the focus, since the focus is at (5,0) and the vertex is at (0,0). Thus, \( p = 5 \).

Step 5: Substitute p into the Equation

Substitute \( p = 5 \) into the standard equation: \[ y^2 = 4(5)x \]which simplifies to \[ y^2 = 20x \].

Step 6: Finalize the Equation

The equation of the parabola with vertex at the origin and focus at (5,0) is\[ y^2 = 20x \].

Key Concepts

Vertex

In the world of parabolas, the vertex is a crucial point. It is essentially the "tip" or turning point of the parabola where it makes its sharpest turn. For any parabola, identifying the vertex gives us a firm foundation from which to build an understanding of its shape and position.

• The vertex defines the point at which the parabola changes direction and provides a central anchor for the curve.
• If you imagine a parabola as a bowl, the vertex would be at the very bottom of the bowl (if it opens upwards) or at the top if it's upside down.
• The vertex of a parabola in the standard form equation can usually be found at coordinates \(h, k\).

However, in our specific problem, the vertex is conveniently located at the origin, (0, 0). This makes our calculations easier and gives us a neat central point from which the parabola expands. The simplicity of having the vertex at the origin allows us to sidestep more complicated algebra involving horizontal or vertical shifts. For this exercise, knowing the vertex is at (0, 0) is our starting point before we delve into other properties of the parabola.

Focus

The focus of a parabola is not a visible point on its graph like the vertex, but it plays an equally key role. It is intimately related to the parabolic shape, controlling the width and direction.

• Think of the focus as a guiding light for the parabola; the entire curve is constructed in such a way that every point on the parabola is equidistant from the focus and a corresponding line known as the directrix.
• In essence, the focus is one of two fundamental points that define a parabola, the other being the vertex.
• The distance from the vertex to the focus, denoted as \( p \), is a determining factor for the parabola's equation form.

In our exercise, the focus is given as (5, 0), which is located on the positive x-axis. This tells us right away that our parabola will open to the right, as it has to "face" this focus point. The focus' position helps us determine the direction of the parabola and fills in the missing information required to form the standard equation. Knowing the focus allows us to calculate the parameter \( p \), which importantly helps us shape the parabola's opening.

Standard Form of Parabola

The standard form of a parabola is a powerful tool. It simplifies the process of graphing and analyzing these curves by providing a consistent format for the equation.

• For different orientations of parabolas, the standard form will differ slightly to accommodate whether they open upwards, downwards, to the right, or to the left.
• For a parabola opening right or left, as in our exercise, the formula is \((y-h)^2 = 4p(x-k)\). With the vertex at the origin \(h=0\) and \(k=0\), it simplifies to \(y^2 = 4px\).
• Here, \( p \) represents the distance from the vertex to the focus, dictating the direction the parabola opens. This makes \( p \) a centerpiece in writing the actual equation of the parabola.

In our specific case, once we know \( p = 5 \) (since the focus is at (5, 0)), we substitute this into our standard form equation to get \( y^2 = 20x \). This straightforward formula captures all the complexity of our parabola and offers a neat way to understand how it behaves. This equation not only represents the curve graphically but also mathematically, making it easy to analyze further or use in calculations.