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From predicting the trajectory of a kicked football to designing satellite dishes and calculating maximum profit in business economics, quadratic functions are among the most essential mathematical models in algebra and calculus.

A quadratic function is a polynomial function of degree 2. When plotted on a Cartesian coordinate grid, its graph creates a distinctive curved U-shaped line known as a parabola.

Understanding how to analyse, factorise, and sketch quadratic functions is a foundational topic across all GCSE and A-Level Mathematics specifications.

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What Defines a Quadratic Function?

The word quadratic originates from the Latin word quadratum, meaning "square." An algebraic function is classified as quadratic if the highest exponent (power) of the independent variable x is exactly 2.

The standard form of a quadratic function is written as:

f(x) = ax^2 + bx + c \quad \text{or} \quad y = ax^2 + bx + c

Where:

a, b, and c are real constant numbers.
$a \neq 0$ (if $a = 0$ , the $x^2$ term disappears, leaving a linear function $y = bx + c$ ).
a is the leading coefficient, b is the linear coefficient, and c is the constant term.

Identifying Quadratic Equations

To determine whether an equation represents a true quadratic function, expand all brackets, collect like terms, and check the highest power of x₂:

Equation	Is it Quadratic?	Reason
$\text{[math]}$	Yes	Highest power of $\text{[math]}$ is 2.
$\text{[math]}$	No	Degree is 4 due to the $\text{[math]}$ term.
$\text{[math]}$	No	Linear function (degree 1).
$\text{[math]}$	Yes	Expands to degree 2 polynomial.
$\text{[math]}$	No	The $\text{[math]}$ terms cancel out, leaving a constant term (degree 0).

Sketching Parabola Graphs

Every quadratic function produces a symmetric curve called a parabola. To sketch a parabola accurately, you need to identify four key geometric features:

graph of a simple parabola with labels for y-intercept, axis of symmetry, vertex and x-intercepts — Image Source: Gianpiero Placidi

Direction of Opening

The sign of the leading coefficient a determines the orientation of the parabola:

If a > 0 (positive): The parabola opens upwards ( $\cup)$ , forming a "u-shape." The vertex represents the absolute minimum point of the curve.
If a < 0 (negative): The parabola opens downwards ( $\cap$ ), forming an "n-shape." The vertex represents the absolute maximum point of the curve.

Vertex (Turning Point)

The vertex is the central peak or trough where the parabola changes direction.

The x-coordinate of the vertex is given by the formula: $x = - \dfrac{b}{2a}$
The y-coordinate is found by evaluating f(x) at $x = -\dfrac{b}{2a}$ :

y = f\left(-\dfrac{b}{2a}\right)

Axis of Symmetry

A parabola is perfectly symmetrical. The axis of symmetry is a vertical line that passes directly through the vertex, dividing the curve into two identical mirror halves:

x = -\dfrac{b}{2a}

Intercepts with the Axes

y-intercept: The point where the curve crosses the vertical y-axis (where x = 0). For $f(x) = ax^2 + bx + c$ , the y-intercept is always (0, c).
x-intercepts (Roots or Zeros): The points where the curve crosses the horizontal x-axis (where y = 0). Found by solving $ax^2 + bx + c = 0$ .

The number of x-intercepts is governed by the discriminant ( $\Delta = b^2 - 4ac$ ):

$\Delta > 0$ : Two distinct real roots (the curve crosses the x-axis twice).
$\Delta = 0$ : Exactly one real root (the vertex sits directly on the x-axis).
$\Delta < 0$ : No real roots (the curve floats entirely above or below the x-axis).

Step-by-Step Graphing Method

To graph any quadratic function $f(x) = ax^2 + bx + c$ , follow these five steps:

Determine direction: Check the sign of a to see if the parabola opens upwards or downwards.
Calculate the vertex: Find $x = -\dfrac{b}{2a}$ and evaluate $y = f(x)$ .
Find the y-intercept: Set $x = 0 \implies (0, c)$ .
Find the x-intercepts: Set $y = 0$ and solve $ax^2 + bx + c = 0$ using factorisation, completing the square, or the quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Plot and sketch: Mark all calculated points on a grid and draw a smooth, continuous U-shaped curve.

Worked Example

Problem: Sketch the graph of the quadratic function $f(x) = x^2 + 6x + 8$

Step-by-Step Solution:

Step 1: Determine orientation: Here a = 1, b = 6, and c = 8. Since a = 1 > 0, the parabola opens upwards ( $\cup$ ).
Step 2: Find the vertex: Calculate the x-coordinate:

x = -\dfrac{b}{2a} = -\dfrac{6}{2(1)} = -\dfrac{6}{2} = -3

Calculate the y-coordinate by substituting x = -3 into f(x):

f(-3) = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1

The vertex is at (-3, -1) and the axis of symmetry is the line x = -3.

