Question

what is vertical line test
and independent vs dependent variable

Answer 1

### Vertical Line Test

The **vertical line test** is a method used to determine whether a given graph represents a function. According to this test, if any vertical line intersects the graph of a relation at more than one point, then the relation is not a function. This is because, by definition, a function assigns exactly one output (dependent variable) for each input (independent variable). So if a vertical line (which represents a specific input value) crosses the graph at more than one point, it implies that there are multiple outputs for that single input, violating the definition of a function.

### Independent vs. Dependent Variables

- **Independent Variable**:
- The independent variable is the variable that is manipulated or changed in an experiment or function. It is considered the "input" variable.
- Typically plotted on the x-axis of a graph.
- In a mathematical context, it is the variable that stands alone and isn't affected by other variables.

- **Dependent Variable**:
- The dependent variable is the variable that is measured or observed in response to changes in the independent variable. It is considered the "output" variable.
- Typically plotted on the y-axis of a graph.
- It relies on the independent variable; its value is dependent on the changes made to the independent variable.

### Example:

In the context of a function \( y = f(x) \):
- \( x \) is the independent variable (input).
- \( y \) is the dependent variable (output).

If you were conducting an experiment on how the amount of sunlight (independent variable) affects plant growth (dependent variable), changes in sunlight would directly impact plant growth. Here, sunlight is manipulated (independent), while plant growth depends on that manipulation (dependent).