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This interactive applet from GeoGebra allows you to input quadratic equations in the form y = ax² + bx + c and visualize the parabola in real-time. Adjust parameters to see how changes affect the vertex, axis of symmetry, and intercepts. Great for understanding transformations over the real set R.

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Desmos offers a free online graphing tool where you can plot quadratic functions, explore their graphs, and interact with sliders to vary coefficients. Experiment with domain and range in R, and create your own quadratic models for real-world scenarios like projectile motion.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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From the University of Colorado, this simulation lets you build and graph quadratic functions interactively. Drag points to see the parabola form and explore roots and vertices. Ideal for hands-on learning of quadratic behavior in the real numbers.

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Discover how changing coefficients changes the shape of a curve. View the graphs of individual terms (e.g. y=bx) to see how they add to generate the polynomial curve. Generate definitions for vertex, roots, and axis of symmetry. Compare different forms of a quadratic function. Define a curve by its focus and directrix.

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An simple yet effective applet to draw and manipulate quadratic graphs. Change a, b, c values and observe the parabola's shape, direction, and position. Helps in studying the standard form and its graphical representation over R.

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Khan Academy: Intro to Quadratic Functions

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Khan Academy: Intro to Quadratic Functions

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A beginner-friendly video series explaining quadratic functions, their graphs, and key features like the vertex form. Includes practice problems to reinforce concepts over the real numbers. Duration: 10-15 minutes per video.

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3,100 possible mastery points

About this unit

We've seen linear and exponential functions, and now we're ready for quadratic functions. We'll explore how these functions and the parabolas they produce can be used to solve real-world problems.

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Quadratic functions & equations | Algebra 1 | Math | Khan Academy

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We've seen linear and exponential functions, and now we're ready for quadratic functions. We'll explore how these functions and the parabolas they produce can be used to solve real-world problems.

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Visually stunning animation exploring the geometry behind quadratic equations. See how parabolas arise from real-world contexts and manipulate them interactively. Perfect for deeper intuition on degree 2 functions in R.

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Step-by-step video on graphing quadratic functions and finding roots using the graph. Includes examples with real coefficients and discussions on the discriminant for solutions in R.

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Fun and engaging animated video breaking down quadratic functions for Grade 9 level. Covers the U-shaped graph, standard form, and basic experiments with plotting points over the real line.

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Projectile Motion Experiment with Quadratics

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Projectile Motion Experiment with Quadratics

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A simple at-home experiment: Drop or throw objects and measure trajectories to fit quadratic models. Use graph paper or a phone app to plot data points and derive the equation y = -16t² + vt + h, exploring physics in R.

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Design a model bridge using quadratic functions to minimize material while maximizing strength. Calculate the parabolic arch using y = ax² + c and test with everyday materials. Relates to real-world applications of degree 2 functions.

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Outdoor activity: Observe parabolic shapes in rainbows, fountains, or bouncing balls. Sketch and approximate quadratic equations based on measurements. Encourages connecting abstract math to real phenomena in the set R.

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Use Microsoft Excel or Google Sheets to input quadratic data (e.g., area vs. side length) and generate graphs. Experiment with formulas to find maximum values, simulating real data analysis for functions over R.

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Printable interactive worksheets with quadratic function problems, including graphing and solving. Includes answer keys for self-checking, focused on real number solutions.

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Comprehensive online guide with examples, formulas, and interactive quizzes on quadratic functions. Covers standard, vertex, and factored forms over the reals, with attractive visuals.

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Quadratic functions are used in different fields of engineering and science to obtain values of different parameters. Graphically, they are represented by a parabola. Depending on the coefficient of the highest degree, the direction of the curve is decided. The word "Quadratic" is derived from the word "Quad" which means square. In other words, a quadratic function is a "polynomial function of degree 2." There are many scenarios where quadratic functions are used. Did you know that when a rocket is launched, its path is described by quadratic function?

In this article, we will explore the world of quadratic functions in math. You will get to learn about the graphs of quadratic functions, quadratic functions formulas, and other interesting facts about the topic. We will also solve examples based on the concept for a better understanding.

What is Quadratic Function?

