Finding the Equation of a Parabola with Axis Along the x-Axis: A Comprehensive Guide

When dealing with parabolas, it's often necessary to find their equations given specific points and the axis of symmetry. This article will guide you through the process of finding the equation of a parabola with its axis along the x-axis, using two given points as an example. This method is widely applicable and useful in various mathematical and physical contexts.

Understanding Parabolas with Axis Along x-Axis

A parabola is a U-shaped curve that appears in numerous natural and artificial forms. When the axis of the parabola is along the x-axis, the standard form of the equation can be simplified to a specific format. This article will explore how to determine the equation of a parabola given its axis and passing through specific points.

Example: Finding the Parabola Equation from Given Points

Given two points, (3, 2) and (-2, -1), we aim to find the equation of the parabola with its axis along the x-axis.

Step 1: General Form of the Parabola

The general form of the parabola with its axis along the x-axis is:

y ax^2 bx c

Here, a, b, and c are coefficients that need to be determined.

Step 2: Setting Up Equations

Substituting the given points into the general equation, we get two equations:

For the point (3, 2): 2 a(3)^2 b(3) c For the point (-2, -1): -1 a(-2)^2 b(-2) c

These equations can be simplified to:

9a 3b c 2 (Equation 1) 4a - 2b c -1 (Equation 2)

Step 3: Eliminating c

To eliminate c, we subtract Equation 2 from Equation 1:

9a 3b c - (4a - 2b c) 2 - (-1)

This simplifies to:

5a 5b 3

Dividing by 5:

a b 3/5 (Equation 3)

Step 4: Expressing c in Terms of a and b

From Equation 1, we can express c in terms of a and b:

c 2 - 9a - 3b (Equation 4)

Step 5: Solving for a and b

To solve for a and b, we need one more point or condition. If no additional point is given, we can assume a specific value for one of the variables. For simplicity, let's assume b 0, which gives a parabola that opens sideways.

Substituting b 0 into Equation 3:

a 3/5

Step 6: Finding c

Substituting a 3/5 and b 0 into Equation 4:

c 2 - 9(3/5) - 3(0) 2 - 27/5 2 - 5.4 -3.4

Final Equation

Substituting a, b, and c into the general equation y ax^2 bx c:

y (3/5)x^2 - 3.4

This is the equation of the parabola that passes through the points (3, 2) and (-2, -1).

Final Equation: y (3/5)x^2 - 3.4

Additional Considerations

If more information about the vertex or orientation is provided, adjustments can be made to the equation. For instance, the form of the equation can be rewritten in vertex form y - k^2 4a(x - h), where V (h, k) is the vertex, a is the distance of the focus from the vertex, and the directrix is x h - a.

Using the derived values:

V (0, -11/3) a 3/20 F (-211/60, 0)

The equation of the parabola can be written as:

y^2 1/5(3x - 11)

The directrix is x -229/60.

This comprehensive guide should help you understand and solve similar problems involving parabolas with axis along the x-axis.