Exponentials

Hong Kong

Stage 4 - Stage 5

Lesson

The most basic exponential function is given by $y=b^x$`y`=`b``x`, where $b$`b` is known as the base of the function.

The base is restricted to positive real numbers not equal to $1$1. By playing with the applet here, can you work out why the base of an exponential function is not 1?

We can introduce more variability to the basic curve by considering the function that includes two interesting constants. This new form is given by $y=A\times b^x+k$`y`=`A`×`b``x`+`k` where $A$`A` is any non-zero real number and $k$`k` is any real number at all.

If $A$`A` is positive and the base is greater than $1$1, then the graph of the function $y=A\times b^x+k$`y`=`A`×`b``x`+`k` monotonically rises ( that is, the curve continually rises across the whole domain). If $A$`A` is positive and the base is less than $1$1 then the curve monotonically falls.

If $A$`A` is negative, the situation reverses. For example, the curve $y=-4\times3^x$`y`=−4×3`x` is a reflection across the $x$`x`-axis of the curve $y=4\times3^x$`y`=4×3`x`. This means that $y=-4\times3^x$`y`=−4×3`x` is a falling curve when $b$`b` is greater than $1$1, and a rising curve when $b$`b` is less than $1$1.

Use the applet below to draw the following graphs, so that you can see how things work. Be sure to make $k=0$`k`=0 before you start changing $A$`A` and $b$`b`.

Graph $1$1 | $y=4\times\left(0.5\right)^x$y=4×(0.5)x |
---|---|

Graph $2$2 | $y=-4\times\left(0.5\right)^x$y=−4×(0.5)x |

Graph $3$3 | $y=4\times3^x$y=4×3x |

Graph $4$4 | $y=-4\times3^x$y=−4×3x |

The addition of the constant $k$`k` will lift the graph or pull it down (depending on the sign of $k$`k`).

So for example the monotonically rising function given by $y=5\times2^x+3$`y`=5×2`x`+3 will have a $y$`y` intercept of $5+3=8$5+3=8. Use the applet to verify this.

We can also make the exponential curve a little more general by including a *translation constant* into the equation. Thus a more general form is given by $y=A\times b^{x-c}+k$`y`=`A`×`b``x`−`c`+`k`. The constant $c$`c` makes the whole graph move sideways (either left or right depending on the sign of $c$`c`). The technical name for this movement is a horizontal translation.

So for example the curve $y=4\times3^{x-2}$`y`=4×3`x`−2 is the same as $y=4\times3^x$`y`=4×3`x` except that it's positioned $2$2 units horizontally to the right of it.

Also, the curve $y=4\times3^{x-2}$`y`=4×3`x`−2 is the same as $y=4\times3^x$`y`=4×3`x` except that it's positioned $2$2 units to the left of it.

Consider the graph given by $y=4\times3^{x-2}$`y`=4×3`x`−2. We can evaluate $y$`y` for any value of $x$`x`, and seven values are shown here in the table. Note that when the exponent $x-2$`x`−2 is negative we must utilise our index rules.

For example, for $x=-1$`x`=−1 we have $y=4\times3^{-1-2}=4\times3^{-3}=\frac{4}{3^3}=\frac{4}{27}$`y`=4×3−1−2=4×3−3=433=427.

$x$x |
$-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
---|---|---|---|---|---|---|---|

$y$y |
$\frac{4}{27}$427 | $\frac{4}{9}$49 | $\frac{4}{3}$43 | $4$4 | $12$12 | $36$36 | $108$108 |

Use the applet to construct the sketch. Then notice the following features:

Firstly, the $y$`y` values are all positive. This is because any power of $3$3 is positive, no matter whether the exponent involved is positive, negative or zero. Since $A=4$`A`=4, and there is no vertical translation constant $k$`k` in the equation, every value of $y$`y` must also be positive.

Secondly, the $y$`y` values are increasing from left to right (or decreasing from right to left). As $x$`x` increases in equal steps, the $y$`y` values increase as well, but in ever increasing steps. that is to say the rate of change of $y$`y` is quickening as $x$`x` increases. For example, the values of $x$`x` stepping up as $1$1, $2$2, $3$3, $4$4, $5$5 show $y$`y` values stepping up as $4$4, $12$12, $36$36 and $108$108.

Going the other way, the values of $y$`y` from right to left are getting closer and closer to zero, but never becoming zero. The curve seems to bend toward the $x$`x`-axis, forever getting closer and closer to it. An asymptote is a line that curve approaches but never reaches, and so the $x$`x`-axis (which is the line $y=0$`y`=0) is an asymptote for this curve.

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

Of the two functions $y=2^x$`y`=2`x` and $y=4\times2^x$`y`=4×2`x`, which is increasing more rapidly for $x>0$`x`>0?

$y=2^x$

`y`=2`x`A$y=4\times2^x$

`y`=4×2`x`B

What is the equation of the curve in the following graph?

Loading Graph...