If by positive you mean that an increase in the independent variable is accompanied by an increase in the dependent variable then this will be indicated by a correlation close to one.
What is considered 'close to one' depends on the field of study. In some fields where it can be quite difficult to establish relationships between variables a correlation of, say, 0.35 might be considered important, provided of course that it has been shown to be statistically significant.
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A linear relationship is one where your equation forms a straight line. A positive linear relationship is one where this line has a positive gradient.
The line that connects the dots is relatively straight.
Whenever you are given a series of data points, you make a linear regression by estimating a line that comes as close to running through the points as possible. To maximize the accuracy of this line, it is constructed as a Least Square Regression Line (LSRL for short). The regression is the difference between the actual y value of a data point and the y value predicted by your line, and the LSRL minimizes the sum of all the squares of your regression on the line. A Correlation is a number between -1 and 1 that indicates how well a straight line represents a series of points. A value greater than one means it shows a positive slope; a value less than one, a negative slope. The farther away the correlation is from 0, the less accurately a straight line describes the data.
It looks like a straight line increasing as the x-values increase. So basically a line that goes from the bottom left area to the top right area on a graph.
Straight line.