Exponential Graphs (Cambridge (CIE) IGCSE International Maths)

Revision Note

Author

Naomi C

Expertise

Maths

Exponential Graphs

What is an exponential graph?

An exponential graph is of the form $y = a^{x}$
- When $a > 1$ , $y = a^{x}$ represents exponential growth
  - Values of y increase as x increases
- When $0 < a < 1$ , $y = a^{x}$ 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 $y = 0$
- do not have a vertical asymptote
- have a $y$ -intercept of $(0, 1)$

Exponential decay can also be identified by a negative power using index laws
- ${(\frac{1}{2})}^{x} = {(2^{- 1})}^{x} = 2^{- x}$
- This has the form $y = a^{- x}$ where $a > 1$

How do I compare exponential graphs?

For $y = a^{x}$ where $a > 1$ :
- The graph with a higher value of $a$ is the “higher” graph for $x > 0$
  - but the "lower" graph for $x < 0$
For $y = a^{x}$ where $0 < a < 1$ :
- The graph with a higher value of $a$ is the “higher” graph for $x > 0$
  - but the "lower" graph for $x < 0$

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: $y = 4^{2 x}$

B: $y = \frac{6}{x}$

C: $y = 0 . 3^{x}$

D: $y = 5 x^{3}$

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: $y = 0 . 3^{x}$ shows exponential decay

(b) On the same set of axes, sketch the graphs of $y = 3^{x}$ and $y = 4^{x}$ .

Both graphs will have the "typical" $y = a^{x}$ exponential growth shape for $a > 1$

$y = 4^{x}$ will be the "higher" graph for $x > 0$ and "lower" graph for $x < 0$

Both graphs go through $(0, 1)$ 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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