is it a line that is slanted
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If I understand the question correctly, the slope line to a curve, at a given point on the curve, is the tangent to the curve at that point.
Your understanding is correct.
For a given curve you differentiate it and apply the x-vaalue at the point you wish to find.
If the line has a slope of 2, then the perpendicular line has a slope of -1/2. The slope of a perpendicular line is the negative reciprocal. Another example would be if the slope of a line is -1/4, then the slope of the perpendicular is 4.
if the slope of a line is 2/3, then the slope of a parallel line would be 2/3.
For two lines to be parallel they must have the same slope. A line parallel to a line with slope -2 would have a slope of -2.
The slope of a line measures the steepness of the line.
The slope is rise over run. If another line was parallel, the slope would be the same.