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2. A quadratic function has a minimum of one term which is of the second degree. It is an algebraic function.

The parent quadratic function is of the form f(x) = x ² and it connects the points whose coordinates are of the form (number, number ²). Transformations can be applied on this function on which it typically looks of the form f(x) = a (x - h) ² + k and further it can be converted into the form f(x) = ax ² + bx + c. Let us study each of these in detail in the upcoming sections.

Standard Form of a Quadratic Function

The standard form of a quadratic function is of the form f(x) = ax ² + bx + c, where a, b, and c are real numbers with a ≠ 0.

Quadratic Function Examples

The quadratic function equation is f(x) = ax ² + bx + c, where a ≠ 0. Let us see a few examples of quadratic functions:

Now, consider f(x) = 4x-11; Here a = 0, therefore f(x) is NOT a quadratic function.

Vertex of Quadratic Function

The vertex of a quadratic function (which is in U shape) is where the function has a maximum value or a minimum value. The axis of symmetry of the quadratic function intersects the function (parabola) at the vertex.

Quadratic Functions Formula

A quadratic function can always be factorized, but the factorization process may be difficult if the zeroes of the expression are non-integer real numbers or non-real numbers. In such cases, we can use the quadratic formula to determine the zeroes of the expression. The general form of a quadratic function is given as: f(x) = ax ² + bx + c, where a, b, and c are real numbers with a ≠ 0. The roots of the quadratic function f(x) can be calculated using the formula of the quadratic function which is:

Different Forms of Quadratic Function

A quadratic function can be in different forms: standard form, vertex form, and intercept form. Here are the general forms of each of them:

Standard form: f(x) = ax² + bx + c, where a ≠ 0.

Vertex form: f(x) = a(x - h)² + k, where a ≠ 0 and (h, k) is the vertex of the parabola representing the quadratic function.

Intercept form: f(x) = a(x - p)(x - q), where a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola representing the quadratic function.

The parabola opens upwards or downwards as per the value of 'a' varies:

If a > 0, then the parabola opens upward.

If a < 0, then the parabola opens downward.

We can always convert one form to the other form. We can easily convert vertex form or intercept form into standard form by just simplifying the algebraic expressions. Let us see how to convert the standard form into each vertex form and intercept form.

Converting Standard Form of Quadratic Function Into Vertex Form

A quadratic function f(x) = ax ² + bx + c can be easily converted into the vertex form f(x) = a (x - h) ² + k by using the values h = -b/2a and k = f(-b/2a). Here is an example.

Example: Convert the quadratic function f(x) = 2x ² - 8x + 3 into the vertex form.

Step - 1: By comparing the given function with f(x) = ax² + bx + c, we get a = 2, b = -8, and c = 3.

Step - 2: Find 'h' using the formula: h = -b/2a = -(-8)/2(2) = 2.

Step - 3: Find 'k' using the formula: k = f(-b/2a) = f(2) = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5.

Step - 4: Substitute the values into the vertex form: f(x) = 2 (x - 2)² - 5.

Converting Standard Form of Quadratic Function Into Intercept Form

A quadratic function f(x) = ax ² + bx + c can be easily converted into the vertex form f(x) = a (x - p)(x - q) by using the values of p and q (x-intercepts) by solving the quadratic equation ax ² + bx + c = 0.

Example: Convert the quadratic function f(x) = x ² - 5x + 6 into the intercept form.

Step - 1: By comparing the given function with f(x) = ax² + bx + c, we get a = 1.

Step - 2: Solve the quadratic equation: x² - 5x + 6 = 0
Byfactoring the left side part, we get
(x - 3) (x - 2) = 0
x = 3, x = 2

Step - 3: Substitute the values into the intercept form: f(x) = 1 (x - 3)(x - 2).

Domain and Range of Quadratic Function

The domain of a quadratic function is the set of all x-values that makes the function defined and the range of a quadratic function is the set of all y-values that the function results in by substituting different x-values.

Domain of Quadratic Function

A quadratic function is a polynomial function that is defined for all real values of x. So, the domain of a quadratic function is the set of real numbers, that is, R.In interval notation, the domain of any quadratic function is (-∞, ∞).

