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Exponential Graphs (Cambridge (CIE) IGCSE International Maths): Revision Note

Exam code: 0607

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Exponential Graphs

What is an exponential graph?

  • An exponential graph is of the form y equals a to the power of x

    • When a greater than 1, y equals a to the power of x represents exponential growth

      • Values of y increase as x increases

    • When 0 less than a less than 1, y equals a to the power of x represents exponential decay

      • a is positive but less than 1

      • Values of y decrease as x increases

  • Graphs of both exponential growth and exponential decay

    • have a horizontal asymptote at y equals 0

    • do not have a vertical asymptote

    • have a y-intercept of open parentheses 0 comma space 1 close parentheses

Exponential Growth: y = 2^x
Exponential Decay: y = (1/2)^x

  • Exponential decay can also be identified by a negative power using index laws

    • open parentheses 1 half close parentheses to the power of x equals open parentheses 2 to the power of negative 1 end exponent close parentheses to the power of x equals 2 to the power of negative x end exponent

    • This has the form y equals a to the power of negative x end exponent where a greater than 1

How do I compare exponential graphs?

  • For y equals a to the power of x where bold italic a bold greater than bold 1:

    • The graph with a higher value of a is the “higher” graph for bold italic x bold greater than bold 0

      • but the "lower" graph for bold italic x bold less than bold 0

    Comparing two exponential graphs, y = 2 to the power x and y = 3 to the power x


  • For y equals a to the power of x where bold 0 bold less than bold italic a bold less than bold 1:

    • The graph with a higher value of a is the “higher” graph for bold italic x bold greater than bold 0

      • but the "lower" graph for bold italic x bold less than bold 0

Comparing two exponential graphs, y = 0.2 to the power x and y = 0.3 to the power x

Worked Example

(a) Identify the equation that shows exponential decay.

A: y equals 4 to the power of 2 x end exponent

B: y equals 6 over x

C: y equals 0.3 to the power of x

D: y equals 5 x cubed

A is an exponential graph as the x is in the power
The base number is greater than 1 so it is exponential growth (not decay)

B has the x as the denominator in a fraction
It is a reciprocal graph

C is an exponential graph as the x is in the power
The base number is between 0 and 1 so it is exponential decay

D has x raised to the power 3
It is a cubic graph

C: bold italic y bold equals bold 0 bold. bold 3 to the power of x shows exponential decay

(b) On the same set of axes, sketch the graphs of  y equals 3 to the power of x  and  y equals 4 to the power of x.

Both graphs will have the "typical" y equals a to the power of x exponential growth shape for a greater than 1

y equals 4 to the power of x will be the "higher" graph for x greater than 0 and "lower" graph for x less than 0

Both graphs go through open parentheses 0 comma space 1 close parentheses and have an asymptote along the x-axis

A sketch of y = 3 to the power x and y = 4 to the power x on the same axes

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.