Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is at \((2,0)\)

Step-by-Step Solution

Answer

The equation of the parabola is y^2 = 8x.

Step 1: Understanding the Problem

We need to find the standard equation of a parabola with its vertex at the origin (0,0) and its focus at (2,0). This means the parabola opens horizontally.

Step 2: Identify Key Components

The vertex of the parabola is at the origin (0,0) and focus is at (2,0). In a horizontally opening parabola, the equation is of the form (y-k)^2 = 4p(x-h), where (h,k) is the vertex and p is the distance from vertex to focus.

Step 3: Determine the Value of p

Since the vertex is at (0,0) and the focus is at (2,0), the distance p is 2. Thus, p = 2.

Step 4: Plug Values into the Equation

Using the general equation (y-k)^2 = 4p(x-h), substitute the values for the vertex and p. Therefore, the equation becomes y^2 = 4(2)x since (h,k) = (0,0).

Step 5: Simplify the Equation

This simplifies to y^2 = 8x. So, the standard equation of the parabola is y^2 = 8x.

Key Concepts

Vertex at Origin

When we talk about a parabola with its vertex at the origin, we are referring to a situation where the turning point of the parabola is located at the coordinate \((0,0)\).This point is very special, as it serves as a central anchor for the shape of the parabola.
In mathematical terms, when the vertex is at the origin, a parabola that opens sideways can easily be expressed in a simple form.This form is \( (y-k)^2 = 4p(x-h) \).
Since both \(h\) and \(k\) are zero, the equation becomes even simpler: \( y^2 = 4px \).Having the vertex at the origin makes calculations easier, because you don't have to worry about any shifts or translations along the axes.

Focus and Directrix

The focus and directrix are essential elements in defining a parabola.The focus is a specific point, and every point on the parabola is equidistant to the focus and a line called the directrix.
In our exercise, the focus is at \((2,0)\).So, the line or directrix is then vertical and can be found at \(x = -2\).Understanding this relationship helps determine the shape and direction of the parabola.
The standard form of our equation \( y^2 = 4px \) directly reflects this, with \( p \) being the distance from the origin to the focus.Here, \( p = 2 \), so the focus being at \((2,0)\) nicely matches \( 4p \) giving \( 8 \), keeping the equation \( y^2 = 8x \).This relationship ensures the parabola opens in the correct direction with the correct width.

Horizontally Opening Parabola

Unlike the common upward or downward opening parabolas, a horizontally opening parabola opens to the right or left.This changes the standard mathematical form of the equation, focusing on \( y^2 \) instead of \( x^2 \).
For a parabola that opens horizontally, the equation morphs into \( y^2 = 4px \).Here, the positioning of the focus to either side of the vertex determines the direction it opens.
In this exercise, with the vertex at \((0,0)\) and focus at \((2,0)\), the parabola opens to the right.This is because \( p \) is positive. Hence, the equation \( y^2 = 8x \) shows that the parabola is opening towards the positive x-axis.
Grasping this concept of horizontal orientation is vital.It helps in comprehending different parabolic trajectories and their applications in real-world problems.