Linear functions

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by the relation $f(x)=ax+b,$ with a, b$\in \mathbb{R}$.

The function $f$ is increasing if a>0, and strictly increasing if a<0.

The function is constant if a=0.

If: 1) $a\ne 0$; ${{x}_{1}}=-\frac{b}{a}$ is the unique root. $S= \!\!\{\!\! -\frac{b}{a} \!\!\}\!\!$.

2) a=0 and $b\ne 0$, the function has no roots. $S=\phi$.

3) a=0 and b=0, then every real number x is a root of the affine function $S=\mathbb{R}$.

The graph of a first-degree (affine) function

Sign of the affine function $f:\mathbb{R}\to \mathbb{R}$, $f(x)=ax+b,$ ($a\ne 0$)

$x$	$-\infty$	$-\frac{b}{a}$	$\infty$
$f(x)$	Opposite sign to a	0	Same sign as a

In the relation $y=ax+b$ a is called the gradient of the line, and b is the y-intercept.

The significance of the gradient (a) can be seen in the figure below:

Particular cases of lines

1. The line parallel to the $Oy$ axis has the equation $x=a,(\forall )y\in \mathbb{R}$;

2. The line parallel to the $Ox$ axis has the equation $y=b,(\forall )x\in \mathbb{R}$;

3. The line with equation $y(x)=x$ is called the first bisector;

4. The line with equation $y(x)=-x$ is called the second bisector.

A function with equation $y=ax+b$ separates the plane into two disjoint half-planes (with no point in common):

1) the half-plane above the line d, made up of the points with the property $y>ax+b$;

2) the half-plane below the line d, made up of the points with the property $y<ax+b$.

The relative position of two lines ${{d}_{1}}$ and ${{d}_{2}}$ in a coordinate system

Let ${{d}_{1}}$ and ${{d}_{2}}$ be two lines with equations ${{d}_{1}} :y={{m}_{1}}x+b$, ${{d}_{2}} :y={{m}_{2}}x+b$

1. Parallel lines. They have the same gradient (${{m}_{1}}={{m}_{2}}$). The two lines do not intersect, so the system formed by them is inconsistent.

2. Intersecting lines. The two lines ${{d}_{1}}$ and ${{d}_{2}}$ form a consistent system with a unique solution S. The geometric image of S is the point where the two lines meet.

Intersecting lines are perpendicular if ${{m}_{1}}\cdot {{m}_{2}}=-1$.

3. Coincident lines. The equations of the two lines ${{d}_{1}}$ and ${{d}_{2}}$ are identical (same gradient and same y-intercept). In this case the system formed by the two lines is consistent and indeterminate.

Try it

Drag the sliders. a is the gradient — how steeply the line climbs — and b is where it crosses the vertical axis.

Watch what happens when a is negative, and what happens when a is exactly 0.