Exponential Graphs
What is an exponential?
- An exponential is a function where the power is a variable, usually
- is an example of an exponential
- In this course exponentials will be in either of the following forms
- Where and are rational numbers, and is a variable
- can be equal to 1, resulting in or
- All of the following are examples of exponentials you may encounter
What does an exponential graph look like?
- A graph of the form where is positive and larger than 1 will be increasing as increases
- is increasing
- A graph of the form where is positive and larger than 1 will be decreasing as increases
- is decreasing
- If is between 0 and 1, then the opposite is true
- is decreasing
- is increasing
- An equation of the form stretches the graph of vertically by scale factor
- If is negative, then this would also reflect the graph in the -axis
- The -intercept of and will be
- You can show this by substituting into the equation
- Substituting into or will reduce both to
- This means that for an exponential in the form or , the -intercept will simply be (0,1)
- The graphs do not cross the -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. or
- 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
Exam Tip
- Remember that the intercept can often be found by inspection, which may save you some working
- For or the -intercept is
- For or the -intercept is
Worked example
Here is a sketch of the curve where and are positive constants.
and lie on the curve.
Work out the values of and .
The value of can be found by inspection. The -intercept is (0, 6) so
The value of can be found by substituting the second coordinate into the equation and solving.
Solve to find .
must be positive, so disregard the negative value.