Exponentials

Hong Kong

Stage 4 - Stage 5

Lesson

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

If we first consider the function $y=2^x$`y`=2`x`, 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.

The function $y=2^x$`y`=2`x` 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`=`b``x` 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`=−`b``x` is simply the mirror image of the curve $y=b^x$`y`=`b``x` . 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`=2`x`+3 or $y=5-2^x$`y`=5−2`x`. 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 following applet allows you to vary the curve given as the general function $y=\pm b^x+k$`y`=±`b``x`+`k`. This means you can create functions like $y=3^x+2$`y`=3`x`+2, $y=-3^x-2$`y`=−3`x`−2, $y=4-3^x$`y`=4−3`x`, etc. Try different combinations to extend your understanding.

Consider the expression $3^x$3`x`.

Evaluate the expression when $x=4$

`x`=4.Evaluate the expression when $x=-4$

`x`=−4. Leave your answer in fractional form.What happens to the value of $3^x$3

`x`as $x$`x`gets larger?it stays the same

Ait gets smaller, approaching zero

Bit gets larger, approaching infinity

CWhat happens to the value of $3^x$3

`x`as $x$`x`gets smaller?it gets larger, approaching infinity

Ait gets smaller, approaching zero

Bit stays the same

C

Consider the graph of the equation $y=4^x$`y`=4`x`.

Loading Graph...

What can we say about the $y$

`y`-value of every point on the graph?The $y$

`y`-value of most points of the graph is greater than $1$1.AThe $y$

`y`-value of every point on the graph is positive.BThe $y$

`y`-value of every point on the graph is an integer.CThe $y$

`y`-value of most points on the graph is positive, and the $y$`y`-value at one point is $0$0.DAs the value of $x$

`x`gets large in the negative direction, what do the values of $y$`y`approach but never quite reach?$4$4

A$-4$−4

B$0$0

CWhat do we call the horizontal line $y=0$

`y`=0, which $y=4^x$`y`=4`x`gets closer and closer to but never intersects?A horizontal asymptote of the curve.

AAn $x$

`x`-intercept of the curve.BA $y$

`y`-intercept of the curve.C

Consider the function $y=8^{-x}$`y`=8−`x`.

Can the value of $y$

`y`ever be negative?Yes

ANo

BAs the value of $x$

`x`increases towards $\infty$∞ what value does $y$`y`approach?$8$8

A$-\infty$−∞

B$\infty$∞

C$0$0

DAs the value of $x$

`x`decreases towards $-\infty$−∞, what value does $y$`y`approach?$0$0

A$\infty$∞

B$8$8

C$-\infty$−∞

DCan the value of $y$

`y`ever be equal to $0$0?Yes

ANo

BDetermine the $y$

`y`-value of the $y$`y`-intercept of the curve.How many $x$

`x`-intercepts does the curve have?Which of the following could be the graph of $y=8^{-x}$

`y`=8−`x`?Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Consider the graph of $y=-3^x$`y`=−3`x`.

Loading Graph...

State the equation of the asymptote of $y=-3^x$

`y`=−3`x`.What would be the asymptote of $y=2-3^x$

`y`=2−3`x`?How many $x$

`x`-intercepts would $y=2-3^x$`y`=2−3`x`have?What is the domain of $y=2-3^x$

`y`=2−3`x`?$x<3$

`x`<3A$x<2$

`x`<2B$x>2$

`x`>2CAll real $x$

`x`DWhat is the range of $y=2-3^x$

`y`=2−3`x`?