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Algebra and Precalculus A Practical Guide

Cameron Geisler

Section 13.1 Intro to Exponential Functions

Example 13.1.1. Graph of \(f(x) = 2^x\).

Consider the graph of \(f(x) = 2^x\text{.}\) First, create a table of values, using \(x\)-values of \(-2, -1, 0, 1\text{,}\) and 2, and plugging them into the equation.
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\begin{align*} \begin{array}{c|l} x \amp f(x) \\ \hline -2 \amp 2^{-2} = \frac{1}{4} = 0.25 \\[4pt] -1 \amp 2^{-1} = \frac{1}{2} = 0.5 \\[4pt] 0 \amp 2^0 = 1 \\ 1 \amp 2^1 = 2 \\ 2 \amp 2^2 = 4 \end{array} \end{align*}
Then, connect these points with a smooth curve. Here is a graph: Graph of \(f(x)=2^x\). Observe the \(y\)-values of \(0.25, 0.5, 1, 2, 4\text{.}\) Notice that each time \(x\) increases by 1, \(y\) is multiplied by 2.
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Example 13.1.2. Graph of \(f(x) = 3^x\).

Consider the graph of \(f(x) = 3^x\text{.}\) First, create a table of values,
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\begin{align*} \begin{array}{c|c} x \amp f(x) \\ \hline -2 \amp 3^{-2} = \frac{1}{9} \approx 0.11 \\[4pt] -1 \amp 3^{-1} = \frac{1}{3} \approx 0.33 \\[4pt] 0 \amp 3^0 = 1 \\ 1 \amp 3^1 = 3 \\ 2 \amp 3^2 = 9 \end{array} \end{align*}
Here is a graph: Graph of \(f(x)=3^x\). Notice that it grows faster than \(2^x\text{,}\) because each time \(x\) increases by 1, \(y\) multiplies by 3 (instead of 2).
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Example 13.1.3. Graph of \(f(x) = \brac{\frac{1}{2}}^x\).

Consider the graph of \(f(x) = \brac{\frac{1}{2}}^x\text{.}\) Creating a table of values,
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\begin{align*} \begin{array}{c|l} x \amp f(x) \\ \hline -2 \amp \brac{\frac{1}{2}}^{-2} = 4 \\ -1 \amp \brac{\frac{1}{2}}^{-1} = 2 \\ 0 \amp \brac{\frac{1}{2}}^0 = 1 \\ 1 \amp \brac{\frac{1}{2}}^1 = \frac{1}{2} = 0.5 \\ 2 \amp \brac{\frac{1}{2}}^2 = \frac{1}{4} = 0.25 \end{array} \end{align*}
Here is a graph: Graph of \(f(x)=\brac{\frac{1}{2}}^x\). Here, each time \(x\) increases by 1, \(y\) is multiplied by \(\frac{1}{2}\text{,}\) which means it gets smaller.
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Example 13.1.4. Graph of \(f(x) = \brac{\frac{3}{4}}^x\).

Consider the graph of \(f(x) = \brac{\frac{3}{4}}^x\text{.}\) Creating a table of values,
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\begin{align*} \renewcommand{\arraystretch}{1.2} \begin{array}{c|l} x \amp f(x) \\ \hline -2 \amp \frac{16}{9} \approx 1.78 \\ -1 \amp \frac{4}{3} \approx 1.33 \\ 0 \amp 1 \\ 1 \amp \frac{3}{4} = 0.75 \\ 2 \amp \frac{9}{16} \approx 0.56 \end{array} \end{align*}
Here is a graph: Graph of \(f(x)=\brac{\frac{3}{4}}^x\). This graph also decreases, but more slowly than \(\brac{\frac{1}{2}}^x\text{.}\)
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Example 13.1.5. Comparison of Graphs.

Example 13.1.6. Desmos Exploration.

