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Q: How can you decide whether a relationship is linear by studying the words used to describe the variables?

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The graph must be a straight line, and it must pass through the origin.

It allows a quick look at the data to establish whether or not there is any relationship between the variables and also an indication of the nature of the relationship: linear, quadratic, power etc.

if it passes through (0,0) then it is a direct variation

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When a question asks you to state the relationship between variables, it is requesting you to describe how the variables are related to each other. This could include whether they have a positive or negative correlation, whether one variable causes a change in the other, or if there is no relationship between the variables.

it depends on who you are

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They can be looking for the shape of the relationship between the variables (whether it is linear, quadratic, reciprocal), and the domain over which that relationship extends. They may be looking for peaks or troughs in the line, they may be looking for harmonics, or periodicity. The answer really depends on what is being studied.

By definition, if you graph the relationship between two variables and the result is a straight line (of whatever slope) that is a linear relationship. If it is a curve, rather than a straight line, then it is not linear.

Yes. It can give insight as to whether there is a relationship between two variables, and if so, whether the relationship is direct or indirect; whether it is linear, polynomial, exponential, logarithmic; whether or not there are asysmptotic values; whether or not there is clustering; etc.

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Words such as "proportional to" "increases as" "decreases as", usually give an indication of a linear relation. If there are words like "Square" "power" "inversely proportional" then most likely not linear.

The graph must be a straight line, and it must pass through the origin.

It allows a quick look at the data to establish whether or not there is any relationship between the variables and also an indication of the nature of the relationship: linear, quadratic, power etc.

if it passes through (0,0) then it is a direct variation

If the points lie on a straight line through the origin, the two variables are in direct proportion.