Range of Quadratic Function

The range of the quadratic function depends on the graph's opening side and vertex. So, look for the lowermost and uppermost f(x) values on the graph of the function to determine the range of the quadratic function. The range of any quadratic function with vertex (h, k) and the equation f(x) = a(x - h) ² + k is:

y ≥ k (or) [k, ∞) when a > 0 (as the parabola opens up when a > 0).

y ≤ k (or) (-∞, k] when a < 0 (as the parabola opens down when a < 0).

Graphing Quadratic Function

The graph of a quadratic function is a parabola. i.e., it opens up or down in the U-shape. Here are the steps for graphing a quadratic function.

Step - 1: Find the vertex.

Step - 2: Compute a quadratic function table with two columns x and y with 5 rows (we can take more rows as well) with vertex to be one of the points and take two random values on either side of it.

Step - 3: Find the corresponding values of y by substituting each x value in the given quadratic function.

Step - 4: Now, we have two points on either side of the vertex so that by plotting them on the coordinate plane and joining them by a curve, we can get the perfect shape. Also, extend the graph on both sides. Here is the quadratic function graph.

Example: Graph the quadratic function f(x) = 2x ² - 8x + 3.
Solution:
By comparing this with f(x) = ax ² + bx + c, we get a = 2, b = -8, and c = 3.

Step - 1: Let us find the vertex.
x-ccordinate of vertex = -b/2a = 8/4 = 2
y-coordinate of vertex = f(-b/2a) = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5.
Therefore, vertex = (2, -5).

Step - 2: Frame a table with vertex written in the middle row.

Step - 3: Fill the first column with two random numbers on either side of 2.

Step - 4: Find y by substituting each x-value in the given quadratic function. For example, when x = 0, y = 2(0)² - 8(0) + 3 = 3.

x

y

0

3

1

-3

2-5

3

-3

4

3

Step - 5: Just plot the above points and join them by a smooth curve.

Note: We can plot the x-intercepts and y-intercept of the quadratic function as well to get a neater shape of the graph.

The graph of quadratic functions can also be obtained using the quadratic functions calculator.

Maxima and Minima of Quadratic Function

Maxima or minima of quadratic functions occur at its vertex. It can also be found by using differentiation. To understand the concept better, let us consider an example and solve it. Let's take an example of quadratic function f(x) = 3x ² + 4x + 7.

Differentiating the function,

⇒f'(x) = 6x + 4

Equating it to zero,

⇒6x + 4 = 0

⇒ x = -2/3

Double differentiating the function,

⇒f''(x) = 6 > 0

Since the double derivative of the function is greater than zero, we will have minima at x = -2/3 (by second derivative test), and the parabola is upwards.

Similarly, if the double derivative at the stationary point is less than zero, then the function would have maxima. Hence, by using differentiation, we can find the minimum or maximum of a quadratic function.

☛ Related Articles
Important Notes on Quadratic Function:

The standard form of the quadratic function is f(x) = ax²+bx+c where a ≠ 0.

The graph of the quadratic function is in the form of a parabola.

The quadratic formula is used to solve a quadratic equation ax² + bx + c = 0 and is given by x = [ -b ± √(b² - 4ac) ] / 2a.

The discriminant of a quadratic equation ax² + bx + c = 0 is given by b²-4ac. This is used to determine the nature of the zeroes of a quadratic function.

Examples of Quadratic Function
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FAQs on Quadratic Function

What is Quadratic Function in Math?

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. In other words, a quadratic function is a " polynomial function of degree 2."

Why is the Name Quadratic Function?

The meaning of "quad" is "square". Hence, a polynomial function of degree 2 is called a quadratic function.

What is Quadratic Function Equation?

A quadratic function is a polynomial of degree 2 and so the equation of quadratic function is of the form f(x) = ax ² + bx + c, where 'a' is a non-zero number; and a, b, and c are real numbers.

What is the Vertex of Quadratic Function?

Vertex of a quadratic function is a point where the parabola changes direction and crosses the axis of symmetry. It is a point where the parabola changes from increasing to decreasing or from decreasing to increasing. At this point, the derivative of the quadratic function is 0.

What Are the Zeros of a Quadratic Function?