Here is a Desmos graph that allows you to graph \(f(x)=b^x\text{,}\) where you can use the slider to change the value of \(b\text{:}\) Exponential Functions.
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  • How does the graph change, based on your choice of \(b\text{?}\)
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In general, for a function like \(f(x)=b^x\text{,}\) each time \(x\) increases by 1, the output is multiplied by the base \(b\text{.}\)
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  • If \(b \gt 1\text{,}\) the \(y\)-values increase.
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  • If \(b \lt 1\text{,}\) the \(y\)-values decrease.
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Also,
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  • The \(y\)-intercept is always \((0,1)\text{,}\) because when \(x=0\text{,}\) the \(y\)-value is \(b^0 = 1\text{.}\)
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  • There are no \(x\)-intercepts, because \(b^x\) is never equal to 0.
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  • The domain is all real numbers, because you can raise a base to any exponent you want, positive or negative.
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  • The range is all positive numbers, or \(y \gt 0\text{,}\) because raising a number \(b\) to the exponent \(x\) always gives a positive output (even if \(x\) is negative).
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Also, notice that all of the functions “flatten out” around the \(x\)-axis. We call this a horizontal asymptote. So, the graph has a horizontal asymptote of the \(x\)-axis, or the line \(y=0\text{.}\)
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Subsection 13.1.1 Exponential Functions

Definition 13.1.7.

An exponential function with base \(b\) is given by,
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\begin{gather*} \boxed{f(x) = b^x} \qquad \text{where $b \gt 0$ and $b \neq 1$} \end{gather*}
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Example 13.1.8. Examples.

Some exponential functions include:
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  • \(\displaystyle f(x) = 2^x\)
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  • \(\displaystyle f(x) = 5^x\)
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  • \(\displaystyle f(x) = 1.13^x\)
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  • \(\displaystyle f(x) = \brac{\frac{1}{2}}^x\)
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  • \(\displaystyle f(x) = 0.78^x\)
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Example 13.1.9. Non-examples.

Non-examples. Some functions that are not exponential functions include:
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  • \(f(x) = x^2\) (\(x\) is not an exponent, this is a polynomial function)
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  • \(f(x) = (-2)^x\) (the base \(b\) must be positive)
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  • \(f(x) = 1^x\) (the base \(b\) cannot be \(1\))
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  • \(f(x) = x^x\) (the base is not a constant)
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In particular, note that a power function, of the form \(f(x) = x^n\text{,}\) is not an exponential function.
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  • With a power function, the base is the variable \(x\text{,}\) and the exponent is a constant \(n\text{.}\)
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  • For exponential functions, the base is a constant \(b\text{,}\) and the exponent is a variable \(x\text{.}\)
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For the exponential function \(f(x)=b^x\text{,}\) its graph looks like an increasing or decreasing curve with a horizontal asymptote.
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  • \(y\)-intercept is \((0,1)\text{.}\)
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  • No \(x\)-intercepts.
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  • Horizontal asymptote is \(y = 0\text{.}\)
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  • Domain is all real numbers, or \(x \in \mathbb{R}\text{.}\)
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  • Range is all positive real numbers, or \(y \gt 0\text{,}\) or \((0,\infty)\text{.}\)
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The base \(b\) is called the growth factor, because it is the number you multiply to see how the function changes.
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  • If \(b \gt 1\text{,}\) it is increasing. We call this exponential growth.
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  • If \(0 \lt b \lt 1\text{,}\) it is decreasing. We call this exponential decay.
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In summary,
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  1. Create a table of values for \(x = -2, -1, 0, 1, 2\text{,}\) and plot the points.
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  2. Connect the points with a smooth curve.
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Remark 13.1.10.

For an exponential function, it is only approached on one side, so you might call it a “half” horizontal asymptote.
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Figure 13.1.11. An exponential graph approaches the horizontal asymptote \(y=0\) only on one side
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Remark 13.1.12.

Exponential functions have no vertical asymptotes, because the domain of an exponential function is all real numbers.
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Exercise Group 13.1.1. Which functions are exponential?

Exercise Group 13.1.2. Determine whether each graph represents an exponential function.

If exponential, say whether it is increasing or decreasing. If not, state what type of function it is.
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(d)
Answer.
No. It is a cubic function (or polynomial function)
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Subsection 13.1.2 Exponential Functions with an Initial Value

Example 13.1.13. Example 1.

\(f(x)=3 \cdot 2^x\)
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Example 13.1.14. Example 2.

\(f(x)=2 \cdot 4^x\)
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Example 13.1.15. Example 3.

