# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/function-modeling/exponential-function
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/function-modeling/exponential-function/en.mdx

Output docs content for large language models.

---

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

export const metadata = {
  title: "Exponential Function",
  description: "Explore exponential growth and decay with real-world applications: population dynamics, radioactive decay, and compound interest. Solve exponential equations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## 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 <InlineMath math="f(x) = a \cdot b^x" /> with <InlineMath math="a \neq 0" />, <InlineMath math="b > 0" />, and <InlineMath math="b \neq 1" />.

**Components of exponential functions:**

In the function <InlineMath math="f(x) = a \cdot b^x" />:

- <InlineMath math="a" /> is the multiplier constant that determines the initial
  value of the function
- <InlineMath math="b" /> is the exponential base that determines the rate of growth
  or decay
- <InlineMath math="x" /> is the independent variable (exponent)

## Characteristics of Exponential Functions

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

### Basic Properties

<div className="flex flex-col gap-4">
  <BlockMath math="f(0) = a \cdot b^0 = a \cdot 1 = a" />
  <BlockMath math="f(x_1 + x_2) = a \cdot b^{x_1 + x_2} = a \cdot b^{x_1} \cdot b^{x_2}" />
  <BlockMath math="f(x_1 - x_2) = a \cdot b^{x_1 - x_2} = \frac{a \cdot b^{x_1}}{b^{x_2}}" />
</div>

### Types of Exponential Functions

**Exponential Growth Function** (<InlineMath math="b > 1" />):

- Function values increase as <InlineMath math="x" /> increases
- Graph rises from left to right
- Example: <InlineMath math="f(x) = 2^x" />

**Exponential Decay Function** (<InlineMath math="0 < b < 1" />):

- Function values decrease as <InlineMath math="x" /> increases
- Graph falls from left to right
- Example: <InlineMath math="f(x) = (0.5)^x" />

## Graphs of Exponential Functions

The following is a visualization of various exponential functions:

<LineEquation
  title={<>Comparison of Exponential Functions</>}
  description={
    <>
      The graph shows the growth function <InlineMath math="f(x) = 2^x" /> and
      decay function <InlineMath math="g(x) = (0.5)^x" />.
    </>
  }
  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 <InlineMath math="f(x) = 2^x" /> and <InlineMath math="g(x) = (0.5)^x" />:

| <InlineMath math="x" />                                    | <InlineMath math="-2" />   | <InlineMath math="-1" />  | <InlineMath math="0" />   | <InlineMath math="1" />   | <InlineMath math="2" />    | <InlineMath math="3" />     |
| ------------------------------------ | ---- | --- | --- | --- | ---- | ----- |
| <InlineMath math="f(x) = 2^x" />     | <InlineMath math="0.25" /> | <InlineMath math="0.5" /> | <InlineMath math="1" />   | <InlineMath math="2" />   | <InlineMath math="4" />    | <InlineMath math="8" />     |
| <InlineMath math="g(x) = (0.5)^x" /> | <InlineMath math="4" />    | <InlineMath math="2" />   | <InlineMath math="1" />   | <InlineMath math="0.5" /> | <InlineMath math="0.25" /> | <InlineMath math="0.125" /> |

## Transformations of Exponential Functions

Exponential functions can be transformed in various ways:

### Vertical Translation

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

<LineEquation
  title={<>Vertical Translation</>}
  description={
    <>
      Comparison of <InlineMath math="f(x) = 2^x" /> with{" "}
      <InlineMath math="g(x) = 2^x + 2" />.
    </>
  }
  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 <InlineMath math="f(x) = a \cdot b^{x-h}" /> shifts the graph to the right (if <InlineMath math="h > 0" />) or to the left (if <InlineMath math="h < 0" />).

## 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 <InlineMath math="P_0" /> and the growth rate is <InlineMath math="r" /> per time period, then:

<BlockMath math="P(t) = P_0 \cdot (1 + r)^t" />

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

- Initial population: 1000 bacteria
- Growth rate: <InlineMath math="r = 0.2" />
- Function: <InlineMath math="P(t) = 1000 \cdot (1.2)^t" />

### Bacterial Growth Table

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

### Radioactive Decay

Radioactive substances decay following exponential functions. If the initial mass is <InlineMath math="M_0" /> and the half-life is <InlineMath math="t_{1/2}" />, then:

<BlockMath 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 <InlineMath math="P" />, interest rate is <InlineMath math="r" /> per year, and time is <InlineMath math="t" /> years:

<BlockMath math="A = P \cdot \left(1 + \frac{r}{n}\right)^{nt}" />

where <InlineMath math="n" /> 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:

<BlockMath math="b^x = c" />

**Method 1: Equalizing Bases**

If <InlineMath math="b^{f(x)} = b^{g(x)}" />, then <InlineMath math="f(x) = g(x)" />

**Example:** Solve <InlineMath math="2^{x+1} = 8" />

<div className="flex flex-col gap-4">
  <BlockMath math="2^{x+1} = 8" />
  <BlockMath math="2^{x+1} = 2^3" />
  <BlockMath math="x + 1 = 3" />
  <BlockMath math="x = 2" />
</div>

**Method 2: Using Logarithms**

To solve <InlineMath math="b^x = c" />, use logarithms:

<BlockMath math="x = \log_b c = \frac{\log c}{\log b}" />

## Exercises

1. Determine the value of <InlineMath math="f(3)" /> if <InlineMath math="f(x) = 5 \cdot 2^x" />

2. Solve the equation <InlineMath math="3^{2x-1} = 27" />

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

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

### Answer Key

1. <InlineMath math="f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40" />

2. <InlineMath math="3^{2x-1} = 27 = 3^3" />, so <InlineMath math="2x - 1 = 3" />
   , therefore <InlineMath math="x = 2" />

3. <InlineMath math="P(10) = 50000 \cdot (1.03)^{10} = 50000 \cdot 1.344 = 67.195" /> people

4. <InlineMath math="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" /> grams
