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Exponential Functions (Cambridge O Level Additional Maths)
Revision Note
Exponential Functions
What is an exponential function?
- An exponential functions in a function where the variable is the power
- They are of the form y = ax with a > 0
What is an exponential graph?
- All graphs of the form y = ax will pass through (0, 1) because a0 = 1
- The x-axis is an asymptote
Exponential graphs when a > 1
- Where x < 0 the higher value of a is the “lower” graph
- Where x > 0 the higher value of a is the “higher” graph
- a > 1 is exponential growth
What about when a ≤ 1?
- You may like to think about why a = 1 is not considered... If a = 1, y = 1x = 1 for all values of x
- 0 < a < 1 represents exponential decay
- Where x < 0 the higher value of a is the “higher” graph
- Where x > 0 the higher value of a is the “lower” graph
Worked example
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"e"
What is e, the exponential function?
- The exponential function is y = ex
- e is an irrational number
- e ≈ 2.718
- As with other exponential graphs y = ex
- passes through (0, 1)
- has the x-axis as an asymptote
What is the big deal with e?
- y = ex has the particular property
- i.e. for every real number x, the gradient of y = ex is also equal to ex (see Differentiating e^x and lnx)
The negative exponential graph
- y = e-x is a reflection in the y-axis of y = ex
What is exponential growth and decay?
- y = Aekx (k > 0) is exponential growth
- y = Ae-kx (k > 0) is exponential decay
- A is the initial value
- k is a (usually positive) constant
- A negative sign is used in the equation making clear whether it is growth or decay
Worked example
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