\(f(x)=8\brac{\frac{1}{4}}^x\)
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Example 13.1.16. Example 4.

\(f(x)=9\brac{\frac{2}{3}}^x\)
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In general, a function of the form \(f(x)=a\cdot b^x\) has a \(y\)-intercept of \(a\text{,}\) and a growth factor of \(b\text{.}\)
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\begin{align*} \boxed{f(x)=a\cdot b^x \quad \begin{cases} a \amp \rightarrow y\text{-intercept} \\ b \amp \rightarrow \text{growth factor} \end{cases} } \end{align*}

Example 13.1.17. Desmos Exploration.

Figure 13.1.18. Exponential functions \(f(x)=a\cdot b^x\text{,}\) all with the same \(y\)-intercept \(a\text{,}\) but with different growth factors \(b\)
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Exercise Group 13.1.3. Sketching Exponential Functions.

Sketch the graph of each exponential function, using its \(y\)-intercept and 1 more point. State the \(y\)-intercept, increasing/decreasing, domain, and range.
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(a)
\(y=10(2^x)\)
Answer.
\(y\)-intercept \((0,10)\text{,}\) increasing, domain \((-\infty,\infty)\text{,}\) range \((0,\infty)\)
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(b)
\(y=6(2^x)\)
Answer.
\(y\)-intercept \((0,6)\text{,}\) increasing, domain \((-\infty,\infty)\text{,}\) range \((0,\infty)\)
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(c)
\(y=27\brac{\frac{1}{3}}^x\)
Answer.
\(y\)-intercept \((0,27)\text{,}\) decreasing, domain \((-\infty,\infty)\text{,}\) range \((0,\infty)\)
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(d)
\(y=4\brac{\frac{1}{2}}^x\)
Answer.
\(y\)-intercept \((0,4)\text{,}\) decreasing, domain \((-\infty,\infty)\text{,}\) range \((0,\infty)\)
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Subsection 13.1.3 Solving Exponential Equations (Numerically)

Trial-and-error: A basic method is to approximate the solution using trial and error. Choose an initial guess for the solution and plug it in, and see if it equals the other side. Then, adjust your solution up or down, depending on if your output is too high or low.
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Exercise Group 13.1.4. Solve each equation by trial and error.

Round answers to 1 decimal place.
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(a)
\(2^x=11\)
Answer.
\(x\approx 3.5\)
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(b)
\(3^x=15\)
Answer.
\(x\approx 2.5\)
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(c)
\(4^x+5=10\)
Answer.
\(x\approx 1.2\)
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(d)
\(3^{x+2}=4\)
Answer.
\(x\approx -0.7\)
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This method really only works for simple equations.
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Graphically: Another method is to solve graphically (just like any other type of equation). Graph each side of the equation, and the solution is the point where they intersect. You can do this using Desmos, or a graphing calculator like the TI-84 Plus CE. In particular, the solution is the \(x\)-coordinate of the intersection point.
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Example 13.1.19. Solving Graphically.

To solve \(3^x=7\) graphically, graph \(y=3^x\) and \(y=7\) on the same graph: Graph of \(y=3^x\) and \(y=7\). The intersection point is about \(x \approx 1.77\text{,}\) which is the answer.
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In general, to solve \(f(x)=k\text{,}\) plot \(y=f(x)\) and \(y=k\text{,}\) and find the \(x\)-value where they intersect,
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For more about solving graphically, see Solving Equations Graphically.
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Solving by scientific calculator. My recommended calculator can solve exponential equations numerically (just like any other equation). For details, see Solving Equations with a Scientific Calculator.
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Exercise Group 13.1.5. Solve each equation using a scientific calculator.

Round answers to 1 decimal place.
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(a)
\(5\cdot 2^x=18\)
Answer.
\(x\approx 1.8\)
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(b)
\(3^{-x}=0.9\)
Answer.
\(x\approx 0.1\)
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(c)
\(2\cdot 3^{x-2}=176\)
Answer.
\(x\approx 6.1\)
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(d)
\(24\cdot (0.95)^x=2\)
Answer.
\(x\approx 48.4\)
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(e)
\(2000\cdot (1.04)^x=5000\)
Answer.
\(x\approx 23.4\)
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