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URL: https://nakafa.com/en/subjects/mathematics/function-modeling/exponential-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-modeling/exponential-function/en.mdx

Explore exponential growth and decay with real-world applications: population dynamics, radioactive decay, and compound interest. Solve exponential equations.

---

## Understanding Exponential Functions

An exponential function is a mathematical function that has a variable as the exponent of a constant number. The general form of an exponential function is $$f(x) = a \cdot b^x$$ with $$a \neq 0$$, $$b > 0$$, and $$b \neq 1$$.

Visible text: An exponential function is a mathematical function that has a variable as the exponent of a constant number. The general form of an exponential function is with , , and .

**Components of exponential functions:**

In the function $$f(x) = a \cdot b^x$$:

Visible text: In the function :

- $$a$$ is the multiplier constant that determines the initial
  value of the function
- $$b$$ is the exponential base that determines the rate of growth
  or decay
- $$x$$ is the independent variable (exponent)

Visible text: - is the multiplier constant that determines the initial
 value of the function
- is the exponential base that determines the rate of growth
 or decay
- is the independent variable (exponent)

## Characteristics of Exponential Functions

Exponential functions have several special properties that distinguish them from other functions:

### Basic Properties

Component: MathContainer
Children:

```math
f(0) = a \cdot b^0 = a \cdot 1 = a
```

```math
f(x_1 + x_2) = a \cdot b^{x_1 + x_2} = a \cdot b^{x_1} \cdot b^{x_2}
```

```math
f(x_1 - x_2) = a \cdot b^{x_1 - x_2} = \frac{a \cdot b^{x_1}}{b^{x_2}}
```

### Types of Exponential Functions

**Exponential Growth Function** ($$b > 1$$):

Visible text: **Exponential Growth Function** ():

- Function values increase as $$x$$ increases
- Graph rises from left to right
- Example: $$f(x) = 2^x$$

Visible text: - Function values increase as increases
- Graph rises from left to right
- Example:

**Exponential Decay Function** ($$0 < b < 1$$):

Visible text: **Exponential Decay Function** ():

- Function values decrease as $$x$$ increases
- Graph falls from left to right
- Example: $$f(x) = (0.5)^x$$

Visible text: - Function values decrease as increases
- Graph falls from left to right
- Example:

## Graphs of Exponential Functions

The following is a visualization of various exponential functions:

Component: LineEquation
Props:
- title: Comparison of Exponential Functions
- description: The graph shows the growth function $$f(x) = 2^x$$ and
decay function $$g(x) = (0.5)^x$$.
  Visible text: The graph shows the growth function and
decay function .
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.2 - 2;
return { x, y: Math.pow(2, x), z: 0 };
}),
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "f(x) = 2^x", at: 15, offset: [2, -0.5, 0] }],
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.2 - 2;
return { x, y: Math.pow(0.5, x), z: 0 };
}),
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "g(x) = (0.5)^x", at: 5, offset: [-2, -0.5, 0] }],
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

Comparison of function values $$f(x) = 2^x$$ and $$g(x) = (0.5)^x$$:

Visible text: Comparison of function values and :

| $$x$$                                    | $$-2$$   | $$-1$$  | $$0$$   | $$1$$   | $$2$$    | $$3$$     |
| ------------------------------------ | ---- | --- | --- | --- | ---- | ----- |
| $$f(x) = 2^x$$     | $$0.25$$ | $$0.5$$ | $$1$$   | $$2$$   | $$4$$    | $$8$$     |
| $$g(x) = (0.5)^x$$ | $$4$$    | $$2$$   | $$1$$   | $$0.5$$ | $$0.25$$ | $$0.125$$ |

Visible text: | | | | | | | |
| ------------------------------------ | ---- | --- | --- | --- | ---- | ----- |
| | | | | | | |
| | | | | | | |

## Transformations of Exponential Functions

Exponential functions can be transformed in various ways:

### Vertical Translation

The function $$f(x) = a \cdot b^x + c$$ shifts the graph upward (if $$c > 0$$) or downward (if $$c < 0$$).

Visible text: The function shifts the graph upward (if ) or downward (if ).

Component: LineEquation
Props:
- title: Vertical Translation
- description: Comparison of $$f(x) = 2^x$$ with{" "}
$$g(x) = 2^x + 2$$.
  Visible text: Comparison of with{" "}
.
- data: [
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.2 - 2;
return { x, y: Math.pow(2, x), z: 0 };
}),
color: getColor("CYAN"),
showPoints: false,
labels: [{ text: "f(x) = 2^x", at: 15, offset: [1.5, -0.5, 0] }],
},
{
points: Array.from({ length: 21 }, (_, i) => {
const x = i * 0.2 - 2;
return { x, y: Math.pow(2, x) + 2, z: 0 };
}),
color: getColor("PINK"),
showPoints: false,
labels: [{ text: "g(x) = 2^x + 2", at: 15, offset: [0, 0.5, 0] }],
},
]
- cameraPosition: [8, 5, 8]
- showZAxis: false

### Horizontal Translation

The function $$f(x) = a \cdot b^{x-h}$$ shifts the graph to the right (if $$h > 0$$) or to the left (if $$h < 0$$).

