Exponential Functions

Master exponential growth, decay, and applications

CAPS Grade 10 Mathematics

This topic covers exponential functions of the form f(x) = a^x. Add points below and the graph will automatically find and display the exponential curve that passes through them. The blue curve updates immediately when you add points.

Exponential Graph: f(x) = a^x

Graph Controls

Plot Points to Define the Exponential Function

Points Added:
No points added yet. Add at least 1 point and click "Fit Exponential Curve".
Current Function: f(x) = 2^x
Hover over graph to see coordinates: x = 0, y = 1

Learning Outcomes

  • Define and identify exponential functions
  • Understand properties of exponential graphs
  • Sketch exponential graphs with transformations
  • Apply exponential functions to real-life problems
  • Solve basic exponential equations

Definition of Exponential Functions

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

Basic Form
f(x) = ax
where a > 0, a ≠ 1

Examples of Exponential Functions

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

Quiz 1 - Exponential Definition

Where does the variable appear in an exponential function?

A) In the base
B) In the exponent
C) As a coefficient
D) As a constant

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)

Quiz 2 - Properties

What is the y-intercept of f(x) = ax?

A) (0, 0)
B) (0, 1)
C) (0, a)
D) No y-intercept

Graphing Exponential Functions

Step-by-Step Graphing

Problem

Sketch f(x) = 2x

Table of Values
x -2 -1 0 1 2
y ¼ ½ 1 2 4

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

Y-intercept: (0, 1), Asymptote: y = 0, Increasing (a=2>1)

Transformations of Exponential Functions

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

Horizontal Shift

f(x) = a(x - p): shift right p units

f(x) = a(x + p): shift left p units

V

Vertical Shift

f(x) = ax + q: shift up q units

New asymptote: y = q

Example

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

  • Shift right 1 unit
  • Shift up 3 units
  • Asymptote: y = 3

Solving Exponential Equations

Example 1

Solve 2x = 8

8 = 2³
x = 3

Example 2

Solve 3x+1 = 9

9 = 3²
x + 1 = 2 → x = 1

Example 3

Solve 5x = 125

125 = 5³
x = 3

Quiz 3 - Solving Equations

Solve 4x = 64

A) x = 2
B) x = 3
C) x = 4
D) x = 5

Applications of Exponential Functions

Compound Interest

Problem

R1000 at 8% compounded annually for 5 years. Find value.

A = P(1 + r)t
A = 1000(1.08)5 = R1469.33

Population Growth

Problem

Bacteria doubles every 3 hours. Start with 100. Find after 12 hours.

P = 100 × 2(12/3)
P = 100 × 24 = 1600

Practice & Assess

Test your knowledge with these interactive games.

Match - Growth or Decay

f(x) = 2x
Growth
f(x) = (½)x
Decay
f(x) = 3x
Growth
f(x) = (¼)x
Decay

Fill - Transformed Form

f(x) = a(x - p) + ___

Practice Questions

Q1

Sketch f(x) = 3x

Q2

Is f(x) = (½)x increasing or decreasing?

Q3

Solve 9x = 27

Answers

Q1: Passes through (0,1), increasing | Q2: Decreasing | Q3: x = 1.5

Summary of Key Concepts

Exponential Growth (a > 1): Increasing, examples: population growth, compound interest
Exponential Decay (0 < a < 1): Decreasing, examples: radioactive decay, depreciation
Transformations: f(x) = a(x-p) + q, asymptote y = q

Key Terms

ExponentialGrowthDecayBaseAsymptoteDomainRangeY-interceptTransformationCompound InterestDoubling Time
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