Problem 76 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Focus \((5,0)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

The problem asks for an equation of a parabola with a given vertex and focus. The parabola's vertex is at the origin \((0,0)\) and the focus is at \((5,0)\).

Step 2: Identify the Parabola Type

Since the focus is on the x-axis to the right of the origin, the parabola opens to the right. We will use the equation of a parabola that opens horizontally.

Step 3: Use the Standard Form for a Horizontal Parabola

The standard form for a horizontally opening parabola is \((y-k)^2 = 4p(x-h)\), where \((h,k)\) is the vertex.

Step 4: Substitute Vertex Coordinates

Substitute the vertex coordinates (0,0) into the equation: \((y-0)^2 = 4p(x-0)\), simplifying to \(y^2 = 4px\).

Step 5: Determine the Value of 'p'

From the vertex at the origin, the focus is 5 units to the right, so \(p = 5\). Substitute this into the equation to get \(y^2 = 4(5)x\).

Step 6: Simplify the Equation

Simplify the equation to get the final form: \(y^2 = 20x\).

Key Concepts

Vertex

The vertex of a parabola is a fundamental point. It is often considered the turning point or the "tip" of the parabola. In this exercise, the vertex is located at the origin, which is the point

• where the parabola is symmetric around.
• where it changes direction.

For a horizontally oriented parabola with the vertex at the origin, the vertex coordinates are

• (0,0).

The location of the vertex defines how the equation is structured. Depending on its location, the general form changes accordingly with

• vertex
• orientation.

Vertex form allows easy identification of the parabola's peak or base.

Focus

The focus of a parabola is another crucial element that lies on the axis of symmetry of the parabola. This essential point influences the curvature and the width of the parabola.
In our example, the focus is given as

• (5,0).

The focus is always inside the parabola, helping determine its shape and opening direction. The standard distance from the vertex to the focus defines the parameter 'p', a key component in forming the equation. In this case, since the distance from

• the vertex (0,0) to focus (5,0) is 5 units,

we conclude that

• p = 5.

The location of the focus helps in ensuring your parabola is accurately positioned.

Standard Form of Parabola

The standard form of a parabola provides a formulaic way to express its equation. This form varies based on the orientation of the parabola. In our case, since the parabola opens horizontally, the formula is

• \((y-k)^2 = 4p(x-h)\).

Here,

• \((h,k)\) determines the coordinates of the vertex.
• 'p' indicates the distance to the focus.

Substituting in the vertex

• (0,0)

we get

• \(y^2 = 4px\).

Once we know 'p', plug it in to complete the parabola equation.

Horizontal Parabola

A horizontal parabola is a type where the parabola opens to the left or right rather than up or down. This orientation is less common than vertical parabolas, but essential in many applications.
In our problem:

• The parabola opens horizontally to the right.

This is because the focus

• lies on the x-axis at (5,0).

The standard form

• for the equation \(y^2 = 4px\)
• guides this design.

The particularly given vertex

• and focus positions cause the parabola formula to be \(y^2 = 20x\),

showing how straightforward calculations help plot the parabola accurately. Understanding these aspects allows you to master both horizontal and vertical orientations.