Graphing Exponential Expressions

5.4.4 Graphing Exponential Expressions

5.4.4 • Graphing Exponential Expressions

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.

Use the graphing tool or technology outside the course. Refer back to your work in the table in activity 5.4.3: Exponential Change: The Growth Factor, where $n = 500 \cdot 2^{t}$ and $p = 100 \cdot 3^{t}$ . Use that information and the given coordinate planes to graph the following using the Desmos tool below.

Use the equation $n = 500 \cdot 2^{t}$ to answer questions 1 – 7

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1. What is the domain?

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The graph of the function shows that all real values of $t$ define the domain. In interval notation this is written as $(- \infty, + \infty)$ or as an inequality, $- \infty < t < + \infty$ .

2. What is the range?

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The graph of the function shows that all real values of $n$ greater than 0 define the range. In interval notation this is written as $(0, + \infty)$ or as an inequality, $0 < t < + \infty$ .

3. The scenario dictates that the domain and range be restricted in comparison to the graph of the function that models it. Write a reasonable domain and range based on the scenario using inequalities.

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Domain: $0 \leq t < + \infty$ ; because we begin measuring time at 0 hours.

Range: $500 \leq n < + \infty$ ; because we begin with 500 bacteria in the lab experiment.

4. Graph $(t, n)$ when $t$ is 0, 1, 2, 3, and 4.

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5. On the graph of $n$ , where can you see each number that appears in the equation?

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500 is the vertical intercept (or the value of $n$ when $t$ is 0). The 2 is the doubling of the $n$ value every time $t$ increases by 1.

6. Graph $(t, p)$ when $t$ is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)

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7. On the graph of $p$ , where can you see each number that appears in the equation?

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100 is the vertical intercept (or the value of $p$ when $t$ is 0). The 3 is the tripling of the $p$ value every time $t$ increases by 1

Video: Graphing Exponential Functions

Watch the following video to learn more about graphing exponential functions.

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Self Check

Which is the graph of

y=2\cdot3^x

Additional Resources

Graphing Exponential Functions

Graph $y = 4^{x}$ .

When graphing any function, it often helps to start with a table. The table should always contain a minimum of 3 to 5 entries. The more information you have, the easier it is to sketch the graph. The information below is generic and applies to the graph of most exponential growth functions.


Domain	$(- \infty, \infty)$
Range	$(0, \infty)$
$x$ -intercept	None
$y$ -intercept	$(0, 1)$
asymptote	none
contains	$(1, a)$ and $(- 1, \frac{1}{a}$ )

Properties of the graph of

f (x) = a^{x}

when

a > 1

This information can be used to create a graph and connect the points with a smooth curve.

GRAPH SHOWING PLOTTED POINTS NEGATIVE 1 COMMA 1 OVER A, 0 COMMA 1, AND 1 COMMA A

Once a table of values is created, plot each of the points on a coordinate plane, remembering that exponential functions are more of a smooth curve. It is possible that your points don’t fit on the graph you have created, but they can be useful to “see” your graph continue off the coordinate plane.

$x$	$y$	$(x, y)$
0	$4^{0} = 1$	$(0, 1)$
1	$4^{1} = 4$	$(1, 4)$
2	$4^{2} = 16$	$(2, 16)$
3	$4^{3} = 64$	$(3, 64)$
4	$4^{4} = 256$	$(4, 256)$
5	$4^{5} = 1, 024$	$(5, 1024)$

GRAPH WITH PLOTTED POINTS AT (0, 1), (1, 4), (2, 16), AND (3, 64).

An asymptote is a line or curve that a function approaches but does not intersect or touch. In other words, as the input values of a function approach a certain value or tend towards infinity or negative infinity, the function values get arbitrarily close to the asymptote without actually crossing it. Asymptotes can provide valuable information about the behavior of a function and can help in understanding its limits and range.

Since equation $y = 4^{x}$ represents an exponential function, we determine the asymptote of this function by considering the behavior of the function as $x$ approaches positive or negative infinity.

As $x$ approaches positive infinity, the values of $4^{x}$ will grow exponentially larger. There is no upper bound, and the function increases without bound as $x$ increases. Therefore, there is no horizontal asymptote in the positive direction.

Similarly, as $x$ approaches negative infinity, the values of $4^{x}$ will become very small but never reach zero. In this case, there is also no horizontal asymptote in the negative direction.

Therefore, the function $y = 4^{x}$ does not have a horizontal asymptote in either direction. The graph of the function will continue to increase, continue off the coordinate plane, indefinitely as $x$ increases or decreases.

Try it

Graphing Exponential Functions

Graph $f (x) = 3^{x}$ .

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Here is how to create the graph of $y = 3^{x}$ :

$x$	$y$	$(x, y)$
0	$3^{0} = 1$	$(0, 1)$
1	$3^{1} = 3$	$(1, 3)$
2	$3^{2} = 9$	$(2, 9)$
3	$3^{3} = 27$	$(3, 27)$
4	$3^{4} = 81$	$(4, 81)$
5	$3^{5} = 243$	$(5, 243)$

GRAPH WITH PLOTTED POINTS AT (0, 1), (1, 3), (2, 9), (3, 27), AND (4, 81).

$THE GRAPH OF AN EXPONENTIAL FUNCTION WITH $y$-intercepts OF 1 AND PASSING THROUGH THE POINTS (1, 3), (2, 9), (3, 27), AND (4, 81).$