Exponential Graphs (AQA GCSE Further Maths)

Revision Note

Written by: Jamie Wood

Reviewed by: Dan Finlay

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
  - $y = 2 \times 0 . 3^{x}$
  - $y = 5.2 \times 3 . 8^{- x}$
  - $y = 4 . 1^{x}$
  - $y = 0 . 34^{- x}$

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

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

Remember that the $y$ intercept can often be found by inspection, which may save you some working
- For $y = b^{x}$ or $y = b^{- x}$ the $y$ -intercept is $(0, 1)$
- For $y = a b^{x}$ or $y = a b^{- x}$ the $y$ -intercept is $(0, a)$

Here is a sketch of the curve $y = a b^{- x}$ where $a$ and $b$ are positive constants.

$(0, 6)$ and $(2, 0.375)$ lie on the curve.

Work out the values of $a$ and $b$ .

The value of $a$ can be found by inspection. The $y$ -intercept is (0, 6) so $a = 6 .$

$y = 6 b^{- x}$

The value of $b$ can be found by substituting the second coordinate into the equation and solving.

$\begin{array}{rcl} y & = & 6 b^{- x} \\ 0.375 & = & 6 b^{- 2} \end{array}$

Solve to find $b$ .

$\begin{array}{rcl} 6 b^{- 2} & = & 0.375 \\ \frac{1}{b^{2}} & = & \frac{0.375}{6} = \frac{1}{16} \\ b^{2} & = & 16 \\ b & = & \pm 4 \end{array}$

$b$ must be positive, so disregard the negative value.

$a = 6, b = 4$

Last updated: 4 October 2024

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