For graphs what does a direct, inverse, linear and power relationships mean?

direct: y = a x

inverse: y = a/x
linear: y = mx + b
power: y = a x^n

a, b, m and n are constants

why do we say that the x-axis is the input and the y-axis is the output?

In the context of graphs, the terms "direct," "inverse," "linear," and "power" describe different types of relationships between variables. Let's break down each term:

1. Direct Relationship: A direct relationship, also known as a positive relationship, exists when two variables change in the same direction. That is, as one variable increases, the other variable also increases. Conversely, as one variable decreases, the other variable decreases as well. In a graph, this is represented by a positive slope, where the line or curve moves upwards from left to right.

2. Inverse Relationship: An inverse relationship, also known as a negative relationship, occurs when two variables change in opposite directions. This means that as one variable increases, the other variable decreases and vice versa. In a graph, this is shown by a negative slope, where the line or curve moves downwards from left to right.

3. Linear Relationship: A linear relationship refers to a straight-line relationship between two variables. In this type of relationship, as one variable changes, the other variable changes at a consistent rate. The graph representing a linear relationship is a straight line.

4. Power Relationship: A power relationship, also known as a nonlinear relationship, is a type of relationship in which the change in one variable is not proportional to the change in the other variable. In this case, the graph is not a straight line but typically exhibits a curve. Power relationships can take various forms depending on the specific equation governing the relationship. Examples include quadratic, exponential, and logarithmic functions.

To determine the type of relationship between variables, you need to examine the pattern of the data points on the graph. If the graph shows a clear upward trend, it indicates a direct relationship. If it demonstrates a downward trend, it suggests an inverse relationship. A straight line, without any significant curve, indicates a linear relationship. Lastly, if the graph shows a curved pattern, you are likely dealing with a power relationship.