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 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.
“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/.