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Exponential Functions

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and introduces exponential functions used in growth and decay situations.

Learning Outcomes

Definition of Exponential Functions

An exponential function is a function in which the variable appears in the exponent.

Basic Form
f(x) = ax

where:

1

Examples of Exponential Functions

Standard Forms
  • f(x) = 2x
  • g(x) = (½)x
  • h(x) = 3x
  • y = 5x

Key Properties

1

Domain and Range

  • Domain: All real numbers (ℝ)
  • Range: y > 0
  • Never touches x-axis
2

Key Points

  • Y-intercept: (0, 1)
  • Asymptote: y = 0
  • Always positive: f(x) > 0
3

Growth vs Decay

  • Growth: a > 1 (increasing)
  • Decay: 0 < a < 1 (decreasing)
  • Special: a = 1 (constant)

Graphing Exponential Functions

2

Step-by-Step Graphing

Problem

Sketch the graph of f(x) = 2x

Solution
  1. Create table of values:
    x = -2: 2⁻² = ¼
    x = -1: 2⁻¹ = ½
    x = 0: 2⁰ = 1
    x = 1: 2¹ = 2
    x = 2: 2² = 4
  2. Plot points:

    (-2, ¼), (-1, ½), (0, 1), (1, 2), (2, 4)

  3. Draw curve:

    Smooth curve approaching x-axis (asymptote y = 0)

  4. Identify:

    Y-intercept: (0, 1)

    Asymptote: y = 0

    Increasing function (a = 2 > 1)

Transformations of Exponential Functions

Transformed Form
f(x) = a(x - p) + q
H

Horizontal Shift

p: Horizontal translation

  • f(x) = a(x - p): shift right p units
  • f(x) = a(x + p): shift left p units
V

Vertical Shift

q: Vertical translation

  • f(x) = ax + q: shift up q units
  • New asymptote: y = q
  • Y-intercept becomes (0, 1 + q)
E

Example Transformation

Graph f(x) = 2(x-1) + 3

  • Shift right 1 unit
  • Shift up 3 units
  • Asymptote: y = 3
  • Y-intercept: f(0) = 2⁻¹ + 3 = 3.5

Solving Exponential Equations

Example 1

Solve 2x = 8

Solution
  1. Express as same base:
    8 = 2³
  2. Set exponents equal:
    2x = 2³
  3. Solve:
    x = 3
Example 2

Solve 3x+1 = 9

Solution
  1. Express as same base:
    9 = 3²
  2. Set exponents equal:
    3x+1 = 3²
  3. Solve:
    x + 1 = 2
    x = 1
Example 3

Solve 5x = 125

Solution
  1. Express as same base:
    125 = 5³
  2. Set exponents equal:
    5x = 5³
  3. Solve:
    x = 3

Applications of Exponential Functions

Compound Interest

Problem

R1000 is invested at 8% annual interest compounded yearly. Find value after 5 years.

Solution
  1. Formula:
    A = P(1 + r)t

    Where A = final amount, P = principal, r = rate, t = time

  2. Substitute values:
    P = 1000, r = 0.08, t = 5
  3. Calculate:
    A = 1000(1 + 0.08)5
    A = 1000(1.08)5
    A = 1000 × 1.46933 ≈ R1469.33

Population Growth

Problem

A bacteria colony doubles every 3 hours. Starting with 100 bacteria, find population after 12 hours.

Solution
  1. Formula:
    P = P₀ × 2(t/d)

    Where P₀ = initial, t = time, d = doubling time

  2. Substitute values:
    P₀ = 100, t = 12, d = 3
  3. Calculate:
    P = 100 × 2(12/3)
    P = 100 × 24
    P = 100 × 16 = 1600 bacteria

Radioactive Decay

Problem

A substance decays by 15% per hour. Starting with 500g, find amount after 4 hours.

Solution
  1. Formula:
    A = A₀(1 - r)t

    Where A₀ = initial, r = decay rate, t = time

  2. Substitute values:
    A₀ = 500, r = 0.15, t = 4
  3. Calculate:
    A = 500(1 - 0.15)4
    A = 500(0.85)4
    A = 500 × 0.522 ≈ 261g

Practice Questions

Question 1

Sketch the graph of f(x) = 3x

Question 2

State whether f(x) = (¼)x is increasing or decreasing

Question 3

Solve 4x = 64

Question 4

Sketch y = 2(x+1) - 3 showing asymptote

Question 5

R2000 invested at 6% compounded annually for 8 years

Question 6

Solve 9x = 27

Summary of Key Concepts

Exponential Growth (a > 1):
  • Increasing function
  • Examples: Population growth, compound interest
  • Graph rises from left to right
Exponential Decay (0 < a < 1):
  • Decreasing function
  • Examples: Radioactive decay, depreciation
  • Graph falls from left to right
Transformations:
  • f(x) = a(x-p) + q
  • p: horizontal shift
  • q: vertical shift (new asymptote y = q)

Download Notes

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