Quarry School

Horizontal and vertical lines

Explain it like I am five

Picture a shelf and a flagpole. Every place on the shelf has the same height, even when you move across it. A horizontal line does that: y stays fixed while x can vary. Every place on the flagpole has the same across position, even when you move up it. A vertical line does that: x stays fixed while y can vary. These descriptions tell you which letter belongs in each equation. A horizontal line is a function because each input gives one height. A vertical line gives several heights at the same input, so it is a line but not a function of x.

HOY: Horizontal, 0 slope, Y equation
VUX: Vertical, Undefined slope, X equation
The memory device follows the coordinate that stays fixed.
Reminder
  • Coordinate order. In (7, 5), x = 7 and y = 5.
  • Subtracting a negative. 5 − (−3) = 8 is the horizontal distance between those inputs.
  • Zero in a fraction. 08 = 0, while 80 is undefined.
  • Function requirement. One input may give one output: input 7 cannot give both 0 and 5 in a function.
  • Slope formula. Slope = changeinychangeinx; a rise of 0 over a run of 8 gives 0.
Why it works. For a horizontal line through two distinct points, the height change is 0 and the across change is nonzero, so slope = 0 divided by a nonzero number = 0. For a vertical line, the across change is 0 while the height changes. Its slope would divide by 0, which has no value. It also breaks the function requirement of one output per input. The vertical line test records that failure: a vertical probe meets more than one point.
RuleRule: A horizontal line has equation y = c and slope 0. A vertical line has equation x = c, undefined slope, and is not a function of x.
The same idea, five ways
Say it

Say: horizontal keeps height; vertical keeps across position.

Write it

A constant coordinate identifies a horizontal or vertical line.

In math
  • horizontal: y = c
  • vertical: x = c
  • horizontal: m = 0
  • vertical: slope undefined
Like

A shelf holds one height; a flagpole holds one across position.

See it
Horizontal: y = −4, points (−3, −4) and (5, −4)
Vertical: x = 7, points (7, 0) and (7, 5)
Read the coordinate that stays the same.
The same idea, other ways
By movement

Walking horizontally changes only x. Walking vertically changes only y. The other coordinate must stay fixed.

square corner
Horizontal and vertical directions form a square corner.
By a rule

The equation y = −4 assigns height −4 to every input. The equation x = 7 permits many heights at one input, so no single output rule can describe it as y = f(x).

705not a function
Input 7 points to more than one output, which fails the function requirement.
LineEquationSlope
Horizontal through (−3, 2) and (5, 2)y = 20
Horizontal through (−3, −4) and (5, −4)y = −40
Vertical through (7, 0) and (7, 5)x = 7undefined
.1Horizontal line

A horizontal line is like a level shelf. Its height is unchanged no matter how far you move left or right. A constant function gives that same output for every input. You can write it as f(x) = c or y = c.

  • Rule: y = c = 0x + c has slope 0.
  • Rule: Its y-intercept is (0, c). If c ≠ 0, it has no x-intercept.
  • Rule: y = 0 is the x-axis itself, so every point on it has output 0.
−4−2246−6−4−22(0, −4)
Every point on this horizontal line has output −4; it never reaches output 0.
Reminder
  • Zero multiplication. 0 × 7 − 4 = −4, which is why a constant function ignores the input.
The same idea, five ways
Say it

Say: every input gets the same height.

Write it

A horizontal line is the graph of a constant function.

In math
  • y = c
  • f(x) = c
  • m = 0
Like

Every spot on a level shelf is the same height.

See it
−4−2246−6−4−22(0, −4)(−3, −4)(5, −4)
Every point on this horizontal line has output −4; it never reaches output 0.
Worked exampleA level line through two points

You need the line through (−3, −4) and (5, −4).

−4−2246−6−4−22(0, −4)firstsecond
Every point on this horizontal line has output −4; it never reaches output 0.
  1. Both heights are −4, so write y = −4.The output stays constant while the input changes.
  2. Compute m = −4−(−4)5−(−3) = 08 = 0.There is no vertical change over the nonzero horizontal change.
  3. Check each point: its second coordinate is −4.The equation describes both original addresses.
Answer
  • y = −4.
  • Slope: 0.
Check Substituting either input in y = 0x − 4 gives −4, verifying the constant function.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: a horizontal line has undefined slope because it does not rise.
A zero numerator is allowed when the denominator is nonzero.
✓ Instead: Its slope is 0: 08 = 0.
Tips and tricks
  • Tip: Memory device HOY: Horizontal, slope 0, equation uses Y. A shelf's fixed height explains the Y.
.2Vertical line

A vertical line is like a flagpole planted at one across position. Heights change but the across position never does. Its equation fixes x. Calling its slope undefined means there is no slope number, not a very large number and not zero.

