Exponential Functions. Lesson Objective: Draw graphs of exponential functions of the form y = ka x and understand ideas of exponential growth and decay. Starter. Suppose you have a choice of two different jobs at graduation Start at £20,000 with a 6% per year increase
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Draw graphs of exponential functions of the form y = kaxand understand ideas of exponential growth and decay
y = kax
Option A y = 20000x1.06x
Option A y = 24000+1000x
y = 2x
y = 5x
y = 0.1x
y = 0.5x
y = 7x
In an exponential function, the variable is in the index. For example:
The general form of an exponential function to the base a is:
y = axwhere a > 0 and a1.
You have probably heard of exponential increase and decrease or exponential growth and decay.
A quantity that changes exponentially either increases or decreases increasingly rapidly as time goes on.
Lets examine exponential functions. They are different than any of the other types of functions weve studied because the independent variable is in the exponent.
Lets look at the graph of this function by plotting some points.
Recall what a negative exponent means:
Reflected about y-axis
This equation could be rewritten in a different form:
So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.
There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.
When 0 < a < 1 the graph of
y = ax has the following shape:
When b > 1 the graph of y = ax has the following shape:
In both cases the graph passes through (0, 1) and (1, a).
This is because:
a0 = 1 and a1 = a
for all a > 0.
y = kax
The value of a new car depreciates at a rate of 15% a year.
The car costs 24 000 in 2010.
How much will it be worth in 2018?
To decrease the value by 18% we multiply it by 0.82.
There are 8 years between 2010 and 2018.
After 8 years the value of the car will be
24 000 0.828 =
4905 (to the nearest pound)
y = 24000x0.82x
Fill in the rest of the table
What is the decay factor?
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing function
Why study exponential functions?
Banking and finance
Compute compound interest
Whenever quantities grow or shrink by a constant factor, such as in radioactive decay, Depreciation
Medicine provides another common situation where exponential functions give an appropriate model. If you take some medicine, the amount of the drug in your system generally decreases over time.
An understanding of exponential functions will aid you in analyzing data particularly in growth and decay
y = kax