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Section7.2Exponential Functions and Their Graphs

A simple exponential function, \(f\text{,}\) is a function that can be expressed as

\begin{equation*} f(x)=b^x \end{equation*}

where \(b \gt 0,\,\,b \neq 1\text{.}\)

Let's consider the function \(f(x)=2^x\text{.}\) Several function values are shown in TableĀ 7.2.1 and the corresponding ordered pairs are plotted in FigureĀ 7.2.2. The function is continuous, and we can infer it's shape from the points plotted in FigureĀ 7.2.2. The continuous function is shown in FigureĀ 7.2.3.

\(x\) \(y\)
\(-3\) \(0.125\)
\(-2\) \(0.25\)
\(-1\) \(0.5\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(8\)
\(4\) \(16\)
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Table7.2.1\(y=2^x\)
Figure7.2.2Points on \(y=2^x\)
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Figure7.2.3\(y=2^x\)

Let's make some observations about the curve in FigureĀ 7.2.3. Let's first focus on the behavior to the left of the \(y\)-axis. As the value of \(x\) moves further and further to the left, the curve seems to hug the \(x\)-axis more and more. We call this behavior asymptotic and, specifically, we say that the \(x\)-axis is the horizontal asymptote for the graph of \(y=2^x\) as \(y \rightarrow -\infty\text{.}\) The last set of symbols is read aloud as "\(x\) approaches negative infinity." In this context, the symbols mean that the value of \(x\) is moving farther and farther to the left with no limit on how far it will get. While this is happening, the value of \(y\) is getting closer and closer to zero, although it will never get to zero as no (real number) value of \(x\) results in \(2^x\) having a value of zero (or a negative value, for that matter).

Let's now turn our attention to the behavior of \(y=2^x\text{.}\) I've plotted the function again in FigureĀ 7.2.4, only this time I've used equal scales on the two axes. In doing so, we see just how incredibly fast the curve rises as it moves rightward away from the \(x\)-axis. The rate at which the value of \(y\) increases relative to small changes in the value of \(x\) is really difficult to fully grasp. Let's ponder one example.

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Figure7.2.4\(y=2^x\)

The Empire State Building in New York is 102 stories tall. It's height is 1250 ft which is equivalent to 15,000 inches. Now suppose that we have a gigantic piece of graph paper that we plaster to the side of the Empire State Building, with the \(x\)-axis lying on the sidewalk. Suppose further that the unit used to scale each axis is one inch, so that the largest \(y\)-coordinate on the graph is 15,000. Let's see how far we'd need to move along the positive \(x\)-axis before we'd no longer be able to plot the point because the \(y\)-coordinate would be larger than 15,000. Turns out not far at all: \(2^{13}=8,192\) and \(2^{14}=16,384\text{.}\) That means that 14 is the first integer \(x\)-coordinate where the \(y\)-coordinate is already higher than the Empire State Building. Think about that ā€” a little more than a foot to the right of the \(y\)-axis and the \(y\)-coordinate is already higher than the Empire State Building. Crazy! But it gets weirder.

The average distance between the Earth and its moon is 238,900 miles which, in turn, is a little more than \(1.5 \times 10^{10}\) inches. On the other hand, \(2^{34}\) is a little more than \(1.7 \times 10^{10}\text{,}\) so on our Empire State Building Grid we've moved less than three feet to the right of the origin and the \(y\)-coordinate on the graph of \(y=2^x\) has already blown past the Moon! To infinity and beyond, indeed.

Now let's consider the graph of \(g(x)=\left(\frac{1}{2}\right)^x\text{.}\) Several function values are shown in TableĀ 7.2.5 and the function is graphed in FigureĀ 7.2.6.

\(x\) \(y\)
\(-4\) \(16\)
\(-3\) \(8\)
\(-2\) \(4\)
\(-1\) \(2\)
\(0\) \(1\)
\(1\) \(0.5\)
\(2\) \(0.25\)
\(3\) \(0.125\)
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Table7.2.5\(y=\left(\frac{1}{2}\right)^x\)
Figure7.2.6\(y=\left(\frac{1}{2}\right)^x\)

Plots of both \(y=2^x\) and \(y=\left(\frac{1}{2}\right)^x\) are shown in FigureĀ 7.2.7 . Notice that the curves are mirror images across the \(y\)-axis. It turns out that any function of form \(y=b^x,\,\,b \gt 1\) has the same basic graphical properties as \(y=2^x\) and any function of form \(y=b^x,\,\,0 \lt x \lt 0\) has the same basic graphical properties as \(y=\left(\frac{1}{2}\right)^x\text{.}\) These properties are summarized below.

