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Such function defines a line that passes through the origin of the coordinate system, that is, the point (,) = (,). In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes b ≠ 0 {\displaystyle b\neq 0} .
Given two different points (x 1, y 1) and (x 2, y 2), there is exactly one line that passes through them. There are several ways to write a linear equation of this line. If x 1 ≠ x 2, the slope of the line is . Thus, a point-slope form is [3]
The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept. If a > 0 then the gradient is positive and the graph slopes upwards. If a < 0 then the gradient is negative and the graph slopes downwards.
If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar: The dotted line goes through the points ( 2 , 4 ) {\displaystyle (2,4)} and ( 3 , 9 ) {\displaystyle (3,9)} , which both lie on the curve y = x 2 {\displaystyle y=x^{2}} .
Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem. Remark 2: The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not related to Pascal's theorem.
This shows that r xy is the slope of the regression line of the standardized data points (and that this line passes through the origin). Since − 1 ≤ r x y ≤ 1 {\displaystyle -1\leq r_{xy}\leq 1} then we get that if x is some measurement and y is a followup measurement from the same item, then we expect that y (on average) will be closer ...
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The line with equation ax + by + c = 0 has slope -a/b, so any line perpendicular to it will have slope b/a (the negative reciprocal). Let (m, n) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point (x 0, y 0). The line through these two points is perpendicular to the original ...