Lines and coordinates

A coordinate plane is the graphing space used to show functions. It’s divided into four quadrants, as shown below:

We use coordinate planes to represent and measure relationships created by functions. To locate a point, you use the $x$-axis (horizontal) and the $y$-axis (vertical). A point is written as an ordered pair: ($x$-value, $y$-value).

In this chapter, you’ll mostly use the coordinate plane to study lines. Some important line relationships include how two lines interact, such as whether they’re perpendicular or parallel.

Definitions

Perpendicular

Perpendicular lines intersect to form a right angle.

Parallel

Parallel lines run side by side and never intersect.

Another key property of a line is its slope. We’ll explore slope next. Later in this chapter, we’ll return to lines to discuss the midpoint and distance formulas.

Slope of a line

The slope of a line tells you how steep the line is, or how quickly $y$ changes as $x$ changes. To find slope, compare two points on the line:

• Find the change in $y$ (the “rise”).
• Find the change in $x$ (the “run”).
• Divide: $\text{slope}=\frac{\text{rise}}{\text{run}}$.

For in-depth instruction on finding the slope of a line, review the chapter Standard form of linear equations.

Parallel lines have the exact same slope. Since they have the same steepness, they stay the same distance apart and never intersect.

Perpendicular lines have slopes that are opposite reciprocals. That means:

• the signs are opposite (positive becomes negative, and vice versa)
• the slopes are reciprocals (you “flip” the fraction)

Equivalently, if a line has slope $m$, a perpendicular line has slope $-\frac{1}{m}$.

Let’s take a look at a few examples.

What is the slope of a line parallel to the line given by the equation $y=3x+7$?

The slope in this equation is $3$. Parallel lines have the same slope, so the answer is $3$.

What is the slope of the line perpendicular to the line given by the equation $y=3x+7$?

This line has slope $3$. To find the opposite reciprocal, change the sign and take the reciprocal:

$$\begin{array}{rl}\text{perpendicular slope}& =-1/\text{slope}\\ \text{perpendicular slope}& =-1/3\end{array}$$

So the answer is $-\frac{1}{3}$.

Midpoint

The midpoint of a line segment is the point exactly halfway between its endpoints. Since a point has both an $x$-coordinate and a $y$-coordinate, you find the midpoint by finding the halfway value in each direction.

In the example below, you can’t just “eyeball” the middle. Instead, you find the middle $x$-value and the middle $y$-value and write them as a coordinate point.

To find the midpoint’s $x$-value, find the number halfway between $-2$ and $1$ by averaging:

$$x_{mid}=\frac{(-2+1)}{2}=-\frac{1}{2}$$

Do the same for the $y$-value:

$$y_{mid}=\frac{(2+0)}{2}=1$$

Putting these together gives the midpoint formula:

$$(x_{mid},y_{mid})=(\frac{(x_1+x_2)}{2},\frac{(y_1+y_2)}{2})$$

This formula matches exactly what you did above: average the two $x$-values, and average the two $y$-values.

Try an example on your own below:

What is the midpoint of the line below?

Distance

The distance between two points is the length of the straight line segment connecting them. This is closely related to the Pythagorean theorem, which is why the distance formula looks similar:

$Distance=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Let’s do an example together:

If a person runs $1$ mile west, then $2$ miles north, and lastly $1$ mile west, what is the distance from his starting point to his final destination?

First, identify the starting and ending points.

• Start at $(0,0)$.
• Moving west decreases $x$.
• Moving north increases $y$.

Follow the directions:

For $x$, you go $1$ mile left, then another $1$ mile left, so the final $x$-value is $-2$.

For $y$, you go $2$ miles up, so the final $y$-value is $2$.

So the ending point is $(-2,2)$. Now substitute into the distance formula:

$$\begin{array}{rl}d& =\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}\\ d& =\sqrt{(-2-0{)}^2+(2-0{)}^2}\\ d& =\sqrt{4+4}\\ d& =\sqrt{8}\\ d& \approx 2.8\end{array}$$

The distance from the starting point to the ending point is $\sqrt{8}$, or about $2.8$ miles.

Key points

Coordinate plane. This is the graphing area for functions. The quadrants are numbered beginning in the top right quadrant and going counterclockwise around the origin.

Perpendicular. These lines form a right angle when they intersect.

Parallel. These lines run together and never intersect.

Slope. This is the rate of change from one point to another. It is found by dividing the movement in the $y$ direction by that of the $x$ direction.

Midpoint. The middle point of a line is the midpoint and is found by identifying the middle between the $x$ and $y$ values of the endpoints of the line. $(x_m,y_m)=(\frac{(x_1+x_2)}{2}+\frac{(y_1+y_2)}{2})$

Distance. This is the distance between any two points. $Distance=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$