Hyperbolic Functions

Master the shape and properties of hyperbolas

CAPS Grade 10 Mathematics

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

Hyperbola Graph: y = a/(x + p) + q

Graph Controls

Adjust Parameters

a (stretch/orientation)

p (horizontal shift)

q (vertical shift)

Current Function: y = 2/(x + 0) + 0

Asymptotes: x = 0, y = 0

Hover over graph to see coordinates: x = 0, y = undefined

Learning Outcomes

Understand the shape and properties of hyperbolas
Determine equations of hyperbolas from given information
Sketch hyperbola graphs with transformations
Identify asymptotes, domain, and range
Apply hyperbola concepts to real-world situations

Introduction to Hyperbolic Functions

Hyperbola functions are rational functions that produce curves with two separate branches. They are characterized by their asymptotes and symmetrical properties.

General Form
y = a/(x + p) + q

a controls shape and orientation
p represents horizontal shift
q represents vertical shift

Quiz 1 - General Form

What is the general form of a hyperbolic function?

A) y = ax + b

B) y = ax² + bx + c

C) y = a/(x + p) + q

D) y = a·bˣ

Key Properties of Hyperbolas

Basic Shape

Two separate branches
Approaches but never touches asymptotes
Symmetrical about asymptote intersection

Asymptotes

Vertical: x = -p
Horizontal: y = q

Domain & Range

Domain: x ≠ -p
Range: y ≠ q

Quiz 2 - Asymptotes

For y = 3/(x - 2) + 4, what is the vertical asymptote?

A) x = 2

B) x = -2

C) x = 3

D) x = 4

Sketching Hyperbola Graphs

Sketching a Hyperbola

Problem

Sketch y = 2/(x - 1) + 3

Solution

1. Vertical asymptote: x = 1
2. Horizontal asymptote: y = 3
3. a = 2 > 0 → branches in quadrants I and III relative to asymptotes
4. Points: (0,1), (2,5), (-1,2), (3,4)

Quiz 3 - Sketching

For y = -3/(x + 2) - 1, what are the asymptotes?

A) x = -2, y = -1

B) x = 2, y = 1

C) x = -2, y = 1

D) x = 2, y = -1

Finding Equations of Hyperbolas

Find Equation from Asymptotes and Point

Example

Asymptotes x = -2, y = 1, passing through (0, 3)

Solution

x = -p = -2 → p = 2, q = 1

y = a/(x + 2) + 1

Substitute (0,3): 3 = a/2 + 1 → a/2 = 2 → a = 4

y = 4/(x + 2) + 1

Effects of Parameters

Parameter a

a > 0: Branches in quadrants I & III
a < 0: Branches in quadrants II & IV
Larger |a| = further from asymptotes

Parameter p

Horizontal shift
Vertical asymptote: x = -p

Parameter q

Vertical shift
Horizontal asymptote: y = q

Comparing Hyperbolas

Function	Vertical Asymptote	Horizontal Asymptote	Quadrants
y = 2/x	x = 0	y = 0	I & III
y = -2/x	x = 0	y = 0	II & IV
y = 2/(x-1)+3	x = 1	y = 3	I & III

Real-World Applications

Boyle's Law (Physics)

Application

Boyle's Law: PV = k. If k = 200, find pressure when volume is 50L.

P = 200/V

P = 200/50 = 4 kPa

Hyperbola with asymptotes V = 0, P = 0

Work Rate Problems

Application

Two workers together take 6 hours. Worker A alone takes 10 hours. Find Worker B's time.

1/10 + 1/x = 1/6

1/x = 1/6 - 1/10 = 2/30 = 1/15

x = 15 hours

Practice & Assess

Test your knowledge with these interactive games.

Match - Parameter Roles

Orientation & stretch

Horizontal shift

Vertical shift

Asymptotes

Lines graph approaches

Fill - Asymptote Formula

For y = a/(x + p) + q, vertical asymptote is x = ___

Practice Questions

Sketch y = 4/(x + 2) showing asymptotes

Find equation with asymptotes x=3, y=-2 through (4, 1)

Domain and range of y = -1/(x - 5) + 2

If y varies inversely with x and y=8 when x=3, find equation

Compare y = 3/x and y = -3/x

Summary of Key Concepts

General Form: y = a/(x + p) + q

Asymptotes: Vertical x = -p, Horizontal y = q

Domain: x ≠ -p, Range: y ≠ q

Orientation: a > 0 (Quadrants I & III), a < 0 (Quadrants II & IV)

Key Terms

Hyperbola Asymptote Branches Domain Range Vertical Shift Horizontal Shift Inverse Proportion Rational Function

Exponential Functions Linear Functions