Step 3: Find the y-intercept: Set x = 0:

f(0) = (0)^2 + 6(0) + 8 = 8 \implies \mathbf{(0, 8)}

Step 4: Find the x-intercepts: Set f(x) = 0:

x^2 + 6x + 8 = 0

Factorise the quadratic expression:

(x + 2)(x + 4) = 0

Solve for x:

x + 2 = 0 \implies x = -2 \quad \text{or} \quad x + 4 = 0 \implies x = -4

The x-intercepts are at (-2, 0) and (-4, 0).

Step 5: Plot the graph:

Practice Questions & Answers

Score:

For the quadratic function $\text{[math]}$

a) State whether the parabola opens upwards or downwards.
b) Calculate the equation of the axis of symmetry.

Solution

a) The leading coefficient is a = -2. Because a <0, the parabola opens downwards.

b) The axis of symmetry formula is

$\text{[math]}$

Substituting a = -2 and b = 8:

$\text{[math]}$

The axis of symmetry is x = 2.

Use the discriminant $\text{[math]}$ to determine the number of real x-intercepts for the quadratic curve:

$\text{[math]}$

Solution

Identify the coefficients: a = 2, b = -4, c = 5.

Calculate the discriminant $\text{[math]}$

$\text{[math]}$

Since:

Delta = -24 <0

the quadratic function has zero real x-intercepts (the curve floats entirely above the x-axis).

A projectile is fired upwards from ground level. Its height h in metres after t seconds is modelled by the quadratic function:

$\text{[math]}$

Calculate the maximum height reached by the projectile and the time taken to reach this peak height.

Solution

Identify the peak coordinates: The maximum height occurs at the vertex of the quadratic function

$\text{[math]}$

Calculate time t at peak:

$\text{[math]}$

Calculate maximum height h(3):

$\text{[math]}$

Final Answer: The projectile reaches its maximum height of 47 metres after 3 seconds.

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function Problems

Domain of a Function

Maxima, Minima and Inflection Points

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Monotone Sequences: Theory and Exercises

Linear Functions Worksheet With Solutions

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Mean Value Theorem Word Problems

Rational Functions Worksheet

L’Hopital’s Rule Problems

Even and Odd Functions Worksheet

Asymptotes Worksheet

Absolute Value Functions Worksheet

Piecewise Functions Worksheet

Optimization Word Problems

Linear Function Word Problems

Finding Maxima and Minima | Worksheet with Worked Solutions

X and Y Intercepts Worksheet | Exercises with Full Solutions

Finite and Infinite Sequences Worksheet: GCSE and A Level Extension

Composition of Functions Worksheet

Maximum and Minimum Word Problems Worksheet with Answers

Quadratic Function Word Problems

Graphing Worksheet

Finding Inverse Functions Worksheet with Answers

Domain of a Function Worksheet

year

Optimization

nth Term

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Kourtney Sison

August 2026

I get it somehow, there are others who are unfamiliar and I have a hard time with it but it’s okay that I get it.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for sharing your thoughts! It’s completely normal to find some parts unfamiliar or more difficult than others. The important thing is that you’re making progress and understanding more as you practise. Keep going! 😊

Kourtney Sison

August 2026

At least I understood some of it, but there are others that I’m confused about and have a hard time understanding, but I get it.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for sharing! It’s great that you’ve understood some of it already. Don’t worry if a few parts are still confusing — keep practising and take your time with the trickier sections. You’ll get there! 😊

Normina Mangogawan

August 2026

The lesson was a little difficult for me, especially the word problems, but the examples helped me understand it better.

Vanessa - Editorial Manager Superprof UK

September 2026

Thanks for your feedback! Word problems can definitely be tricky, so we’re glad the examples helped make the lesson clearer. Keep practising — it gets easier with time! 😊

Kourtney Sison /Normina Mangogawan

August 2026

Thankyouuu po

Summarise with AI:

Chapters

What Defines a Quadratic Function?
Sketching Parabola Graphs
Step-by-Step Graphing Method
Practice Questions & Answers

Quadratic Function Graphs: Theory and Exercises

What Defines a Quadratic Function?

Identifying Quadratic Equations

Sketching Parabola Graphs

Direction of Opening

Vertex (Turning Point)

Axis of Symmetry

Intercepts with the Axes

Step-by-Step Graphing Method

Worked Example

Step-by-Step Solution:

Practice Questions & Answers

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function Problems

Domain of a Function

Maxima, Minima and Inflection Points

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Monotone Sequences: Theory and Exercises

Linear Functions Worksheet With Solutions

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Mean Value Theorem Word Problems

Rational Functions Worksheet

L’Hopital’s Rule Problems

Even and Odd Functions Worksheet

Asymptotes Worksheet

Absolute Value Functions Worksheet

Piecewise Functions Worksheet