Free Study Materials for Grades 10 to 12

Access structured notes, past papers, and revision guides aligned with the South African CAPS curriculum.

Quadratic Functions

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and introduces quadratic functions used in various mathematical and real-world contexts.

Learning Outcomes

Identify and understand the standard form of quadratic functions
Find roots using factorization, quadratic formula, and completing the square
Determine axis of symmetry, vertex, and y-intercept
Sketch graphs of quadratic functions accurately
Apply quadratic functions to solve real-world problems

Introduction to Quadratic Functions

A quadratic function is a polynomial function of degree two. Its general form is given by:

General Form
f(x) = ax² + bx + c

where:

a, b, c are constants
a ≠ 0 (if a = 0, it becomes a linear function)
The graph of a quadratic function is called a parabola

📊 Real-World Applications: Quadratic functions appear in projectile motion, optimization problems, curve fitting, economics, and engineering.

Standard Form and Coefficients

Students should be able to identify the coefficients a, b, and c in a given quadratic function and understand how they affect the graph.

Identifying Coefficients

Problem

Given the quadratic function f(x) = 2x² - 5x + 3, identify the coefficients a, b, and c, and determine the direction the parabola opens.

Solution

Identify coefficients:
a = 2, b = -5, c = 3
Determine direction:
Since a = 2 > 0, the parabola opens upwards.
Conclusion:
The quadratic function has a minimum point (vertex) because it opens upwards.

Finding the Roots (x-intercepts)

The roots of a quadratic function are the values of x for which f(x) = 0. These are also known as the x-intercepts of the parabola.

Factorization Method

Express the quadratic as a product of two linear factors.

Example

Find roots of f(x) = x² - 4x + 3

Solution

Factorize: x² - 4x + 3 = (x - 1)(x - 3)
Set factors to zero: x - 1 = 0 or x - 3 = 0
Solve: x = 1 or x = 3
Roots: x = 1, x = 3

Quadratic Formula

Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a

Example

Find roots of f(x) = 2x² + 3x - 2

Solution

Identify: a = 2, b = 3, c = -2
Apply formula:
x = (-3 ± √(3² - 4(2)(-2))) / (2(2))
Simplify:
x = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4 = (-3 ± 5) / 4
Solve:
x = (-3 + 5)/4 = 1/2

x = (-3 - 5)/4 = -2

Completing the Square

Rewrite in the form a(x - h)² + k

Example

Find roots of f(x) = x² + 6x + 5

Solution

Complete square:
x² + 6x + 5 = (x² + 6x + 9) - 9 + 5 = (x + 3)² - 4
Set to zero:
(x + 3)² - 4 = 0
Solve:
(x + 3)² = 4

x + 3 = ±2

x = -1
or
x = -5

Axis of Symmetry and Vertex

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

Axis of Symmetry Formula
x = -b / 2a

Vertex Coordinates
(-b / 2a, f(-b / 2a))

Finding Axis of Symmetry and Vertex

Problem

Find the axis of symmetry and vertex of the quadratic function f(x) = x² - 2x + 4.

Solution

Identify coefficients:
a = 1, b = -2, c = 4
Find axis of symmetry:
x = -b / 2a = -(-2) / (2 × 1) = 2 / 2 = 1

Axis of symmetry: x = 1
Find vertex:
x-coordinate = 1

y-coordinate = f(1) = (1)² - 2(1) + 4 = 1 - 2 + 4 = 3

Vertex: (1, 3)

y-intercept

The y-intercept is where the parabola intersects the y-axis (when x = 0).

Finding y-intercept

Problem

Find the y-intercept of the quadratic function f(x) = 3x² + x - 2.

Solution

Set x = 0:
f(0) = 3(0)² + (0) - 2 = -2
Identify y-intercept:
The y-intercept is the point (0, c) where c = -2
Conclusion:
y-intercept: (0, -2)

Sketching Quadratic Graphs

Step-by-Step Method:

Find roots (x-intercepts)
Find y-intercept (0, c)
Find axis of symmetry x = -b/2a
Find vertex (-b/2a, f(-b/2a))
Determine shape (upward if a > 0, downward if a < 0)
Plot points and draw smooth parabola

Sketching a Quadratic Function

Problem

Sketch the graph of the quadratic function f(x) = x² - 6x + 8.

Solution

Find roots:
x² - 6x + 8 = (x - 2)(x - 4)

Roots: x = 2 and x = 4
Find y-intercept:
f(0) = (0)² - 6(0) + 8 = 8

y-intercept: (0, 8)
Find axis of symmetry:
x = -(-6) / (2 × 1) = 6 / 2 = 3
Find vertex:
f(3) = (3)² - 6(3) + 8 = 9 - 18 + 8 = -1

Vertex: (3, -1)
Determine shape:
a = 1 > 0, so parabola opens upwards
Sketch:
Plot points: (2,0), (4,0), (0,8), (3,-1)

Draw smooth curve through points

Applications of Quadratic Functions

Real-World Problem Solving:

📈 Projectile Motion

Real-World Problem

A ball is thrown vertically upwards from the ground with an initial velocity of 20 m/s. The height h(t) of the ball after t seconds is given by h(t) = -5t² + 20t. Find:

The maximum height reached by the ball
The time when the ball hits the ground

Solution

Maximum height (vertex):
t = -b / 2a = -20 / (2 × -5) = -20 / -10 = 2 seconds

h(2) = -5(2)² + 20(2) = -20 + 40 = 20 meters

Maximum height: 20 meters
Time when ball hits ground (roots):
-5t² + 20t = 0

-5t(t - 4) = 0

t = 0
(start) or
t = 4
seconds
Ball hits ground after 4 seconds

📊 Optimization Problem

Real-World Problem

A farmer has 100 meters of fencing to create a rectangular garden. Find the dimensions that maximize the area of the garden.

Solution

Define variables:
Let length = x meters

Then width = (100 - 2x)/2 = 50 - x meters
Area function:
A(x) = x(50 - x) = -x² + 50x
Find maximum area (vertex):
x = -b / 2a = -50 / (2 × -1) = 25 meters

Width = 50 - 25 = 25 meters
Maximum area:
A(25) = -25² + 50(25) = 625 m²

Maximum area: 625 m² with dimensions 25m × 25m

Assessment Guidelines

Assessment of quadratic functions in Grade 10 should include:

Tests and Examinations

Identifying coefficients and parabola properties
Finding roots using different methods
Sketching graphs accurately
Solving application problems

Assignments and Projects

Problem-solving tasks and investigations
Real-world application projects
Mathematical modeling exercises
Group investigations of quadratic patterns

Download Notes

Download a printable PDF summary of quadratic functions for offline study.

Download Quadratic Functions Notes (PDF)

← Linear Functions Exponential Functions →