Linear functions and equations.

Forms of lines, or linear equations of two variables:
y=mx+b is the slope-intercept form.
y-y1=m(x-x1) is the slope-point form.
y-y1=[(y2-y1)/(x2-x1)](x-x1) is the two-points form.
Ax + By + C = 0 is the standard form equation of a line. m=-A/B and b=-C/B
These forms are interconvertible and thus equivalent.

ƒ(x) = mx + b   and/or   y = mx + b

A subclass of polynomial function where the variable x is only in the first power (i.e. a degree one polynomial in one variable (i.e. univariate)). m and b are any real numbers.

m is the slope.
b is the y-intercept.

Its graph in the xy Cartesian plane is a (straight) line. This is why linear functions are called line-ar. (A graph shows all the points (i.e. pairs of x and y values) that make the equation true. And shows, for each value of x, its corresponding ƒ(x) (i.e. y) function value.)

Slope is the steepness/shallowness, grade, gradient, tilt, pitch, slant, incline/decline of the line. The slope is the same at all points of the line. Slope is how much the function is increasing or decreasing, i.e. its rate of change, which is how much the real-world process/activity is changing. A linear function's rate of change is unchanging, i.e. it is the same.
Between any two points:
--the rise over the run;
--the change in the vertical divided by the change in the horizontal;
--the change in y per one unit change in x, i.e. the rate of change of the y.
m = Δy / Δx = (y2-y1)/(x2-x1) for any two points on the line.

Slope Explorer demos slope.

Line crosses the y axis at the y-intercept: (0,b). If b is 0, graph crosses origin (0,0). A nonzero b is an offset from the origin, often representing an initial value.

Line crosses the x axis at the x-intercept, also called the zero of the function: (-b/m,0). X-intercept's x coordinate is the solution to the linear equation mx+b=0.

Every linear function's line crosses both axes. (Except the special case of horizontal lines, see below). A line through the origin has x-intercept and y-intercept at the same point, i.e. (0,0).


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°
If α is an angle in raidians: m=tan(α) and m=tan(angle°*(π/180))
α=arctan(m) and angle°=arctan(m)*(180/π)
Slope as the change in y as a percentage of x: %=m*100% (e.g. y=2x y=200%x; y=1/2x y=50%x)
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. 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.

Extra Special case: any vertical line is not even a function. The graph does not pass the vertical line test. 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 have same slope m. Linear functions with same slope m are parallel lines. They differ in their y-intercept b.

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.

Various lines:

Tangent line to a curve at a point. 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.

Calculus' differentiation of the function f is the function f' (i.e. the derivative of f) whose evaluation for each x yields the slope of the tangent line at the (x,y) point on the curve of f.
f(x) = y points on the curve of f. Graph above: f(x)=x2-1
f'(x) = m slope of the line tangent to curve at (x,f(x)). Graph above: f'(x)=2x
tangent tangible touchable

Examples of linear functions

Systems of linear equations

Inverses of linear functions