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Figure7.2.7\(\highlight{y=2^x}\,\,\text{and}\,\,\highlightr{y=\left(\frac{1}{2}\right)^x}\)
Properties of the function \(f(x)=b^x,\,\,b \gt 0\)

  • The domain is \((-\infty,\infty)\text{.}\)

  • The range is \((0,\infty)\text{.}\)

  • \(f(0)=1\)
  • As \(x \rightarrow -\infty\text{,}\) \(y \rightarrow 0\)

  • As \(x \rightarrow \infty\text{,}\) \(y \rightarrow \infty\)

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Figure7.2.8\(y=b^x,\,\,b \gt 1\)
Properties of the function \(f(x)=b^x,\,\,0 \lt b \lt 1\)

  • The domain is \((-\infty,\infty)\text{.}\)

  • The range is \((0,\infty)\text{.}\)

  • \(f(0)=1\)
  • As \(x \rightarrow -\infty\text{,}\) \(y \rightarrow \infty\)

  • As \(x \rightarrow \infty\text{,}\) \(y \rightarrow 0\)

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Figure7.2.9\(y=b^x,\,\,0 \lt b \lt 1\)

We now turn our attention to graphical transformations as they apply to exponential functions. While on paper the transformations are the same for exponential functions as they are for any other type of functions, the apparent effect can sometimes be extremely subtle when the transformation is applied to an exponential function. Let's recall the transformations and see them applied to exponential functions.

Transformations of type \(g(x)=f(x \pm h)\)

Assuming that \(h\) is a positive number, the graph of \(y=f(x-h)\) shifts every point on \(y=f(x)\) rightward by \(h\) units while the graph of \(y=f(x+h)\) shifts every point on \(y=f(x)\) leftward by \(h\) units.

The graphs of \(y=2^x\) and \(y=2^{x-3}\) are shown in FigureĀ 7.2.10. While the rightward shift of the latter function is very apparent for positive values of \(x\text{,}\) it is not nearly as apparent to the left of \(-2\) and it is not at all apparent once you get to the left of \(-4\text{.}\)

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Figure7.2.10\(\highlight{y=2^x}\,\,\text{and}\,\,\highlightr{y=2^{x-3}}\)
Transformations of type \(g(x)=f(x) \pm k\)

Assuming that \(k\) is a positive number, the graph of \(y=f(x)+k\) shifts every point on \(y=f(x)\) upward by \(k\) units while the graph of \(y=f(x)-k\) shifts every point on \(y=f(x)\) downward by \(k\) units.

The graphs of \(y=2^x\) and \(y=2^x-2\) are shown in FigureĀ 7.2.11. The downward shift of the latter function is very apparent below the \(x\)-axis ā€” most notably the horizontal asymptote has moved from the line \(y=0\) to the line \(y=-2\text{.}\) You have to be much more deliberate to identify the downward shift above the \(x\)-axis, and once you get very far to right of the \(y\)-axis the specific amount of downward shift becomes almost impossible to discern.

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Figure7.2.11\(\highlight{y=2^x}\text{,}\) \(\highlightr{y=2^x-2}\) and \(y=-2\)
Transformations of type \(g(x)=-f(x)\) or \(g(x)=f(-x)\)

The graph of \(y=-f(x)\) reflects every point on \(y=f(x)\) across the \(x\)-axis whereas the graph of \(y=f(-x)\) reflects every point on \(y=f(x)\) across the \(y\)-axis. As seen in FigureĀ 7.2.12 and FigureĀ 7.2.13, these transformations are just as dramatic for exponential functions as the are for any other function.

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Figure7.2.12\(\highlight{y=2^x}\) and \(\highlightr{y=-2^x}\)
Figure7.2.13\(\highlight{y=2^x}\) and \(\highlightr{y=2^{-x}}\)

When you look at FigureĀ 7.2.13, you might get a sense that you've seen that graph before. If so, you are mostly correct. We saw the same sort of reflection when we graphed \(y=2^x\) and \(\left(\frac{1}{2}\right)\) on the same set of axes. How can this be? Lets see.

\begin{align*} 2^{-x}\amp=2^{-1 \cdot x}\\ \amp=\left(2^{-1}\right)^x\\ \amp=\left(\frac{1}{2}\right)^x\text{.} \end{align*}

So not only could it be, it had to be. Exponential functions are funny this way, and things will get even stranger when we explore the next type of transformation.