The zeroes of a quadratic function are points where the graph of the function intersects the x-axis. At the zeros of the function, the y-coordinate is 0 and the x-coordinate represents the zeros of the quadratic polynomial function. The zeros of a quadratic function are also called the roots of the function.

What is a Quadratic Functions Table?

A quadratic functions table is a table where we determine the values of y-coordinates corresponding to each x-coordinates and vice-versa. The table consists of the coordinates of the graph of the quadratic functions. We usually write the vertex of the quadratic functions in the quadratic functions in one of the rows of the table.

How to Draw Quadratic Graph?

The graph of a quadratic function is a parabola. It can be drawn by plotting the coordinates on the graph. We plug in the values of x and obtain the corresponding values of y, hence obtaining the coordinates of the graph. After plotting the coordinates on the graph, we connect the dots using a free hand to obtain the graph of the quadratic functions. Finding the vertex helps in drawing a quadratic graph.

How to Find the x-intercept of a Quadratic Function?

The X-intercept of a quadratic function can be found considering the quadratic function f(x) = 0 and then determining the value of x. In other words, the x-intercept is nothing but zero of a quadratic equation.

Is Parabola is a Quadratic Function?

A parabola is a graph of a quadratic function. A quadratic function is of the form f(x) = ax ² + bx + c with a not equal to 0. Parabola is a U-shaped or inverted U-shaped graph of a quadratic function.

How to Find the Inverse of a Quadratic Function?

The inverse of a quadratic function f(x) can be found by replacing f(x) by y. Then, we switch the roles of x and y, that is, we replace x with y and y with x. After this, we solve y for x and then replace y by f⁻¹(x) to obtain the inverse of the quadratic function f(x).

What are the Forms of Quadratic Function?

A quadratic function can be in different forms: standard form, vertex form, and intercept form. Here are the general forms of each of them:

Standard form: f(x) = ax² + bx + c, where a ≠ 0.

Vertex form: f(x) = a(x - h)² + k, where a ≠ 0 and (h, k) is the vertex of the parabola representing the quadratic function.

Intercept form: f(x) = a(x - p)(x - q), where a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola representing the quadratic function.

We can convert one of these forms into the other forms. For more information, click here.

What is the Difference Between Quadratic Function and Quadratic Equation?

A quadratic function is of the form f(x) = ax ² + bx + c, where a ≠ 0. Each point on its graph is of the form (x, ax ² + bx + c). This is for the graphing purpose. On the other hand, a quadratic equation is of the form ax ² + bx + c = 0, where a ≠ 0. This is for finding the solution and it gives definite values of x as solution.

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Quadratic Function - Standard Form, Formula, Examples

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Quadratic Functions are polynomial functions with one or more variables in which the highest power of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2.

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Inside Mathematics: Quadratic Video Lesson

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Inside Mathematics: Quadratic Video Lesson

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A classroom-style video lesson for 9th-11th graders on quadratic functions, emphasizing problem-solving strategies and interactive discussions.

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Think Academy: Solving Quadratics Methods

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Think Academy: Solving Quadratics Methods

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Article with embedded interactives on various ways to solve quadratic equations, including graphing and the formula. Updated with recent examples for Grade 9.

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If you've ever opened Google and seen its logo come alive with animation, you've experienced a Google Doodle-those playful redesigns that celebrate everything from holidays to history-makers. Since the very first Doodle in 1998, more than 5,000 have appeared, from Pac-Man to the Olympics. On September 8, 2025, Google gave math its moment in the spotlight with the Quadratic Equation Doodle. The letters of "Google" became part of a basketball shot tracing a perfect parabola, a clever nod to how the quadratic formula describes real-world motion.

But let's be real-we're not here to analyze Google's logo. We're an online math class brand, and today's spotlight is on the quadratic equation itself. Quadratic equations are a cornerstone of Algebra, showing up in everything from projectiles to area optimization.

In this guide, we'll explain what a quadratic equation is, what the quadratic graph (parabola) looks like, when students typically learn quadratics in U.S. schools, why they can be challenging, and the three main solving methods-factoring, using the quadratic formula, and completing the square-with examples of each step.

