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Algebra 1

5.4.4 Graphing Exponential Expressions

Algebra 15.4.4 Graphing Exponential Expressions

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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 · 2 t n = 500 · 2 t and p = 100 · 3 t p = 100 · 3 t . Use that information and the given coordinate planes to graph the following using the Desmos tool below.

Use the equation n = 500 · 2 t n = 500 · 2 t to answer questions 1 – 7

1. What is the domain?

2. What is the range?

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.

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

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

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

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

Video: Graphing Exponential Functions

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

Self Check

Which is the graph of y = 2 3 x ?
  1. GRAPH OF AN EXPONENTIAL FUNCTION WITH Y-INTERCEPT OF 2 AND PASSING THROUGH THE POINT (1, 6).
  2. GRAPH OF A LINE WITH Y-INTERCEPT OF 2 AND PASSING THROUGH THE POINT (1, 6).
  3. GRAPH OF A LINE PASSING THROUGH THE POINTS (0, 0) AND (1, 6).
  4. GRAPH OF AN EXPONENTIAL FUNCTION WITH Y-INTERCEPT OF 1 AND PASSING THROUGH THE POINT (1, 6).

Additional Resources

Graphing Exponential Functions

Graph y = 4 x 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 ( , ) ( , )
Range ( 0 , ) ( 0 , )
x x -intercept None
y y -intercept ( 0 , 1 ) ( 0 , 1 )
asymptote none
contains ( 1 , a ) ( 1 , a ) and ( 1 , 1 a ( 1 , 1 a )
Properties of the graph of f ( x ) = a x f ( x ) = a x when a > 1 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 x y y ( x , y ) ( x , y )
0 4 0 = 1 4 0 = 1 ( 0 , 1 ) ( 0 , 1 )
1 4 1 = 4 4 1 = 4 ( 1 , 4 ) ( 1 , 4 )
2 4 2 = 16 4 2 = 16 ( 2 , 16 ) ( 2 , 16 )
3 4 3 = 64 4 3 = 64 ( 3 , 64 ) ( 3 , 64 )
4 4 4 = 256 4 4 = 256 ( 4 , 256 ) ( 4 , 256 )
5 4 5 = 1 , 024 4 5 = 1 , 024 ( 5 , 1024 ) ( 5 , 1024 )
GRAPH WITH PLOTTED POINTS AT (0, 1), (1, 4), (2, 16), AND (3, 64). Insert Alt Text

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

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

Similarly, as x x approaches negative infinity, the values of 4 x 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 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 x increases or decreases.

Try it

Try It: Graphing Exponential Functions

Graph f ( x ) = 3 x f ( x ) = 3 x .

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