Quadratic equations are second-degree algebraic expressions and are of the form *ax ^{2} + 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 = ax ^{2} + 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 = x ^{2}*. You can apply transformations to the graph of

*y = x*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

^{2}*y = a(x−h)*where:

^{2}+ k*|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 Convert Standard Form to Vertex Form?

### Method 1: Completing the Square

### Method 2: Using the “Quick Tidbit” to Convert to Vertex Form

## How to Graph a Parabola in Vertex Form

See the following webpages:

- “How to Graph a Parabola in Vertex Form”. 2022.
*study.com*. https://study.com/skill/learn/how-to-graph-a-parabola-in-vertex-form-explanation.html. - “Graphing Quadratic Functions Using Vertex Form and Intercept Form”. 2022.
*Mathleaks*. https://mathleaks.com/study/graphing_Quadratic_Functions_Using_Vertex_Form_and_Intercept_Form. - Stapel, Elizabeth. 2023. “Using The Leading Coefficient & Finding The Vertex”.
*Purplemath*. https://www.purplemath.com/modules/grphquad2.htm.

## References

“3.3 Vertex Form of a Quadratic Function”. 2022. *CK-12*. https://flexbooks.ck12.org/cbook/ck-12-interactive-algebra-2/section/3.3/related/lesson/vertex-form-of-a-quadratic-function-alg-i-hnrs/.

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.

“Quadratic Equation”. 2021. *Mathematical Mysteries*. https://mathematicalmysteries.org/quadratic-equation/.

“Quadratic Equation – Formula, Examples | Quadratic Formula”. 2023. * CUEMATH*. https://www.cuemath.com/algebra/quadratic-equations/.

“Quadratic Equation – Wikipedia”. 2023. *en.wikipedia.org*. https://en.wikipedia.org/wiki/Quadratic_equation.

“Quadratic Equations”. 2023. *mathsisfun.com*. https://www.mathsisfun.com/algebra/quadratic-equation.html.

“Quadratic Formula – Wikipedia”. 2023. *en.wikipedia.org*. https://en.wikipedia.org/wiki/Quadratic_formula.

Roberts, Donna. 2023. “Vertex Form Of Quadratic Equation – MathBitsNotebook (A1 – CCSS Math)”. *mathbitsnotebook.com*. https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html.

“Standard Form To Vertex Form – Formula, Examples, FAQs”. 2023. *CUEMATH*. https://www.cuemath.com/algebra/standard-form-to-vertex-form/.

Stapel, Elizabeth. 2023. “The Quadratic Formula Explained | Purplemath”. *Purplemath*. https://www.purplemath.com/modules/quadform.htm.

Stapel, Elizabeth. 2023. “Using The Leading Coefficient & Finding The Vertex”. *Purplemath*. https://www.purplemath.com/modules/grphquad2.htm.

The feature image on this page is from the webpage “Python Program To Solve Quadratic Equation – GeeksForGeeks”. 2020. *GeeksForGeeks*. https://www.geeksforgeeks.org/python-program-to-solve-quadratic-equation/.