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, written as:

𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0

𝑎, 𝑏, 𝑐 are constants, 𝑎 is not equal to zero.

𝑥 is the variable.

The solutions (roots) are the 𝑥-values where the parabola crosses the x-axis.

How to Graph a Quadratic Function (Parabola Explained)

The graph of a quadratic equation is a U-shaped curve called a parabola. Every parabola has key features that make it easier to analyze and graph.

Key Features of a Parabola

Parabola: The U-shaped curve formed by graphing a quadratic function.

Vertex: The turning point of the parabola, which is the minimum (if it opens upward) or maximum (if it opens downward).

Axis of Symmetry: A vertical line that divides the parabola into two mirror halves. If the vertex is (ℎ, 𝑘), the axis of symmetry is 𝑥 = ℎ.

Opening Direction:

If 𝑎 > 0 in the quadratic equation 𝑓(𝑥) = 𝑎𝑥² + 𝑏𝑥 + 𝑐, the parabola opens upward.

If 𝑎 < 0, the parabola opens downward.

Intercepts:

Y-intercept: The point where the parabola crosses the y-axis (set 𝑥 = 0).

X-intercepts (Roots): The points where the parabola crosses the x-axis. These can be found by solving the quadratic equation.

Types of Quadratic Functions

Quadratic equations can be expressed in three common forms. Each highlights different properties of the parabola.

When Do Students Learn Quadratic Equations?
Quadratic equations are usually introduced in Algebra 1.

Public schools: Typically taught in high school (9th-10th grade), or as early as 8th grade in accelerated tracks.

Private/advanced programs: Sometimes introduced as early as 7th grade.

Quadratics return in later courses such as Geometry, Algebra II, Pre-Calculus, and also appear on standardized tests like the SAT and ACT.

Why Are Quadratic Equations Challenging?

Students often struggle because:

Multiple solving methods can feel confusing.

Negative signs and square roots lead to mistakes.

Graphing links algebra with geometry

The discriminant introduces abstract cases with complex solutions.

Set real world model by quadratic equations

How to Solve Quadratic Equations

1. Factoring

If the quadratic factors neatly, this is the quickest method.

Example:

𝑥² + 5𝑥 + 6 = 0

(𝑥 + 2)(𝑥 + 3) = 0 ⇒ 𝑥 = −2, −3

2. Quadratic Formula

The universal method for solving a quadratic equation is:

\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

The part under the square root, 𝑏² − 4𝑎𝑐, is called the discriminant.
Why does the discriminant matter?

If 𝑏² − 4𝑎𝑐 > 0 → two distinct real roots

If 𝑏² − 4𝑎𝑐 = 0 → one repeated real root

If 𝑏² − 4𝑎𝑐 < 0 → two complex roots

3. Completing the Square

This rewrites the quadratic as a perfect square trinomial.

Example:

𝑥² − 2𝑥 − 3 = 0

𝑥² − 2𝑥 = 3

𝑥² − 2𝑥 + 1 = 4

(𝑥 − 1)² = 4 ⇒ 𝑥 = 1 ± 2

Final result: 𝑥 = 3, −1

Example: Three Ways of Solving the Same Quadratic
Problem:
Solve the equation

𝑥² − 2𝑥 − 3 = 0
Solution:
Factoring

(𝑥 − 3)(𝑥 + 1) = 0 ⇒ 𝑥 = 3, −1
Apply Quadratic Formula
\[ x = \frac{-(-2) \pm \sqrt{(-2)^{2} - 4(1)(-3)}}{2(1)} \]

⇒ 𝑥 = 3, −1
Completing the Square
(𝑥 − 1)² = 4 ⇒ 𝑥 = 3, −1

Summary: Key Takeaways of Quadratic Equations

Quadratics have the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.

Factoring is fastest but not always possible

The quadratic formula always works

The discriminant shows whether solutions are two real, one repeated, or complex

Completing the square connects algebra with graphing

Additional Math Topics for Algebra I - Free Worksheets

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How to Solve Quadratic Equations: Forms and Methods - Math Learning | Think Academy US

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Learn quadratic equations step by step: definition, graph of a parabola, common challenges, and solving methods including factoring and formula.