Exponential Graphs (Cambridge International IGCSE 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

  • $( 1 \frac{1}{2} )^{x = ( 2^{-1} } )^{x = 2^{-x} }$
  • 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 $x<0$

Worked Example (a) Identify the equation that shows exponential decay.

A: $y=4^{2x}$

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

C: $y=0.3^{x}$

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

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

Final answer: **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