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Hyperbola Functions

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and is assessed under Functions and Graphs.

Learning Outcomes

Introduction to Hyperbola 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

where:

Key Properties of Hyperbolas

1

Basic Shape

  • Two separate branches
  • Approaches but never touches asymptotes
  • Symmetrical about asymptote intersection
  • Quadrants depend on sign of a
2

Asymptotes

  • Vertical: x = -p
  • Horizontal: y = q
  • Lines that graph approaches but never crosses
3

Domain & Range

  • Domain: x ≠ -p
  • Range: y ≠ q
  • All real numbers except asymptote values

Sketching Hyperbola Graphs

1

Sketching a Hyperbola

Problem

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

Solution
  1. Identify asymptotes:
    Vertical: x = 1 (from x - 1 = 0)
    Horizontal: y = 3
  2. Determine orientation:

    Since a = 2 > 0, branches are in quadrants I and III

  3. Plot points:
    x = 0: y = 2/(0-1) + 3 = -2 + 3 = 1 → (0, 1)
    x = 2: y = 2/(2-1) + 3 = 2 + 3 = 5 → (2, 5)
    x = -1: y = 2/(-1-1) + 3 = -1 + 3 = 2 → (-1, 2)
    x = 3: y = 2/(3-1) + 3 = 1 + 3 = 4 → (3, 4)
  4. Draw graph:

    Draw asymptotes at x=1 and y=3

    Plot points and draw smooth curves

    Curves approach but don't cross asymptotes

Finding Equations of Hyperbolas

Example 1

Find equation with asymptotes x = -2 and y = 1, passing through (0, 3)

Solution
  1. Identify p and q:
    x = -p = -2 → p = 2
    y = q = 1 → q = 1
  2. Write general form:
    y = a/(x + 2) + 1
  3. Substitute point (0, 3):
    3 = a/(0 + 2) + 1
    3 = a/2 + 1
    a/2 = 2 → a = 4
  4. Final equation:
    y = 4/(x + 2) + 1
Example 2

Find domain and range of y = -3/(x + 4) - 2

Solution
  1. Vertical asymptote:
    x + 4 = 0 → x = -4

    Domain: x ≠ -4

  2. Horizontal asymptote:
    y = -2

    Range: y ≠ -2

  3. Orientation:

    Since a = -3 < 0, branches are in quadrants II and IV

Example 3

Hyperbola passes through (2, 1) with asymptotes x=3, y=-1. Find equation.

Solution
  1. Identify parameters:
    p = -3, q = -1
    y = a/(x - 3) - 1
  2. Substitute (2, 1):
    1 = a/(2 - 3) - 1
    1 = a/(-1) - 1
    1 = -a - 1 → -a = 2 → a = -2
  3. Final equation:
    y = -2/(x - 3) - 1

Effects of Parameters

a

Parameter a

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

Parameter p

  • Horizontal shift
  • Vertical asymptote: x = -p
  • Positive p: shift left
  • Negative p: shift right
q

Parameter q

  • Vertical shift
  • Horizontal asymptote: y = q
  • Positive q: shift up
  • Negative q: shift down
2

Comparing Different Hyperbolas

Analysis

Compare these hyperbolas:

  1. y = 2/x
  2. y = -2/x
  3. y = 2/(x - 1) + 3
Comparison
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 (shifted)

Real-World Applications

Boyle's Law (Physics)

Application

Boyle's Law states that pressure (P) and volume (V) of a gas are inversely proportional at constant temperature: PV = k. If k = 200, find pressure when volume is 50L and sketch the relationship.

Solution
  1. Equation:
    PV = 200 → P = 200/V

    This is a hyperbola with a = 200

  2. Calculate for V = 50L:
    P = 200/50 = 4 kPa
  3. Properties:

    Asymptotes: V = 0 (vertical), P = 0 (horizontal)

    Domain: V > 0 (positive volume)

    Range: P > 0 (positive pressure)

  4. Interpretation:

    As volume increases, pressure decreases

    Curve shows inverse relationship

Work Rate Problems

Application

Two workers complete a job together in 6 hours. Worker A alone takes 10 hours. How long would Worker B take alone?

Solution
  1. Work rates:
    A: 1/10 job per hour
    B: 1/x job per hour
    Together: 1/6 job per hour
  2. Equation:
    1/10 + 1/x = 1/6
  3. Solve:
    1/x = 1/6 - 1/10
    1/x = 5/30 - 3/30 = 2/30 = 1/15
    x = 15 hours
  4. Hyperbolic relationship:

    Time together vs individual times shows inverse relationship

Practice Questions

Question 1

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

Question 2

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

Question 3

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

Question 4

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

Question 5

Sketch and compare y = 3/x and y = -3/x

Question 6

Two pipes fill a tank in 4 hours together. Pipe A takes 6 hours alone. Find Pipe B's time.

Summary of Key Concepts

Hyperbola Form:
  • y = a/(x + p) + q
  • Vertical asymptote: x = -p
  • Horizontal asymptote: y = q
Properties:
  • Two separate branches
  • Never crosses asymptotes
  • Domain: x ≠ -p
  • Range: y ≠ q
  • Symmetrical about point (-p, q)
Orientation:
  • a > 0: Quadrants I & III
  • a < 0: Quadrants II & IV
  • |a| controls distance from asymptotes

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