# Difference Between Linear Function and Exponential Function

The main difference between linear and exponential functions is their rate of change. Linear functions have a constant rate, while exponential functions have a rate that continuously increases or decreases.

**Comparative Analysis of Linear Function and Exponential Function**

### 1: **Rate of Change**

**Linear**:Â In linear functions, the rate of change isÂ constant. This means that for every equal increase in the input (x), the output (y) increases by the same amount. Imagine climbing a staircase – each step takes you the same distance higher.

**Exponential**:Â In exponential functions, the rate of changeÂ continuously increases or decreases. This means that for every equal increase in the input, the output increases (or decreases) by aÂ larger multiplier. Picture bacteria multiplying or radioactive decay – the change gets faster or slower over time.

**2: Visually**

**Linear function**:Â Graph is a straight line with a constant slope.

**Exponential function**:Â Graph is a curve that either bends upwards or downwards, getting steeper or shallower as it progresses.

**Examples**

**Linear**:Â Distance traveled at a constant speed, cost of groceries with a fixed price per item.

**Exponential**:Â **Population growth** with constant birth rate, **radioactive decay**, and compound interest.

**Linear Function vs Exponential Function**

Here is a table summary showing Difference Between Linear Function and Exponential Function:

Property | Linear Function | Exponential Function |
---|---|---|

Definition | f(x) = mx + b | f(x) = a^x, where a is a positive constant and a â‰ 1 |

Rate of Change | Constant | Increases or decreases exponentially depending on the value of a |

Graph | Straight line | Non-linear curve that approaches the x-axis asymptotically as x approaches negative infinity and increases/decreases without bound as x approaches positive/negative infinity depending on the value of a |

Examples | y = 2x + 3, distance traveled at a constant speed | y = 2^x, population growth over time |

Applications | Modeling linear relationships between variables, representing uniform motion | Modeling exponential growth or decay, representing compound interest, radioactive decay |

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