The Coordinate Plane

A point is written $(x,y)$: first move $x$ units left or right from the origin $(0,0)$, then $y$ units up or down. $(5,-2)$ is 5 right and 2 down. Points with $x=0$ sit on the $y$-axis, like $(0,-4)$; points with $y=0$ sit on the $x$-axis.

A rule such as $y=2x+1$ produces a table: substitute each $x$. At $x=3$, $y=2(3)+1=7$, giving the point $(3,7)$. Going the other way, look at how $y$ changes as $x$ increases by 1. If $y$ goes 3, 5, 7, 9 the rule contains $2x$; check the first pair to find the constant: $y=2x+1$.

• Same $x$-coordinates: the distance is the difference of the $y$-values. $(2,5)$ to $(2,-3)$ is $5-(-3)=8$.
• Midpoint: average each coordinate. Midpoint of $(-2,6)$ and $(4,2)$ is $\left(\frac{-2+4}{2},\frac{6+2}{2}\right)=(1,4)$.
• Slope $=\frac{\text{change in }y}{\text{change in }x}$. From $(0,3)$ to $(2,7)$: $\frac{7-3}{2-0}=2$.

A relation is any set of ordered pairs. It is a function when each input $x$ has exactly one output $y$. $\{(1,2),(2,2),(3,2)\}$ is a function (outputs may repeat). $\{(1,2),(1,3),(2,4)\}$ is not, because $x=1$ has two outputs. On a graph, a vertical line that crosses twice shows a relation is not a function.

Three vertices of a rectangle are $(1,1)$, $(5,1)$ and $(5,4)$. The missing vertex must share $x=1$ with the first point and $y=4$ with the third: $(1,4)$. Width $5-1=4$, height $4-1=3$, area $12$ square units.

• Plotting $(x,y)$ as $(y,x)$.
• Putting $(-4,0)$ on the $y$-axis.
• Finding a rule from the first pair only.
• Calling a relation "not a function" because $y$-values repeat.
• Keeping a negative sign after squaring: $(-2)^2=4$.