Exponential Graphs (AQA GCSE Further Maths): Revision Note

Exponential graphs

What is an exponential?

• An exponential is a function where the power is a variable, usually $x$

  • $y=3^x$ is an example of an exponential

• In this course exponentials will be in either of the following forms

  • $y=a{b}^x$

  • $y=a{b}^{-x}$

  • Where $a$ and $b$ are rational numbers, $b>0$ and $x$ is a variable

  • $a$ can be equal to 1, resulting in $y=b^x$ or $y=b^{-x}$

  • All of the following are examples of exponentials you may encounter

    • $y=2\times 0.3^x$

    • $y=5.2\times 3.8^{-x}$

    • $y=4.1^x$

    • $y=0.{34}^{-x}$

What does an exponential graph look like?

• A graph of the form $y=b^x$ where $b$ is positive and larger than 1 will be increasing as $x$ increases

  • $y=5^x$ is increasing

• A graph of the form $y=b^{\mathit{-}\mathit{x}}$ where $b$ is positive and larger than 1 will be decreasing as $x$ increases

  • $y=7^{-x}$ is decreasing

• If $b$ is between 0 and 1, then the opposite is true

  • $y=0.4^x$ is decreasing

  • $y=0.6^{-x}$ is increasing

• An equation of the form $y=a{b}^x$ stretches the graph of $y=b^x$ vertically by scale factor $a$
  

  • If $a$ is negative, then this would also reflect the graph in the $x$-axis

• The $\mathit{y}$-intercept of $y=a{b}^x$ and $y=a{b}^{-x}$ will be $(0,\mathit{a})$

  • You can show this by substituting $x=0$ into the equation

  • Substituting $x=0$ into $y=a{b}^x$ or $y=a{b}^{-x}$ will reduce both to $y=a\times b^0=a\times 1=a$

  • This means that for an exponential in the form $y=b^x$ or $y=b^{-x}$, the $y$-intercept will simply be (0,1)

• The graphs do not cross the $x$-axis anywhere

• Exponential graphs do not have any minimum or maximum points

  • They are either always increasing, or always decreasing

How can I find the equation of an exponential graph?

• A typical exam question may give you one or two co-ordinates that lie on a curve, and an approximate form for the equation of the graph

  • e.g. $y=a{b}^{-x}$ or $y=k^x$

• Remember that all co-ordinates on the curve must satisfy the equation

• You can therefore substitute each coordinate into the given equation, and solve to find any unknown constants