Horizontal and Vertical Lines
Picture a city laid out on a grid. If you walk straight east along Maple Street, your north-south address never changes, and only your east-west position does. If you walk north along 5th Avenue, it is the other way around. These two kinds of lines are so simple that you can write their equations without any formula. Let's find out how, and what happens when Maple Street and 5th Avenue cross.
Horizontal lines
A horizontal line never rises or falls. Every point on it is at the same height, so every point has the same $y$-coordinate.
Slide the value of $k$ below and watch the line move up and down while it stays flat.
Write down the equation of the line parallel to the x-axis ($X$-axis) that passes through the point $A(3, -5)$.
Write down the equation of the line parallel to the x-axis ($X$-axis) that passes through the point $A(3, -5)$.
Write down the general equations of the coordinate axes $X$ and $Y$.
Write down the general equations of the coordinate axes $X$ and $Y$.
Vertical lines
Now turn the picture a quarter turn. On a vertical line you can move up and down as much as you like, but your left-right position never changes.
A vertical line is not the graph of a function: one $x$-value goes with many $y$-values, so it fails the vertical line test. That is why it cannot be written as $y = mx + b$.
Write down the equation of the line parallel to the y-axis ($Y$-axis) that passes through the point $B(-2, 7)$.
Write down the equation of the line parallel to the y-axis ($Y$-axis) that passes through the point $B(-2, 7)$.
A delivery drone moves through a city with grid-like streets. Its path follows a straight line parallel to the $Y$-axis passing through the coordinate $(6, 2)$. Write…
A delivery drone moves through a city with grid-like streets. Its path follows a straight line parallel to the $Y$-axis passing through the coordinate $(6, 2)$. Write down the equation of its path and the coordinates of two points it will pass through.
Direction vector and slope
Every line has a direction. A direction vector $\langle a, b\rangle$ of a line is a vector that points along the line: it says how far to move right ($a$) and how far to move up ($b$) to stay on it. Any nonzero multiple of a direction vector works too.
The slope is the rise over the run, so a line with direction vector $\langle a, b\rangle$ has slope $m = \dfrac{b}{a}$. Some problems call the slope the gradient. It is the same thing.
State the direction vector and the gradient of the line with equation $y = 4$.
State the direction vector and the gradient of the line with equation $y = 4$.
State the direction vector and the gradient of the line with equation $x = -1$.
State the direction vector and the gradient of the line with equation $x = -1$.
Parametric equations
A line can also be written by saying where a moving point is at time $t$. Start at a point $(x_0, y_0)$ and move along a direction vector $\langle a, b\rangle$:
$$x = x_0 + a t, \qquad y = y_0 + b t$$For lines parallel to the axes, one coordinate never changes. That is how you recognize them.
Given the line in parametric form $x = -3$, $y = t$, state which axis it is parallel to and write down its general equation.
Given the line in parametric form $x = -3$, $y = t$, state which axis it is parallel to and write down its general equation.
The printhead of a 3D printer must draw a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$-axis. Write down the parametric equations and the…
The printhead of a 3D printer must draw a constant horizontal support line at $15\text{ mm}$ from the origin on the $Y$-axis. Write down the parametric equations and the general equation of this line.
Recognizing them from two points
You don't always get the equation. Sometimes you get two points. Look for the coordinate that repeats.
Drag the point and watch which coordinate stays fixed as you move along each axis.
A line passes through the points $P(4, 2)$ and $Q(-3, 2)$. Is it parallel to either of the axes? Write down its equation.
A line passes through the points $P(4, 2)$ and $Q(-3, 2)$. Is it parallel to either of the axes? Write down its equation.
Spotting them in standard form
A line in the form $Ax + By + C = 0$ (you may also see $Ax + By = C$) is parallel to an axis exactly when one of the two variables disappears. Some problems call this form the general equation.
Determine the value of $k$ so that the line with general equation $kx + 3y - 6 = 0$ is parallel to the $X$-axis. Write down the resulting equation.
Determine the value of $k$ so that the line with general equation $kx + 3y - 6 = 0$ is parallel to the $X$-axis. Write down the resulting equation.
Determine the value of $a$ so that the line with equation $4x + (a+1)y - 8 = 0$ is parallel to the $Y$-axis. Write down the resulting equation.
Determine the value of $a$ so that the line with equation $4x + (a+1)y - 8 = 0$ is parallel to the $Y$-axis. Write down the resulting equation.
Find the value of $m$ so that the line $(m^2-9)x + (m+3)y + 5 = 0$ is parallel to the $X$-axis. Is there any value of $m$ for which this same line is parallel to the…
Find the value of $m$ so that the line $(m^2-9)x + (m+3)y + 5 = 0$ is parallel to the $X$-axis. Is there any value of $m$ for which this same line is parallel to the $Y$-axis? Justify your answer.
Intersections, perpendicular lines, and distance
A horizontal line and a vertical line always cross at a right angle. That makes several questions easy.
Determine the coordinates of the point of intersection of the lines $r: x = 5$ and $s: y = -2$.
Determine the coordinates of the point of intersection of the lines $r: x = 5$ and $s: y = -2$.
Find the equation of the line perpendicular to $x = 7$ that passes through the point $M(-4, 9)$.
Find the equation of the line perpendicular to $x = 7$ that passes through the point $M(-4, 9)$.
