Varying the (positive) slope m, holding b at 0 (i.e. y-intercept is (0,0)).
Positive slope m means function is increasing (rising when going from left to right on graph). As x increases, y increases.

The larger the slope m, the steeper the line, the faster it rises/grows.
If m=1, line is the main diagonal, 45°. (this y=x is the identity function ƒ(x)=x.)
If m>1, line is >45°. If m<1, line is <45°

Varying the (negative) slope m, holding b at 0 (i.e. y-intercept is (0,0) )
Negative slope m means function is decreasing (falling when going from left to right on graph). As x increases, y decreases.

The larger the slope |m|, the steeper the line, the faster it falls.
If m=-1, line is the other diagonal, y=-x.

Varying the y-intercept b only, holding slope m at 1. Positive b raises/shifts the line vertically up, negative b lowers it.

Varying both the slope and the y-intercept: y = ±2x ± 3

Special case: any horizontal (flat) line has function ƒ(x)=b or y=b, where b is y-intercept. There is no x term, so some people don't consider this a linear function. Or it is a degree 0 polynomial function. Slope m is 0: y=0x+b, which is y=b. The function is neither increasing nor decreasing. There is no change in the y. For every x, the function outputs b. It's a constant function, the simplest kind of function, always the same, never changing. Whatever the x input is, the output is the same, b. It doesn't do any arithmetic. The range is a single number.
Special special case: ƒ(x)=0 or y=0, the zero function.   Is horizontal line at/on the x-axis. Is the no-degree polynomial function. Is the only function that is both even and odd.

Extra Special case: any vertical line is not even a function. The graph does not pass the vertical line test. Its equation is x=c. Slope is undefined (there is no change in the x, so the slope equation would be m=(y₂-y₁)/0, which would be infinite).

Parallel lines (i.e. don't intersect; no points in common) have same slope m. Linear functions with same slope m are parallel lines. They differ in their y-intercept b.
The distance of the gap between two parallel lines is:
cos(arctan(m))*|b₁-b₂|, and also
|b₁-b₂| / √(m²+1)
Between two parallel lines are an infinite number of parallel lines.

Perpendicular lines (i.e. lines that meet at right angles) have slopes that are the negative reciprocal of each other. i.e. product of slopes of two perpendicular lines is -1: if the slope of a line is m, then all lines perpendicular to it have slopes -1/m.

Every horizontal line is perpendicular to every vertical line, and vice versa.

Various lines:

The sum, difference, and composition of two linear functions is a linear function.
Multiplying or dividing two linear functions takes you non-linear into quadratic and rational functions, respectively.

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Distance between two points (x₁,y₁) and (x₂,y₂), i.e. the length of the line segment connecting the points:
d = √((x₁-x₂)²)+(y₁-y₂)²))
i.e. the square root of the sum of the squares of the differences of the coordinates.
From Pythagorean theorem: length of hypotenuse is square root of sum of sides squared: h = √(a²+b²)
Horizontal line: |x₁-x₂|
Vertical line: |y₁-y₂|

Midpoint of a line segment:
(x_m,y_m) = ((x₁+x₂)/2,(y₁+y₂)/2)
i.e. = (average of the x's, average of the y's)

Intersection point (x_i,y_i) of two lines y=m₁x+b₁ and y=m₂x+b₂
x_i = (b₂-b₁)/(m₁-m₂)
y_i = m₁x_i + b₁    or    m₂x_i + b₂

Given a line y=mx+b and a point (x₁,y₁) not on the line, the line perpendicular to the given line and going through the point is:
y = -1/m x + (x₁/m + y₁)

Given a line y=mx+b and a point (x₁,y₁) not on the line, the line parallel to the given line and going through the point is:
y = mx + (-x₁/m + y₁)

Given a line y=mx+b and a point (x₁,y₁) on the line, the two points on the line a distance d from the given point
(x₁±d/√(1+m²), y₁±dm/√(1+m²))

Given two lines y=m₁x+b₁ and y=m₂x+b₂, the angle between them is:
|arctan(m₁) - arctan(m₂)| * 180/π
(Do the calculator in radians mode)

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Identity function: f(x) = x or id(x)=x or I(x)=x


The input is the output. No arithmetic is done.
If a function ƒ has an inverse ƒ^-1, then their composition equals the identity function: ƒ ∘ ƒ^-1 = id(x)

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Two non-parallel lines m₁x+b₁ and m₂x+b₂ intersect where
the two expressions are equal: x_i=(b₂-b₁)/(m₁-m₂)
     

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Secant line: On the curve of a function ƒ the slope of the secant line connecting any two points (a,ƒ(a)) and (b,ƒ(b)) is the average rate of change over the interval [a,b]. (On a curve, the rate of change changes, or differs, at every point.)

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Tangent line to a curve at a point; the one line just touching the curve at that point. Every point on a curve has one tangent line; each point's tangent line is different than the tangent lines of all other points on the curve; the tangent line is on the convex side of the curve. Slope of the tangent line at the point is the instantaneous change of that function at that point, i.e. how much it is changing at that point, its rate of change. (Some points' different tangent lines might have the same slope.)
   
Calculus's differentiation of the function ƒ is the function ƒ' (i.e. the derivative of ƒ) whose evaluation for each x yields the slope of the tangent line at the (x,y) point on the curve of ƒ.
ƒ(x) = y    points on the curve of ƒ. Graph above: f(x)=x²-1
ƒ'(x) = m    slope of the line tangent to curve at (x,ƒ(x)). Graph above: ƒ'(x)=2x
tangent tangible touchable

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Linear regression line through a bunch of points (data): is the "best" line for approximating the linear relationship between the X and Y variables.

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Asymptote: a line that a curve approaches but never reaches.
Vertical where x is not in domain of the function (typically that would make the denominator 0).
horizontal as x approaches +∞ and/or -∞
slant/oblique