The graph below shows the exponential function \(f(x) = b^x\) along with the so-called tangent line at the \(y\)-intercept. This line is the best-fit of a linear function to the exponential function’s graph at the point \((0,1)\text{.}\) It just “touches” the curve at this point.
Figure7.7.Graph of \(y=b^x\text{.}\)
(a)
Move the slider to adjust the base of the exponential function \(b\text{.}\) Observe how the slope of the tangent line changes with the slider.
(b)
Move the slider to find the largest value of \(b\) that results in a slope of the tangent which is less than \(1\text{.}\) Record this value of \(b\text{.}\)
(c)
Move the slider to find the smallest value of \(b\) that results in a slope of the tangent which is greater than \(1\text{.}\) Record this value of \(b\text{.}\)
(d)
There must is a special value of \(b\) called \(e\) for which the slope of the tangent is exactly \(1\text{.}\) In what interval is this special number \(e\) in?
Definition7.8.
There is a real number \(e\) for which the slope of the tangent line to \(y = e^x\) at \((0,1)\) is exactly \(1\text{.}\) We call \(e\) the natural base and the function \(y = e^x\) the natural exponential function.
In fact, \(e\approx 2.718 \gt 1\) is an irrational number and at best can be only approximated with a finite number of digits. The graph of the natural exponential function grows exponentially with domain \((-\infty,\infty)\) and range \((0,\infty)\text{.}\) Like all exponential functions, it has a \(y\)-intercept at \((0,1)\text{.}\) The graph decays to the left towards its horizontal asymptote \(y = 0\text{.}\)
Figure7.9.Graph of \(y=e^x\) and its tangent line at \(x=0\text{.}\)
Example7.10.
Sketch the graph of each transformed exponential function. Label the \(y\)-intercept and at least one other point on the graph.
\(\displaystyle \displaystyle y = 2e^{x}\)
\(\displaystyle \displaystyle y = e^{2x}\)
\(\displaystyle \displaystyle y = e^{x+2}\)
\(\displaystyle \displaystyle y = e^{x}+2\)
\(\displaystyle \displaystyle y = -e^{x}\)
\(\displaystyle \displaystyle y = e^{-x}\)
Solution.
The graph of \(\displaystyle y = 2e^{x}\) is obtained by stretching the graph of \(y=e^{x}\) vertically by a factor or 2. This changes the \(y\)-intercept, but does not change the horizontal asymptote.
Figure7.11.Graph of \(y=2e^x\)
The graph of \(\displaystyle y = e^{2x}\) is obtained by compressing the graph of \(y=e^x\) horizontally by a factor of 2. This does not change the \(y\)-intercept nor the horizontal asymptote, but it does increase the rate at which the graph grows.
Figure7.12.Graph of \(y=e^{2x}\)
The graph of \(\displaystyle y = e^{x+2}\) is obtained by translating the graph of \(y=e^x\) two units to the left. This changes the \(y\)-intercept, but not the overall shape of the graph.
Figure7.13.Graph of \(y=e^{x+2}\)
The graph of \(\displaystyle y = e^{x}+2\) is obtained by translating the graph of \(y=e^x\) two units up. This changes the \(y\)-intercept to \(0,3\) and the horizontal asymptote to \(y=2\text{.}\)
Figure7.14.Graph of \(y=e^x+2\)
The graph of \(\displaystyle y = -e^{x}\) is obtained by reflecting the graph of \(y=e^x\) across the \(x\)-axis.
Figure7.15.Graph of \(y=-e^x\)
The graph of \(\displaystyle y = e^{-x}\) is obtained by reflecting the the graph of \(y=e^x\) across the \(y\)-axis.