Calculate the minimum distance separating the parallel lines $r: y = 8$ and $s: y = -3$.
Calculate the minimum distance separating the parallel lines $r: y = 8$ and $s: y = -3$.
Paths, altitudes, and parameters
Now combine the ideas.
Stepped paths
A path made of one horizontal move and one vertical move lies on two axis-parallel lines.
Altitudes
An altitude of a triangle is the segment from a vertex perpendicular to the opposite side. If that side is horizontal, the altitude is vertical, and vice versa.
Lines with a parameter
When a line has a parameter $k$, split into cases: the values of $k$ that make a coefficient vanish usually behave differently from all the others. Substitute the axis-parallel line into the other equation and see what is left.
Now try a vertical line with a parameter. This time the case $k = 0$ behaves differently.
A robotic arm must move from the point $P(-2, -5)$ to the point $Q(4, 7)$ following a stepped path composed of two segments: first a movement parallel to the $X$-axis…
A robotic arm must move from the point $P(-2, -5)$ to the point $Q(4, 7)$ following a stepped path composed of two segments: first a movement parallel to the $X$-axis and then a movement parallel to the $Y$-axis. Write the equations of the lines that contain these two segments.
Given the triangle with vertices $A(-3, 2)$, $B(5, 2)$ and $C(1, 8)$, find the equation of the altitude from vertex $C$. Which axis is this altitude parallel to?
Given the triangle with vertices $A(-3, 2)$, $B(5, 2)$ and $C(1, 8)$, find the equation of the altitude from vertex $C$. Which axis is this altitude parallel to?
Determine the relative position of the line $r: x = 4$ and the line $s: 2x + ky - 8 = 0$ depending on the parameter $k$.
Determine the relative position of the line $r: x = 4$ and the line $s: 2x + ky - 8 = 0$ depending on the parameter $k$.
Rectangles aligned with the axes
Every rectangle with its sides parallel to the axes is bounded by two horizontal and two vertical lines. This is how courts, screens, and floor plans are described in coordinates.
For the court in the figure, the length of $94$ ft lies along the x-axis and the width of $50$ ft along the y-axis. So the sides are $x = \pm 47$ and $y = \pm 25$.
On a soccer field, we represent the center of the field at the origin $(0,0)$. If the field is $100\text{ m}$ long (located on the $X$-axis) and $60\text{ m}$ wide…
On a soccer field, we represent the center of the field at the origin $(0,0)$. If the field is $100\text{ m}$ long (located on the $X$-axis) and $60\text{ m}$ wide (located on the $Y$-axis), write down the equations of the four straight lines that bound the playing field.
In the development of a video game, the screen has a resolution of $1920 \times 1080$ pixels. If we place the origin $(0,0)$ at the bottom-left corner, write the…
In the development of a video game, the screen has a resolution of $1920 \times 1080$ pixels. If we place the origin $(0,0)$ at the bottom-left corner, write the equations of the two lines (one horizontal and one vertical) that divide the screen exactly into four equal quadrants.
A square has its center at the point $C(3, 3)$ and its sides are parallel to the coordinate axes. If the side of the square measures $4\text{ u}$, write the equations of…
A square has its center at the point $C(3, 3)$ and its sides are parallel to the coordinate axes. If the side of the square measures $4\text{ u}$, write the equations of the four lines that contain its sides.
Constants in real situations
When a quantity stays the same no matter what the other one does, its graph is a horizontal line. When a position is fixed while the other coordinate varies, its graph is a vertical line. In a time graph, time goes on the x-axis, so a constant quantity gives $y = k$ with direction vector $\langle 1, 0\rangle$.
If a problem gives a temperature or other quantity, use the number exactly as stated and label the axis with it. For example, an oven held at $450^\circ$F for all time $t$ gives $T = 450$, a horizontal line with direction vector $\langle 1, 0\rangle$.
In a chemistry experiment, the temperature $T$ (in degrees Fahrenheit) is kept constant at $35^\circ\text{C}$ over time $t$ (in minutes). Representing time on the…
In a chemistry experiment, the temperature $T$ (in degrees Fahrenheit) is kept constant at $35^\circ\text{C}$ over time $t$ (in minutes). Representing time on the $X$-axis and temperature on the $Y$-axis, write down the equation of the line and its direction vector.
A drip irrigation system is installed along a pipe parallel to the $Y$-axis. We know that the pipe must water a tree located at the point $A(5, 12)$ and that the exact…
A drip irrigation system is installed along a pipe parallel to the $Y$-axis. We know that the pipe must water a tree located at the point $A(5, 12)$ and that the exact distance from the pipe to the $Y$-axis is $5\text{ m}$. Write down the equation of the line representing the pipe.
An electronic components factory has fixed monthly costs of $12000\text{ \$}$, regardless of the quantity $x$ of parts manufactured. The revenue is given by the line $y…
An electronic components factory has fixed monthly costs of $12000\text{ \$}$, regardless of the quantity $x$ of parts manufactured. The revenue is given by the line $y = 30x$. Write the equation of the fixed cost line on the $(x, y)$ plane and find the exact point where the revenue equals the fixed costs.
Summary
Back to the street grid: Maple Street is a horizontal line such as $y = 3$ and 5th Avenue is a vertical line such as $x = 5$. They are perpendicular and meet at exactly one corner, $(5, 3)$.