  • Rule: x = c has undefined slope because the run is 0.
  • Rule: A vertical line fails the vertical line test and is not a function of x.
  • Rule: x = c meets the x-axis at (c, 0). If c ≠ 0, it never meets the y-axis; x = 0 is the y-axis itself.
Vertical line: x = 7
Run between any two points: 0
A zero run prevents a defined slope.
Reminder
  • Function requirement. A rule assigning both 0 and 5 to input 7 is not a function.
  • Division by zero. 50 has no value because no number multiplied by 0 makes 5.
The same idea, five ways
Say it

Say: every height uses the same across position.

Write it

A vertical line fixes the input and permits many outputs.

In math
  • x = c
  • run = 0
  • nonzerorise0 is undefined
Like

Different places up a flagpole are all above its same base.

See it
x = 7
(7, 0), (7, 5)
rise = 5, run = 0
slope: undefined
The fixed first coordinate identifies the vertical line.
Worked exampleAn upright line through two points

You need the line through (7, 0) and (7, 5).

x = 7
(7, 0) and (7, 5) both fit
Same input, different outputs
The point addresses show why this is an upright line.
  1. Both input coordinates are 7, so write x = 7.The across position stays fixed while the heights differ.
  2. Compute rise 5 − 0 = 5 and run 7 − 7 = 0. The slope is undefined.The slope calculation would divide by zero.
  3. Check both points: each has x = 7.They both lie on the equation's fixed across position.
  4. On the x-axis, y = 0. Keeping x = 7 gives its x-intercept point (7, 0).The point's zero height puts it on the x-axis, and the original equation keeps its input 7.
  5. On the y-axis, x must be 0. That would require 0 = 7 in this equation, which is false, so this line has no y-intercept.The fixed input 7 cannot also be the y-axis input 0.
Answer
  • x = 7.
  • Slope: undefined.
  • It is not a function of x.
  • x-intercept point: (7, 0).
  • No y-intercept.
Check The different outputs 0 and 5 at input 7 confirm the function failure independently of the slope calculation.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: the line through (7, 0) and (7, 5) is y = 7.
Neither point has height 7. The shared number 7 is the first coordinate.
✓ Instead: Write x = 7.
✗ Not this: The vertical line x = 0 has only one point on the y-axis.
Its equation describes the entire y-axis. Both (0, 1) and (0, −2), along with every (0, y), satisfy it.
✓ Instead: Every point of x = 0 lies on the y-axis; it has no unique crossing with that same axis.
Tips and tricks
  • Tip: Memory device VUX: Vertical, Undefined slope, equation uses X. The flagpole fixes the across position.
Strategy: step by step
  1. 1. Compare the coordinates of two distinct points.
  2. 2. If their y-values match, keep that output fixed: y = c.
  3. 3. If their x-values match, keep that input fixed: x = c.
  4. 4. Check both given points in the equation.
  5. 5. Do not force a vertical line into y = mx + b.
Strategy
Strategy: identify the fixed coordinate
1
Do the points have equal y-coordinates?
YesWrite y = that common height; slope is 0.
NoCheck the x-coordinates.
↓
2
Do the points have equal x-coordinates?
YesWrite x = that common input; slope is undefined.
NoUse the ordinary two-point slope method.
  1. Read the coordinates of at least two distinct points.
  2. Keep the repeated coordinate letter and value in the equation.
  3. Compute rise and run if a slope is requested.
  4. Use the function requirement to decide whether it can be y = f(x).
Worked exampleKeep the correct coordinate fixed

You need equations for the horizontal line through (−3, −4) and (5, −4), and the vertical line through (7, 0) and (7, 5).

Horizontal: y = −4
Vertical: x = 7
The unchanged coordinate goes in the equation.
  1. For the horizontal pair, both outputs are −4: y = −4.The second coordinate stays fixed, as on a shelf.
  2. Its slope is −4−(−4)5−(−3) = 08 = 0.Zero rise over a nonzero run is zero.
  3. For the vertical pair, both inputs are 7: x = 7.The first coordinate stays fixed, as on a flagpole.
  4. Its slope would be 5−07−7 = 50, so it is undefined.A zero denominator has no division value.
  5. Check the pairs in their equations: the first pair has y = −4, and the second pair has x = 7.Each equation must include both given points.
Answer
  • Horizontal equation: y = −4.
  • Horizontal slope: 0.
  • Vertical equation: x = 7.
  • Vertical slope: undefined.
  • The vertical line is not a function of x.
Check The horizontal pair has two different inputs with one output each; the vertical pair gives two outputs to input 7. That checks the function distinction.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: any straight line is the graph of a function of x.
A vertical line contains many points with the same input and different outputs.
✓ Instead: Every nonvertical straight line is a graph y = mx + b. Vertical lines require x = c.
Tips and tricks
  • Tip: Use HOY and VUX, then check a supplied point to make sure the remembered letter fits.
  • Tip: The y-axis x = 0 does not have one special crossing with itself. Every point on it is already on the y-axis.
Trap. Writing y = 7 for the vertical line. Say what stays fixed before choosing the letter. Fixed across position means x; fixed height means y.