Understanding the Equation of a Parabola with Given Characteristics

The equation of a parabola is a crucial concept in mathematics, especially when it has specific given points or properties. For a parabola that opens horizontally along the x-axis with its vertex at the origin, the general form of the equation is:

General Form of Parabola Equation

The standard form of an equation for a parabola opening horizontally along the x-axis, with its vertex at the origin, is:

${y^2 4px}$

where p is the distance from the vertex to the focus.

Deriving the Specific Equation

Let's consider a parabola that passes through the point (3, 4) and has its vertex at the origin (0, 0). To find the value of p, we can substitute the point (3, 4) into the general equation:

${4^2 4p(3)}$

This simplifies to:

$16 12p$

Solving for p:

${p frac{16}{12} frac{4}{3}}$

Substituting the value of p back into the general equation gives us:

${y^2 4left(frac{4}{3}right)x$

Which simplifies to:

${y^2 frac{16}{3}x$

Thus, the equation of the parabola is:

${y^2 frac{16}{3}x$

Alternative Form Using Directrix and Focus

Another approach is to use the directrix and focus of the parabola. Let's consider the equation (y^2 2ax). In this form, the focus is at ((frac{a}{2}, 0)) and the directrix is at (x -frac{a}{2}).

Given that the parabola passes through the point (3, 4), we can plug these values into the equation to solve for a:

${4^2 2a(3)}$

This simplifies to:

$16 6a$

Solving for a:

${a frac{16}{6} frac{8}{3}}$

Substituting this value of a back into the equation:

${y^2 2left(frac{8}{3}right)x$

This simplifies to:

${y^2 frac{16}{3}x$

Therefore, the equation of the parabola is:

${y^2 frac{16}{3}x$

Conclusion and Practical Applications

The process of determining the equation of a parabola with given points or properties is a fundamental concept in algebra and calculus. Understanding these equations helps in various fields such as physics, engineering, and computer graphics. For instance, in physics, the motion of projectiles can be described using parabolic equations, while in computer graphics, the shape of certain objects can be rendered accurately using these equations.