Problem 37 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin. focus at \((0,100)\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y = 0.0025x^2\).

Step 1: Understand the Vertex

The vertex of our parabola is at the origin of the coordinate system, this is the point \((0,0)\).

Step 2: Understand the Focus

The focus of our parabola is at \((0, 100)\). Since the focus is above the vertex, our parabola opens upwards.

Step 3: Write the General Form of the Equation

The equation of a parabola that opens upwards or downwards is of the form \(y = a(x-h)^2 + k\) where \((h,k)\) is the vertex. In our case, the vertex is at the origin so in our case it would be the simpler: \(y = ax^2\)

Step 4: Find the Value of 'a'

We know that the distance from the vertex to the focus is given by \(1/4a\). However, in our case the distance from the vertex at \((0, 0)\) to the focus at \((0, 100)\) is just 100 units. Therefore, we have \(1/4a = 100\) or \(a = 1/(4*100)\) or \(a = 0.0025\).

Step 5: Write the Final Equation

Now, substitute 'a' into the equation, so the final equation of the parabola is \(y = 0.0025x^2\).

Key Concepts

Vertex

In a parabola, the vertex is a crucial point that represents the tip or the highest/lowest point, depending on the parabola's orientation. The vertex serves as a kind of turning point where the direction changes. When dealing with parabolas, the vertex provides stability and direction.
In this exercise, the vertex is located at the origin,

• Origin: This means it's at the point (0,0) in the coordinate system.
  
• Simplification: Having the vertex at the origin simplifies the equation because it eliminates additional variables related to horizontal and vertical shifts.
  

Understanding the vertex's position is essential when developing the equation of the parabola.

Focus of a Parabola

The focus of a parabola is a point that helps in determining the shape and direction of the parabola. It's one of the fixed points from which distances are measured, influencing the parabola's "width" and position.
In our problem, the focus is at (0, 100).

• Direction: Since the focus is above the vertex, the parabola opens upwards.
  
• Role: The focus impacts the parabola's equation, as the distance between the vertex and the focus directly relates to the 'steepness' of the curve.
  

The focus is essential to construct the parabola accurately and adds meaning to its geometric form.

Coordinate System

The coordinate system is like a grid that helps us place and identify points in space. In most mathematics, especially when dealing with parabolas, we use a two-dimensional coordinate system with x and y axes.
The coordinate system's origin,

• Reference Point: This is (0,0), where the x and y axes intersect, and plays a vital role in our exercise because both the vertex and focus are aligned vertically.
  
• Graphical Understanding: Being able to plot points on this grid helps visualize geometric shapes like parabolas.
  

A solid grasp of the coordinate system enables us to connect algebraic equations with geometric representations.

Equation of a Parabola

Writing the equation of a parabola involves translating geometric information into an algebraic form. A parabola that opens vertically uses the equation format

• General Equation: \[ y = a(x-h)^2 + k \] where (h,k) is the vertex.
  
• Simplified Form: With the vertex at the origin in our problem, this simplifies to \[ y = ax^2 \].
  

The coefficient 'a' determines how "wide" or "narrow" the parabola is based on its distance to the focus.
In this exercise, we calculated 'a' as 0.0025 based on the distance to the focus.
Substituting this back, our final equation is \[ y = 0.0025x^2 \]. This showcases how spatial and algebraic formats are interrelated in mathematics.