Quadratic equations are second-degree algebraic expressions and are of the form ax2 + bx + c = 0. The word “quadratic” is derived from the word “quad” which means square. In other words, a quadratic equation is an “equation of degree 2.” There are many scenarios where a quadratic equation is used. Did you know that when a rocket is launched, its path is described by a quadratic equation? Further, a quadratic equation has numerous applications in physics, engineering, astronomy, etc.

The quadratic equations are second-degree equations in x that have a maximum of two answers for x. These two answers for x are also called the roots of the quadratic equations and are designated as (α, β).

## Standard Form of a Quadratic Function

The standard form of a parabola is: y = ax2 + bx + c

Here a, b, and c are real numbers (constants) where a ≠ 0, and x and y are variables where (x, y) represents a point on the parabola.

## Vertex Form of a Quadratic Function

The equation for a basic parabola with a vertex at (0, 0) is y = x2. You can apply transformations to the graph of y = x2 to create a new graph with a corresponding new equation. This new equation can be written in vertex form. The vertex form of a quadratic function is y = a(x−h)2 + k where:

• |a| is the vertical stretch factor. If a is negative, there is a vertical reflection and the parabola will open downwards.
• k is the vertical translation.
• h is the horizontal translation.

Given the equation of a parabola in vertex form, you should be able to sketch its graph by performing transformations on the basic parabola.

## Transformations of Function f(x)

Transformation of functions means that the curve representing the graph either “moves to left/right/up/down” or “it expands or compresses” or “it reflects”. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.

## How to Graph a Parabola in Vertex Form

See the following webpages:

## References

Lloyd, Philip. “How do I reformat y = ax^2 + bx + c to y = a(x-h)^2 + k without a graph?” 2023. Quora. https://qr.ae/prODIm.

Lloyd, Philip. “What Is The Best Way To Explain The Quadratic Formula?” 2023. Quora. https://qr.ae/prODp4.