Exponential Graph

Exponential Graph is a curve that represents the exponential function. Exponential function graph is a graph with a horizontal asymptote that can have increasing and decreasing slope depending on the case. Graphing exponential function is very important as it represents various aspects of life such as the growth of population in a country, the spread of viruses, etc.

In this article, we will learn about, Exponential Function graphs, How to Plot Exponential Functions, Examples of Exponential Functions, and others in detail.

What is Exponential Graph?

An exponential graph is the graphical representation of exponential functions.

Graph of exponential functions are non-linear and their slope is always changing. In an exponential graph the independent variable acts as the exponent for some real number as its base. In an exponential graph, the rate of growth/decay is very rapid as the value of the independent variable increases.

Exponential functions are of the form

f(x) = ka^x

where,

• x is Independent Variable
• k is Some Constant
• a is Base of Exponential Graph

Exponential Function Formula

The exponential functions are the functions of the from f(x) = a^x, where x is the independent variable and acting as an exponent. The graph of the exponential function depends on the exponential function which depends on the value of the independent variable.

f(x) = ka^x

where,

• x is Independent Variable
• k is Some Constant
• a is Base of Exponential Graph

Since the value of the graph depend on the value of a and k(majorly on a, because sometimes k can be one also), so we can say that:

• Value of ‘a‘ cannot be 0, because in that case the f(x) = 0, will be a line which in no other than the x-axis.

• Value of ‘a‘ cannot be 1, because in that case, f(x) = k, will be a line parallel to the x-axis.

Since value of f(x) can never be 0 for any real value of x, so we can say that the graph will not cut the x-axis, but it will always cut the y-axis (if extended in direction). Therefore the graph can have x-intercept but it will always have a y-intercept.

Value of f(x) initially starts with a horizontal line then changes at a slow rate and then a rapid change occurs with the increase in the value of x, the change can be either increase or decrease.

Graphing Exponential Function

Exponential graph are plots of exponential functions, where the exponential functions are of type, f(x) = a^x. The value of f(x) can never be zero for any real value of x, so the x-axis will always be an asymptote (a line which touches the at infinity) to the graph.

Consider an example of f(x) = a^x + b, for drawing any graph, x and y intercept plays an important role, so let’s understand that. Steps to draw the exponential graph:

Step 1: Find the asymptote of the graph, which shift by an amount in case of bias(quantity added/subtracted) in the function, it will be shifted vertically by an amount ‘b’ in this case.

Step 2: Find the y-intercept of the graph by putting the value x = 0, which will be the point where the graph cuts the y-axis (b in this case).

Step 3: In a tabular format find the value of f(x) for 4-5 values of x (consider both positive and negative values), and plot them in the graph.

Step 4: Join all the plotted point with raw hand making the x-axis as asymptote.

Exponential Growth Graph

Based on how the exponential graph changes it can be categorized as exponential decay graph and exponential growth graph. When the value of the base is greater than 1, then the value of function first increase slow with the increase in the value of x, then as the value of x keeps increasing the value of function increases very rapidly with the increase in the value of x, such graphs are called exponentially increasing graph.

When the value of base ‘a‘ of the graph is a > 1, it will have a increasing nature, showing exponential increase with a steep upward slope.

Example: Population Growth, Spread of Disease etc.

• Condition of exponentially increasing graph: f(x) = a^x , where (a > 1)

Example of Exponentially Increasing Graph: 2^x , 3^x, e^x etc. The graph of e^x is added below,

Exponential Decay Graph

When the value of function decrease very rapidly in the beginning and the as the value of the exponent increase the decrease becomes steady, such graphs are called exponentially increasing graph.

When the value of base ‘a‘ of the graph ranges 0 < a < 1, the graph will show an exponential decrease and will have a downward curve.

Example: Radioactive Decay, Cooling of Sunstances, etc.

• Condition of exponentially increasing graph: f(x) = ax , where (a < 1)

Example of Exponentially Increasing Graph: -2^x , (0.11)^x, (1/e)^x etc. The graph of e^-x is added below,

Exponential Function Derivative

The derivatives of exponential functions can be given as,

f(x) = e^x

f'(x) = e^x

Hence, derivative of exponnetial function is also, e^x.

Exponential Series

An exponential series is a mathematical series that represents the sum of terms which converges at infinity. The general term of exponential series is given by:

t_{(r + 1)} = x^r/r!

Some of commonly used exponential series is given below:

• e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + …

Difference Between Exponential Graph and Logarithmic Graph

Difference between Exponential Graph and Logarithmic Graph:

Exponential Graph Vs Logarithmic Graph
Point of Difference	Exponential Graph	Logarithmic Graph
Based On	These are result of Exponential Functions.	These are result of Logarithmic Functions.
Function Form	f(x) = ax	f(x) = logb(x)
Nature	Shown Rapid growth (a > 1) or decay (a < 1).	Always shows slow increase.

Read More,

• Laws of Exponents
• Negative Exponents
• How to multiply and divide exponents
• Adding and Subtracting of  Exponents

Exponential Graph Examples

Example 1: Consider an example of f(x) = 3^x – 3. Draw the graph.

Solution:

Step1: Find Asymptote:

We can see that in the above given f(x) the bias is added so the asymptote will not simply be the x-axis, but will be shifted by a value of b(-3 in this case).

Step2: Calculate y-intercept:

For finding y-intercept we have to put the x=0,

f(0) = 30 – 3 = 1 – 3 = -2

Step3: Take 3-4 points randomly to draw the graph:

x	f(x) = 3x – 3
-3	⇒ 3-3 – 3 = -2.96
-2	⇒ 3-2 – 3 = -2.88
2	⇒ 32 – 3 = 6
3	⇒ 33 – 3 = 24
4	⇒ 34 – 3 = 78

Plot these points to see the graph

Example 2: Draw Graph of e^-x

Solution:

The graph of e^-x is added below,

Exponential Graph – Practice Questions

Problem 1: Draw a graph for e^x and find out whether it is a decay or a growth graph.

Probelm 2: Mention the asymptote and draw the graph for f(x) = 80 – 9^x.

Problem 3: Find the y-intercept and draw the graph for f(x) = (0.1)^x and find out whether it is a decay or a growth graph.

Exponential Graph – Frequently Asked Questions

What is an Exponential Graph?

Exponential Graphs are graphs that represent Exponential Functions.

How to Graph Exponential Function?

To follow the Exponential Functions follow the steps added above in the article.

What are Asymptotes of Exponential Graph?

Asymptotes are the lines which appear to touch the graph at infinity or in other words Asymptotes are the tangent to the curve at infinity. In case of Exponential Graph x-axis act as the asymptote.

Is an Exponential Graph a Parabola?

No, the Exponential Graph are the result of the functions of the form f(x) = a^x where as the parabolas are result of the quadratic function of the f(x) = ax²+ bx+c.