Exponential functions

In a nutshell

Exponential functions are of the form $$y=a^x$$ where $$a$$ is a constant greater than $$0$$ and $$x$$​ is the exponent. Exponential functions are always greater than zero, that is $$y=a^x > 0$$​ for all values of $$x$$. Whenever $$a>1$$, $$y=a^x$$​ is an increasing function that grows without limit as $$x$$​ increases. Whereas, for $$0<a<1$$​, $$y=a^x$$​ is a decreasing function that grows without limit as $$x$$​ decreases.

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Sketching exponentials 

Consider the function $$y=2^x$$. The table below contains the values of this function evaluated at some integers close to zero.

The graph of $$y=2^x$$ is a smooth curve that looks like this:

As $$x$$​ decreases, the value of $$2^x$$ tends towards zero. The dotted line $$y=0$$​ above is an asymptote for $$y=2^x$$​. As the value of $$x$$ increases the function grows without a limit.

Example 1

Sketch the graphs of $$y=3^x$$​ and $$y=1.5^x$$​ on the same axes.

Here the constants $$3$$​ and $$1.5$$​ are greater than $$1$$, so both functions are increasing functions that tend to infinity as $$x$$​ increases, and in this case both tend toward $$0$$​ as ​$$x$$​ decreases.

For both graphs, $$y=1$$​ when $$x=0$$​. 

When $$x>0$$, $$3^x > 1.5^x$$​.

When $$x<0$$​, $$3^x < 1.5^x$$​.

Example 2

Sketch the graphs of $$y=\left(\dfrac12\right)^x$$ and $$y=2^x$$.

Since $$0<\dfrac12<1$$,  $$y=\left(\dfrac12\right)^x$$​ is a decreasing function that tends to infinty as $$x$$ decreases, and in this case it tends towards $$0$$​ as $$x$$​ increases.

 $$y=\left(\dfrac12\right)^x$$​ is a reflection in the $$y$$-axis of the graph of $$y=2^x$$.

If $$f\left(x\right)=2^x$$​, then:

$$y=\left(\dfrac12\right)^x = \left(2^{-1}\right)^x = \left(2\right)^{-x} =f\left(-x\right)$$​

Note: the graph of $$y=f\left(-x\right)$$​ is a relection of the graph of $$y=f\left(x\right)$$​ in the $$y$$-axis.

Example 3

Sketch the graph $$y=\left(\dfrac13\right)^x+2$$$$.$$​  Find the coordinates of the point where the graph crosses the y-axis and give the function of the asymptote.

If $$f\left(x\right)=\left(\dfrac13\right)^x$$, then

$$y=\left(\dfrac13\right)^x+2 = f\left(x\right)+2$$​​

The graph is a translation of the graph $$y=\left(\dfrac13\right)^x$$ upwards by $$2$$​.

The graph crosses the $$y$$​​-axis when $$x=0$$.

Substituting $$x=0$$​ into $$y=\left(\dfrac13\right)^x+2$$, gives

$$\begin{aligned}y&=\left(\dfrac13\right)^0+2\\&=1+2\\&=3\end{aligned}$$​​

Therefore, the graph crosses the $$y$$-axis at $$\underline{(0,3)}$$ and the asymptote is given by $$\underline{y =2}$$.