Understanding the Equation of a Horizontal Parabola

A parabola with an axis parallel to the x-axis is called a horizontal parabola. The standard form of a horizontal parabola is given by the equation: x a(y - k)2 h, where (h, k) is the vertex of the parabola.

In this article, we will explore the process of finding the equation of a horizontal parabola given its vertex and a point it passes through. Specifically, we will examine the parabola with a vertex at (-3, -3) and a point (17, 7) on its curve. Let's begin by outlining the steps and then solving the problem step by step.

The Given Equation and Steps Involved

The general equation for a horizontal parabola is:

[ x a(y - k)^2 h]

Given that the vertex (h, k) is at (-3, -3), the equation transforms to:

[ x a(y - (-3))^2 - 3 quad text{or} quad x a(y 3)^2 - 3]

Substituting the Point (17, 7)

The point (17, 7) is given to be on the parabola. By substituting this point into the equation, we can solve for the coefficient 'a' which will give us the specific equation of the parabola.

Substituting (17, 7) into the equation:

[ 17 a(7 3)^2 - 3]

Simplifying the equation:

[ 17 a(10)^2 - 3 quad text{or} quad 17 100a - 3]

Rearranging to solve for 'a':

[ 17 3 100a quad text{or} quad 20 100a] [ a frac{20}{100} frac{1}{5}]

Therefore, the equation of the parabola is:

[ x frac{1}{5}(y 3)^2 - 3]

Axis of Symmetry and Further Analysis

The axis of symmetry for a horizontal parabola is a vertical line that passes through the vertex. For our example, the axis of symmetry is given by:

[ y -3]

This vertical line acts as the mirror for the parabola, splitting the curve into two symmetric halves.

Conclusion

In summary, we have determined that the equation of the horizontal parabola with a vertex at (-3, -3) and passing through the point (17, 7) is:

[ x frac{1}{5}(y 3)^2 - 3]

Understanding the properties and solving such equations is crucial for comprehending the behavior and characteristics of parabolas. Whether you're working with mathematical problems or engineering applications, this knowledge forms a strong foundation in conic sections.