Growth, Decay, Examples | Graphing Exponential Function - Cuemath

Exponential Graph

Exponential graph is the graph of an exponential function. It always has a horizontal asymptote but no vertical asymptote. Graphing of exponential function can be done by plotting the horizontal asymptote, intercepts, and a few points on it.

Let us see how to draw an exponential graph in detail and let us see what the exponential growth graph and exponential decay graph would look like.

1.	What is Exponential Graph?
2.	Graphing Exponential Function
3.	Exponential Growth Graph and Exponential Decay Graph
4.	FAQs on Exponential Graph

What is Exponential Graph?

An exponential graph is a curve that represents an exponential function. An exponential graph is a curve that has a horizontal asymptote and it either has an increasing slope or a decreasing slope. i.e., it starts as a horizontal line and then it first increases/decreases slowly and then the growth/decay becomes rapid. It always cuts the y-axis at some point but it may or may not cut the x-axis. i.e., an exponential graph always has a y-intercept but it may or may not have the x-intercept. The exponential graph may look in one of the following ways.

Graphing Exponential Function

Graphing exponential function is the process of drawing the curve representing it. An exponential function is of the form f(x) = ax, where 'a' is a constant and a > 0. The value of ax is never 0 for any value of x and so y = 0 is the horizontal asymptote of the exponential function f(x) = ax. The horizontal asymptote plays an important role in the process of the graphing exponential function.

The horizontal asymptote of an exponential function is nothing but its vertical shift (i.e., it is a number that is being added to ax). For example, the horizontal asymptote of f(x) = 2x is y = 0 and the horizontal asymptote of g(x) = 2x - 3 is y = -3. Here are the steps to draw the exponential graph in the easiest way.

• Step 1: Find the horizontal asymptote.
• Step 2: Find the y-intercept by substituting x = 0 in the function. Every exponential graph has a horizontal asymptote.
• Step 3: Find the x-intercept by substituting y = 0 in the function. An exponential graph may or may not have an x-intercept.
• Step 4: Create a table with two columns x and y; take some random numbers for x, say -1, 0, and 1; substitute each of these numbers in the function to get corresponding values of y.
• Step 5: Plot all the above information and join all the points obtained above by a curve without touching but reaching the horizontal asymptote.

Here is an example of graphing exponential function.

Example: Graph the exponential function f(x) = 2x - 3.

Solution:

The horizontal asymptote is y = -3.

For y-intercept, put x = 0. Then we get y = 20 - 3 = 1 - 3 = -2. So the y-intercept is (0, -2).

For x-intercept, put y = 0. Then we get 0 = 2x - 3 ⇒ 2x = 3 ⇒ x = log2 3 ≈ 1.6. So the x-intercept is (1.6, 0).

Now, we will create the table of the exponential function.

x	y
-1	2-1 - 3 = (1/2) - 3 = -2.5
1	21 - 3 = 2 - 3 = -1
2	22 - 3 = 4 - 3 = 1

Let us plot all this information to obtain the exponential graph.

Here, the graph has a negative y-intercept and a positive slope (increasing curve).

Exponential Growth Graph and Exponential Decay Graph

The above graph is increasing (see from left to right always) and hence that graph represents exponential growth. Note that the function that represents the above graph is f(x) = 2x - 3 where the base is "2" and is "greater than 1". So in general, for any exponential function f(x) = ax,

• the exponential graph shows growth when a > 1
• the exponential graph shows decay when 0 < a < 1.

For example,

1. f(x) = 2x shows exponential growth as 2 > 1.
2. g(x) = 0.5x shows exponential decay as 0 < 0.5 < 1.

We can see both graphs in teh figure below.

Note some other cases here.

• Does f(x) = -2x represent growth or decay? Its decay as g(x) = 2x represents exponential growth.
• Does f(x) = 2-x represent growth or decay? It represents exponential decay as we can represent it as f(x) = (2-1)x (by power of a power property of exponents) = (1/2)x = 0.5x.
• Does f(x) = -2-x represent growth or decay? It represents exponential growth as 2-x represents decay.

Important Notes on Exponential Graph:

• For graphing exponential function, plot its horizontal asymptote, intercept(s), and a few points on it.
• f(x) = ax is an exponential growth if a > 1 and is an exponential decay when 0 < a < 1.
• (0, 1) and (1, a) are always two points on f(x) = ax and they help in graphing exponential graph.
• An exponential function never has a vertical asymptote (as it is defined for all values of x).

☛ Related Topics:

• Logarithmic Functions
• Exponential Equations
• Exponential Function Calculator