Transformations of type \(g(x)=a \cdot f(x), a \gt 0\)

The graph of \(y=a \cdot f(x), a \gt 0\) moves the \(y\)-coordinate of every point on \(f\) by the factor of \(a\text{.}\) When \(a \gt 1\text{,}\) the transformation is called a vertical stretch. When \(0 \lt a \lt 1\text{,}\) the transformation is called a vertical compression.

The graphs of \(y=2^x\) and \(y=4 \cdot 2^x\) are shown in FigureĀ 7.2.14. Three vertical shifts by a factor of 4 on the graph.

\begin{equation*} (0,1) \rightarrow (0,4) \end{equation*}
\begin{equation*} (1,2) \rightarrow (1,8) \end{equation*}
\begin{equation*} (2,4) \rightarrow (2,16) \end{equation*}
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Figure7.2.14\(\highlight{y=2^x}\) and \(\highlightr{y=4 \cdot 2^x}\)

The graphs of \(y=2^x\) and \(y=4 \cdot 2^x\) are also shown in FigureĀ 7.2.15. Three horizontal shifts have been highlighted.

\begin{equation*} (4,16) \rightarrow (2,16) \end{equation*}
\begin{equation*} (3,8) \rightarrow (1,8) \end{equation*}
\begin{equation*} (2,4) \rightarrow (0,4) \end{equation*}
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Figure7.2.15\(\highlight{y=2^x}\) and \(\highlightr{y=4 \cdot 2^x}\)

So, apparently, a vertical stretch is also a horizontal shift. In this specific case, the vertical stretch of \(y=2^x\) by a factor of 4 of the function \(y=2^x\) is equivalent to a leftward shift of 2 units for the same function. What's the mathematics behind this?

\begin{align*} 4 \cdot 2^x\amp=2^2 \cdot 2^x\\ \amp=2^{x+2} \end{align*}

In a similar vein, \(y=\frac{1}{8} \cdot 2^x\) affects upon \(y=2^x\) a vertical compression by a factor of \(\frac{1}{8}\) which in turn is also a rightward shift of 3 units. This is established below.

\begin{align*} \frac{1}{8} \cdot 2^x\amp=2^{-3} \cdot 2^x\\ \amp=2^{x-3} \end{align*}

Let's consider \(y=7 \cdot 2^x\text{.}\) It is not "obvious" what power of 2 results in 7, but it turns out that the appropriate power is approximately 2.807. The functions \(y=2^x\) and \(y=7 \cdot 2^x\) are both shown in FigureĀ 7.2.16. Three pairs of points have been highlighted to show the attendant leftward shift of a little less than 3 units.

\begin{align*} 7 \cdot 2^x \amp\approx 2^{2.807} \cdot 2^x\\ \amp=2^{x+2.807} \end{align*}
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Figure7.2.16\(\highlight{y=2^x}\) and \(\highlightr{y=7 \cdot 2^x}\)

As a side note, a process used to determine the power of 2 that results in 7 is discussed in the section about applying the properties of logarithms.

Transformations of type \(g(x)=f(ax), a \gt 0\)

The graph of \(y=f(ax), a \gt 0\) moves the \(x\)-coordinate of every point on \(f\) by the factor of \(\frac{1}{a}\text{.}\) When \(a \gt 1\) the result is a horizontal compression. When \(0 \lt a \lt 1\text{,}\) the result is a horizontal stretch. When algebraically simplified, these type of transformations always result in an exponential function with a different base. For example, consider \(f(x)=2^x\) and \(g(x)=f(3x)\text{.}\)

\begin{align*} g(3x)\amp=2^{3x}\\ \amp=\left(2^3\right)^x\\ \amp=8^x \end{align*}

Graphs of \(y=2^x\) and \(y=8^x\) are shown in FigureĀ 7.2.17. Three pairs of points have been highlight to demonstrate that \(y=8^x\) is indeed a horizontal compression of the function \(y=2^x\) by a factor of \(\frac{1}{3}\) .

\begin{equation*} (12,4096) \rightarrow (4,4096) \end{equation*}
\begin{equation*} (9,512) \rightarrow (3,512) \end{equation*}
\begin{equation*} (6,64) \rightarrow (2,64) \end{equation*}
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Figure7.2.17\(\highlight{y=2^x}\) and \(\highlightr{y=8^x}\)