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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.
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.
|-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.
- f(x) = 2x shows exponential growth as 2 > 1.
- 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:
Exponential Graph Examples
Example 1: What are the horizontal asymptotes of the exponential graphs of the following exponential functions? a) f(x) = (1/3)x - 2 + 5 b) g(x) = (-0.5)x + 2 - 8.
The vertical shift of an exponential function itself would give the horizontal asymptote. i.e., we should just look at the number that is being added to the exponent part.
a) The horizontal asymptote of f(x) = (1/3)x - 2 + 5 is y = 5.
b) The horizontal asymptote of g(x) = (-0.5)x + 2 - 8 is y = -8.
Answer: a) y = 5 b) y = -8.
Example 2: Explain the process of graphing exponential function f(x) = -6 (1/2)x. Draw the exponential graph as well.
We follow the steps below for graphing the given exponential function.
Its horizontal asymptote is y = 0.
For finding the y-intercept, y = -6 (1/2)0 = -6(1) = -6. So its y-intercept is (0, -6).
For finding the x-intercept, 0 = -6 (1/2)x. But there is no x that satisfies this equation. So there is no x-intercept.
Now, let us find some points on it.
x y 1 -6 (1/2)1 = -3 2 -6 (1/2)2 = -1.5 3 -6 (1/2)3 = -0.75
Just plot all the information without crossing or touching the horizontal asymptote.
Answer: The process of graphing exponential function is explained and the exponential graph is also drawn.
Example 3: Do each of the following exponential functions represent growth or decay? a) f(x) = 3x - 5 + 7 b) g(x) = 0.53x - 2 - 8.
In general, an exponential function f(x) = ax shows growth if a > 0 and decay if 0 < a < 1.
a) In the function f(x) = 3x - 5 + 7, a = 3 > 1. Hence it represents exponential growth graph.
b) In the function g(x) = 0.53x - 2 - 8, a = 0.5 which lies between 0 and 1. Hence it represents exponential decay graph.
Answer: a) Exponential growth b) Exponential decay.
Practice Questions on Exponential Graph
FAQs on Exponential Graph
What is an Exponential Graph?
Exponential graph is the graph of an exponential function. It always has a horizontal asymptote but no vertical asymptote. Graphing exponential function is the process of drawing the curve representing it.
How to Graph Exponential Function?
To draw an exponential graph, follow the steps mentioned below:
- Find the horizontal asymptote.
- Find the y-intercept and the x-intercept (if any).
- Find some points on it by taking some random values for x.
- Plot all of them on the graph and join all the points.
How Would an Exponential Graph of Growth look like?
The exponential growth graph is always increasing if we see it from left to right. i.e., it always has a positive slope. An exponential function f(x) = ax shows growth if a > 1.
What are the Asymptotes of Exponential Graph?
An exponential graph has a horizontal asymptote (HA). The HA of an exponential function f(x) = ax + b is y = b. It doesn't have any vertical asymptote.
Is an Exponential Graph a Parabola?
No, the parabola is the equation of a quadratic function. The graph of an exponential function either starts slowly and increases rapidly or it is the other way round.
What is the Difference Between Exponential Graph and Logarithmic Graph?
Mathematically, the exponential function and the logarithmic function are inverses of each other. So their graphs are reflections of each other with respect to the line y = x.
How Would an Exponential Graph of Decay look like?
The exponential decay graph is always decreasing if we see it from left to right. i.e., it always has a negative slope. An exponential function f(x) = ax shows decay if 0 < a < 1.