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.
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.
“Shifting Graphs Up/down Left/Right.” 2025. onemathematicalcat.org. 2025. https://www.onemathematicalcat.org/Math/Precalculus_obj/shiftUDLR.htm.
“Horizontal and Vertical Stretching/Shrinking.” 2025. onemathematicalcat.org. 2025. https://www.onemathematicalcat.org/Math/Precalculus_obj/horizVertScaling.htm.
“Reflecting about Axes, and the Absolute Value Transformation.” 2025. onemathematicalcat.org. 2025. https://www.onemathematicalcat.org/Math/Precalculus_obj/reflectAbsVal.htm.
“Functions Transformations – Graphing, Rules, Tricks.” 2020. CUEMATH. 2020. https://www.cuemath.com/calculus/transformation-of-functions/.
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.
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/.
Dincer, Baris. “Quadratic Functions — Applications I”. 2023. Medium. https://brsdincer.medium.com/quadratic-functions-applications-i-ef58936cb81a.
Dincer, Baris. “Quadratic Functions — Applications II”. 2023. Medium. https://brsdincer.medium.com/quadratic-functions-applications-ii-154b61d5eb0e.
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/.
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