Identify Characteristics of Exponential Functions

The mathematical function which models natural growth and decay phenomena is known as the exponential model. What distinguishes it from other functions you may have encountered is that the independent variable is in the exponent, (the exponent is also called the power or index). 

Examples include $y=2^x$y=2x, $y=3\times2^{-x}$y=3×2−x, $y=8\times\left(0.5\right)^x+1$y=8×(0.5)x+1, $y=4-5^x$y=4−5x, etc.

If we first consider the function $y=2^x$y=2x, and substitute the consecutive integer value of $x$x into it, starting from $x=-3$x=−3 and finishing at $x=3$x=3, we could develop a table of values for the function. 

So for $x=-3$x=−3, $y=2^{-3}=\frac{1}{2^3}=\frac{1}{8}$y=2−3=123​=18​. For $x=-2$x=−2, $y=\frac{1}{4}$y=14​ and continuing:

$x$x	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2	$3$3
$y$y	$\frac{1}{8}$18​	$\frac{1}{4}$14​	$\frac{1}{2}$12​	$1$1	$2$2	$4$4	$8$8

Typical of exponential growth we see that initially there are small increases in $y$y , but as we continue to increase $x$x, the rate of increase in the size of $y$y is increasing. That is to say the $y$y values are getting larger faster as $x$x steadily increases. Populations, by and large, behave like this - the more people that live in a city, the faster the total population grows. Investments in bank accounts are similar - the more money you have in the bank, the more interest you earn. 

Things can work in reverse as well. Consider the function given by $y=120\times2^{-x}$y=120×2−x. This table shows what happens to $y$y for integer values between $0$0 and $6$6.

$x$x	$0$0	$1$1	$2$2	$3$3	$4$4	$5$5	$6$6
$y$y	$120$120	$60$60	$30$30	$15$15	$7.5$7.5	$3.75$3.75	$1.875$1.875

 

We can use the table to model depreciation. For example, imagine an expensive car, say $$120000$120000, that halves its value every $4$4 years. After the first four years ($x=1$x=1), it would be worth $$60000$60000. After the second $4$4 years ($x=2$x=2) it would be worth $$30000$30000. After $24$24 years ($x=6$x=6), the car's value is reduced to $$1875$1875.

Understanding the shape of the exponential graph. 

The function $y=2^x$y=2x is a continuously rising curve. Its rate of rising quickens as $x$x gets larger and larger. It has a $y$y-intercept of $1$1 because,at $x=0$x=0, $y=2^0=1$y=20=1. To the left of the $y$y-axis, the curve draws closer and closer to the x-axis the further it is from the origin. The $x$x-axis becomes what is known as an asymptote - a line that a curve approaches without ever touching it.

 

On the right side the curve moves upward away from the $x$x-axis. As it moves upward the bend becomes more pronounced as the rate of change in the y values becomes larger and larger. 

Because the curve is always above the $x$x-axis, the range of y values is restricted to  $y>0$y>0. 

The function $y=2^{-x}$y=2−x on the other hand is a continuously falling curve. Its rate of falling slows as $x$x gets larger and larger.

 

It too has a $y$y-intercept of $1$1 because,at $x=0$x=0, $y=2^{-0}=1$y=2−0=1. On the left side the curve moves upward away from the $x$x-axis. Again, as it moves upward the bend becomes more pronounced as the rate of change in the y values becomes larger and larger. To the right of the $y$y-axis, the curve draws closer and closer to the $x$x-axis the further it is from the origin. Once again the $x$x-axis becomes  an asymptote. 

The same behaviour is exhibited by any exponential graph of the forms $y=b^x$y=bx or $y=b^{-x}$y=b−x where $b$b is called the base of the function. The base can be any positive number not equal to $1$1.

There are some simple variations that we can discuss with these functions.

For example the curve given by $y=-b^x$y=−bx is simply the mirror image of the curve $y=b^x$y=bx . It is an image reflected across the x-axis. The same applies to $y=-b^{-x}$y=−b−x. 

We can also consider adding a constant to the function to produce, for example, curves given by $y=2^x+3$y=2x+3 or $y=5-2^x$y=5−2x. In such instances the whole curve is translated upward or downward according to the value of the added constant. Everything else remains the same. 

The Applet

The following applet allows you to vary the curve given as the general function $y=\pm b^x+k$y=±bx+k. This means you can create functions like $y=3^x+2$y=3x+2, $y=-3^x-2$y=−3x−2, $y=4-3^x$y=4−3x, etc. Try different combinations to extend your understanding.  

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3

QUESTION 4