Parabolic Functions

Master the properties and equations of parabolas

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and is assessed under Functions & Graphs. Use the interactive graph below to visualize how changing parameters affects the parabola.

Parabola Graph: (x - h)² = 4p(y - k)

Graph Controls

Adjust Parameters

Current Function: (x)² = 4(y)
Vertex: (0, 0) | Focus: (0, 1) | Directrix: y = -1
Hover over graph to see coordinates: x = 0, y = 0

Learning Outcomes

  • Understand the definition and structure of a parabola
  • Identify key properties: vertex, focus, directrix
  • Work with standard equations in different orientations
  • Find vertex, focus, and directrix from given equations
  • Apply parabolas to real-world situations

Introduction to Parabolas

A parabola is a U-shaped curve that appears in physics, engineering, architecture, and many real-life applications.

Definition: A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

Key Components of a Parabola

F

Focus

  • Fixed point inside the curve
  • All points on parabola are equidistant from focus and directrix
D

Directrix

  • Fixed line outside the curve
  • Never touches the parabola
V

Vertex

  • Turning point of the parabola
  • Midpoint between focus and directrix
A

Axis of Symmetry

  • Line through vertex and focus
  • Divides parabola into two equal halves
p

Parameter p

  • Distance from vertex to focus
  • Distance from vertex to directrix
  • p > 0: opens upward/right
  • p < 0: opens downward/left

Quiz 1 - Parabola Components

What is the fixed point inside a parabola called?

A) Vertex
B) Focus
C) Directrix
D) Axis

Standard Equations of Parabolas

Vertical Parabola
(x - h)² = 4p(y - k)
  • Vertex: (h, k)
  • p > 0: opens upward
  • p < 0: opens downward
  • Focus: (h, k + p)
  • Directrix: y = k - p
Horizontal Parabola
(y - k)² = 4p(x - h)
  • Vertex: (h, k)
  • p > 0: opens right
  • p < 0: opens left
  • Focus: (h + p, k)
  • Directrix: x = h - p
Vertex at Origin
x² = 4py (vertical)
y² = 4px (horizontal)

Quiz 2 - Standard Form

For (x - h)² = 4p(y - k), where is the focus?

A) (h, k)
B) (h, k + p)
C) (h + p, k)
D) (h, k - p)

Finding Components from Equations

Vertical Parabola Example

Problem

Find vertex, focus, directrix of (x - 2)² = 8(y + 1)

Solution
  1. Vertex: (2, -1)
  2. 4p = 8 → p = 2 (opens upward)
  3. Focus: (2, -1 + 2) = (2, 1)
  4. Directrix: y = -1 - 2 = -3

Horizontal Parabola Example

Problem

Find components of y² = -12x

Solution
  1. Vertex: (0, 0)
  2. 4p = -12 → p = -3 (opens left)
  3. Focus: (-3, 0)
  4. Directrix: x = 3

Quiz 3 - Finding p

For (x - 1)² = 16(y - 2), what is p?

A) 2
B) 4
C) 8
D) 16

Finding Equations from Components

From Vertex and Focus

Example

Vertex (3, -2), focus (3, 0)

Solution
p = 0 - (-2) = 2
(x - 3)² = 4(2)(y + 2)
(x - 3)² = 8(y + 2)

Real-World Applications

Projectile Motion

Physics Application

A ball's path follows a parabolic trajectory. The vertex represents maximum height.

Satellite Dishes

Engineering Application

Parabolic shape focuses signals at the receiver (focus).

Practice & Assess

Test your knowledge with these interactive games.

Match - Parabola Orientation

(x-h)²=4p(y-k), p>0
Opens upward
(x-h)²=4p(y-k), p<0
Opens downward
(y-k)²=4p(x-h), p>0
Opens right
(y-k)²=4p(x-h), p<0
Opens left

Fill - Parabola Formula

For vertical parabola: (x - h)² = __(y - k)

Practice Questions

Q1

Find focus and directrix of y² = 12x

Q2

Equation with vertex (0,3) and p = -2

Q3

Does (x-1)² = -4(y+2) open up or down?

Summary of Key Concepts

Vertical: (x-h)² = 4p(y-k), Focus (h,k+p), Directrix y = k-p
Horizontal: (y-k)² = 4p(x-h), Focus (h+p,k), Directrix x = h-p
Orientation: p > 0 opens up/right, p < 0 opens down/left

Key Terms

ParabolaVertexFocusDirectrixAxis of SymmetryParameter pLatus Rectum
Linear FunctionsExponential Functions