## 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

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