1. **Problem Statement:** Explain how to determine if a relation is linear, quadratic, or exponential given a table of values, an equation, or a graph, and provide real-life examples. 2. **Determining from a Table of Values:** - **Linear:** The differences between consecutive y-values are constant. - **Quadratic:** The first differences are not constant, but the second differences (differences of differences) are constant. - **Exponential:** The ratios of consecutive y-values are constant (each y-value is multiplied by the same factor). 3. **Determining from an Equation:** - **Linear:** The equation is of the form $y = mx + b$ where $m$ and $b$ are constants, and the variable $x$ is to the first power. - **Quadratic:** The equation is of the form $y = ax^2 + bx + c$ where $a \neq 0$ and $x$ is squared. - **Exponential:** The equation is of the form $y = ab^x$ where $a$ and $b$ are constants, and $x$ is in the exponent. 4. **Determining from a Graph:** - **Linear:** The graph is a straight line with constant slope. - **Quadratic:** The graph is a parabola, typically U-shaped. - **Exponential:** The graph is a curve that increases or decreases rapidly, showing exponential growth or decay. 5. **Real-life Examples:** - **Linear:** Distance traveled at a constant speed (e.g., walking 3 km every hour). - **Quadratic:** The height of a ball thrown upwards over time (due to gravity). - **Exponential:** Population growth where the population doubles every fixed period.