Linear Equations & Functions

The coordinate plane is formed by two number lines that cross at the origin $(0,0)$.

• The horizontal line is the $x$-axis; the vertical line is the $y$-axis.
• Every point is named by an ordered pair $(x,y)$. The first number $x$ tells how far right ($+$) or left ($-$) you go; the second number $y$ tells how far up ($+$) or down ($-$).
• The axes split the plane into four quadrants, numbered counterclockwise: @@BLOCK0@@

A function is a rule that assigns to each input exactly one output. Think of it as a machine: put in $x$, get out one value.

We write $f(x)$ (read “$f$ of $x$”) for the output when the input is $x$. Here $x$ is the input and $f(x)$ is the output. The letters $f$, $g$, $h$ are just names for the rule.

The domain is the set of all allowed inputs (the $x$-values). The range is the set of all resulting outputs (the $y$-values).

Slope $m$ measures steepness: how much $y$ changes for each unit that $x$ increases. Given two points $(x_1,y_1)$ and $(x_2,y_2)$,

Slope is the rate of change of the line.

In slope-intercept form

the number $m$ is the slope and $b$ is the $y$-intercept (the $y$-value where the line crosses the $y$-axis, at the point $(0,b)$).

If a line has slope $m$ and passes through $(x_1,y_1)$, its equation is

This is the fastest way to write a line when you know a point and the slope.

Standard form is $Ax+By=C$, where $A$, $B$, $C$ are integers. It is especially handy for finding both intercepts:

• $x$-intercept: set $y=0$ and solve for $x$. Point $(x,0)$.
• $y$-intercept: set $x=0$ and solve for $y$. Point $(0,y)$.

• Slope $+$ point: plug into point-slope $y-y_1=m(x-x_1)$, then simplify.
• Two points: first find $m=\dfrac{y_2-y_1}{x_2-x_1}$, then use point-slope with either point.
• From a graph: read $b$ (where it crosses the $y$-axis) and count rise/run for $m$; write $y=mx+b$.

• Parallel lines never meet: they have equal slopes, $m_1=m_2$.
• Perpendicular lines meet at a right angle: their slopes are negative reciprocals, $m_1\cdot m_2=-1$, so $m_2=-\dfrac{1}{m_1}$.

• A horizontal line has the form $y=c$. Every point has the same $y$-value. Slope $=0$.
• A vertical line has the form $x=c$. Every point has the same $x$-value. Slope is undefined. (A vertical line is not a function.)

Every non-vertical line can be written in three equivalent ways. Fluency means switching to whichever form a problem rewards.

• Slope-intercept $y=mx+b$ — best for graphing and reading slope/intercept.
• Point-slope $y-y_1=m(x-x_1)$ — best when you know a point and the slope.
• Standard $Ax+By=C$ (integers, $A\ge 0$) — best for intercepts and for vertical lines $x=c$.

Translation rules: to leave point-slope, distribute and solve for $y$; to reach standard form, collect $x$ and $y$ on one side and clear denominators; to leave standard form, solve for $y$ so that $m=-\tfrac{A}{B}$ and $b=\tfrac{C}{B}$.

The distance from a point $(x_0,y_0)$ to the line $Ax+By+C=0$ is the length of the perpendicular segment from the point to the line:

Move everything to one side first so the line reads $Ax+By+C=0$. The absolute value keeps the distance positive; the denominator $\sqrt{A^2+B^2}$ scales for the line's slant.

Three points $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ form a triangle whose area is given by the shoelace formula

Collinearity test: the three points lie on one line exactly when this area is $0$ (a “flat” triangle). Equivalently, the slope $AB$ equals the slope $BC$.

Parallel lines have equal slopes. Take $y=m_1x+b_1$ and $y=m_2x+b_2$. They intersect where $m_1x+b_1=m_2x+b_2$, i.e. $(m_1-m_2)x=b_2-b_1$. If $m_1\ne m_2$ this has exactly one solution, so the lines cross. Parallel lines never cross, forcing $m_1=m_2$.

Perpendicular lines have negative-reciprocal slopes. A line of slope $m$ points along the direction vector $(1,m)$. Rotating a direction by $90^\circ$ sends $(1,m)\mapsto(-m,1)$, whose slope is $\dfrac{1}{-m}=-\dfrac{1}{m}$. Hence a perpendicular line has slope $-\tfrac{1}{m}$, and $m\cdot\left(-\tfrac{1}{m}\right)=-1$.

A piecewise function uses different rules on different parts of the domain. Each rule is paired with the interval where it applies:

At the “seam” $x=1$ both pieces give $2$, so the graph connects. A step function is a piecewise function whose pieces are constant, producing a staircase of flat segments (for example, a parking rate that jumps to the next whole-hour price).

Real data rarely lands perfectly on one line. A line of best fit (or trend line) is the straight line $y=mx+b$ that passes as close as possible to all the points at once. The standard method, least squares, chooses $m$ and $b$ to make the total of the squared vertical gaps (the residuals) between the data and the line as small as possible.