Visible text: The function shifts the graph to the right (if ) or to the left (if ).

## Applications of Exponential Functions

Exponential functions are widely used in daily life:

### Population Growth

Living organism populations often follow exponential growth patterns. If the initial population is $$P_0$$ and the growth rate is $$r$$ per time period, then:

Visible text: Living organism populations often follow exponential growth patterns. If the initial population is and the growth rate is per time period, then:

```math
P(t) = P_0 \cdot (1 + r)^t
```

**Example:** Bacterial population that reproduces every hour at a rate of $$20\%$$:

Visible text: **Example:** Bacterial population that reproduces every hour at a rate of :

- Initial population: $$1000 \text{ bacteria}$$
- Growth rate: $$r = 0.2$$
- Function: $$P(t) = 1000 \cdot (1.2)^t$$

Visible text: - Initial population: 
- Growth rate: 
- Function:

### Bacterial Growth Table

| Time (hours) | $$0$$ | $$1$$ | $$2$$ | $$3$$ | $$4$$ | $$5$$ |
| ------------ | ---- | ---- | ---- | ---- | ---- | ---- |
| Population   | $$1000$$ | $$1200$$ | $$1440$$ | $$1728$$ | $$2074$$ | $$2488$$ |

Visible text: | Time (hours) | | | | | | |
| ------------ | ---- | ---- | ---- | ---- | ---- | ---- |
| Population | | | | | | |

### Radioactive Decay

Radioactive substances decay following exponential functions. If the initial mass is $$M_0$$ and the half-life is $$t_{1/2}$$, then:

Visible text: Radioactive substances decay following exponential functions. If the initial mass is and the half-life is , then:

```math
M(t) = M_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}}
```

### Compound Interest

Investments with compound interest grow exponentially. If the initial capital is $$P$$, interest rate is $$r$$ per year, and time is $$t \text{ years}$$:

Visible text: Investments with compound interest grow exponentially. If the initial capital is , interest rate is per year, and time is :

```math
A = P \cdot \left(1 + \frac{r}{n}\right)^{nt}
```

where $$n$$ is the frequency of interest compounding per year.

Visible text: where is the frequency of interest compounding per year.

## Exponential Equations

An exponential equation is an equation that contains a variable in the exponent. General form:

```math
b^x = c
```

**Method $$1$$: Equalizing Bases**

Visible text: **Method : Equalizing Bases**

If $$b^{f(x)} = b^{g(x)}$$, then $$f(x) = g(x)$$

Visible text: If , then

**Example:** Solve $$2^{x+1} = 8$$

Visible text: **Example:** Solve

Component: MathContainer
Children:

```math
2^{x+1} = 8
```

```math
2^{x+1} = 2^3
```

```math
x + 1 = 3
```

```math
x = 2
```

**Method $$2$$: Using Logarithms**

Visible text: **Method : Using Logarithms**

To solve $$b^x = c$$, use logarithms:

Visible text: To solve , use logarithms:

```math
x = \log_b c = \frac{\log c}{\log b}
```

## Exercises

1. Determine the value of $$f(3)$$ if $$f(x) = 5 \cdot 2^x$$

2. Solve the equation $$3^{2x-1} = 27$$

3. A city's population is $$50{,}000 \text{ people}$$ and grows $$3\%$$ per year. What will the population be after $$10 \text{ years}$$?

4. A radioactive substance has a half-life of $$5 \text{ years}$$. If the initial mass is $$100 \text{ grams}$$, how much mass remains after $$15 \text{ years}$$?

Visible text: 1. Determine the value of if 

2. Solve the equation 

3. A city's population is and grows per year. What will the population be after ?

4. A radioactive substance has a half-life of . If the initial mass is , how much mass remains after ?

### Answer Key

1. $$f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40$$

2. $$3^{2x-1} = 27 = 3^3$$, so $$2x - 1 = 3$$
   , therefore $$x = 2$$

3. $$P(10) = 50000 \cdot (1.03)^{10} = 50000 \cdot 1.344 = 67.195 \text{ people}$$

4. $$M(15) = 100 \cdot \left(\frac{1}{2}\right)^{15/5} = 100 \cdot \left(\frac{1}{2}\right)^3 = 100 \cdot \frac{1}{8} = 12.5 \text{ grams}$$

Visible text: 1. 

2. , so 
 , therefore 

3. 

4.