Exponential Functions

Key Concepts

• Define an exponential function.
• Draw the graph of an exponential function.
• Write an exponential function represented by the graph.
• Write an exponential function represented by the table.

Exponential Functions

1. Exponential function 

The product of an initial amount and a constant ratio raised to a power is an exponential function. 

Exponential functions are modeled using f(x)=a.b^x, where a is a non-zero constant, b>0 b≠1.

2. Graph of exponential functions 

If the value of b lies between 0 and 1, the graph is decreasing. 

f the value of b is greater than 1, the graph is increasing. 

3. Steps to draw the graph of an exponential function 

Example: Draw the graph of the function 2(5)^x

Step 1: Make a table 

The values of f(x) vary for different values of x. 

Step 2: Draw the graph: 

4. Steps to write the exponential function from the data represented by a table 

Step 1: Find the initial amount from the table of values given. 

Step 2: Calculate the constant ratio from the y-values. 

Step 3: Substitute in the standard form of an exponential function. 

Example: Write the exponential function for the data given: 

Step 1: Find the initial value. 

The initial value of the function is 8. 

Step 2: Find the constant ratio. 

The constant ratio is 4. 

Step 3: Write the exponential function. 

In f(x) = a.b^x, substitute 8 for a and 4 for b. 

Therefore, the function is f(x) = 8(4)^x

5. Steps to write/ frame an exponential function for the data represented by a graph 

Step 1: Find the initial amount from the graph given. 

Step 2: Calculate the constant ratio from the y-values. 

Step 3: Substitute in the standard form of an exponential function. 

Example: Write the exponential function represented by the graph. 

Step 1: Find the initial value. 

The initial value of the function is 7. 

Step 2: Find the constant ratio. 

56 ÷ 28 = 2 

28 ÷ 14 = 2 

14 ÷ 7 = 2 

The constant ratio is 2. 

Step 3: Write the exponential function. 

In f(x) = a.b^x, substitute 7 for a and 2 for b. 

Therefore, the function is f(x) = 7(2)^x

6. Comparison of linear and exponential functions 

The linear function increases at a constant rate, whereas the exponential function increases at a constant ratio. 

Concept Map

• The product of an initial amount and a constant ratio raised to a power is an exponential function.  

• Graph of an exponential function is a horizontal asymptot

What we have learned

The product of an initial amount and a constant ratio raised to a power is an exponential function.
Exponential functions are modeled using f(x)=a.b^x, where a is a non-zero constant, b>0,b≠1