Graph exponentials

 

The basic curve

The most basic exponential function is given by $y=b^x$y=bx, 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? 

 

Introducing A and k

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×bx+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×bx+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.

Negative a

If $A$A is negative, the situation reverses. For example, the curve $y=-4\times3^x$y=−4×3x is a reflection across the $x$x-axis of the curve $y=4\times3^x$y=4×3x. This means that $y=-4\times3^x$y=−4×3x 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=5×2x+3 will have a $y$y intercept of $5+3=8$5+3=8. Use the applet to verify this.

 

Introducing a horizontal translation

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×bx−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=4×3x−2 is the same as $y=4\times3^x$y=4×3x except that it's positioned $2$2 units horizontally to the right of it.

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

 

How does the exponential rise and fall?

Consider the graph given by $y=4\times3^{x-2}$y=4×3x−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 exponent 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.

 

 

 

 

Worked Examples

Question 1

Question 2

Question 3