The Graphs of Horizontal and Vertical Lines

The Graphs of Horizontal and Vertical Lines exercise

• Identify the characteristics of horizontal lines.

  Hints

  Keep the slope tree in mind. It helps you to remember the slope of different lines.

  In general a linear equation is given by $y=mx+b$, where $m$ is the slope and $b$ the $y$-intercept.

  Solution

  Here you see the slope tree. The part of the tree which is parallel to the bottom of the tree has a slope of zero.

  Any linear equation like $y=c$, where $c$ is constant, is parallel to the bottom of the tree and thus has a slope of zero. So, the horizontal line corresponding to $y=3$, is parallel to the bottom of the tree, or the $x$-axis, and thus has a slope of zero.

  Let us check this using the slope formula for two given points on the line, for example $(1,3)$ and $(4,3)$. You subtract the $y$-coordinates and divide the resulting difference by the difference of the $x$-coordinates in the appropriate order:

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{3-3}{4-1}=\frac03=0$.

  This line has some further properties:

  • The $y$-intercept is $(0,3)$. The line has no $x$-intercept.
  • The $x$-values change while the $y$-values stay constant ($y=3$).
  • The line is perpendicular to the $y$-axis.

• Describe vertical lines.

  Hints

  Keep the slope formula in mind: $m=\frac{y_2-y_1}{x_2-x_1}$, and remember that you can't divide by zero!

  Find the part of the slope tree which represents a vertical line.

  Solution

  Here you see the slope tree. A line parallel to the trunk of the tree has an undefined slope. Any linear equation like $x=c$, where $c$ is constant, is parallel to the trunk of the tree and thus has an undefined slope. The horizontal line $x=2$ in particular is parallel to the trunk and thus has an undefined slope.

  This line has some further properties:

  • The $x$-intercept is $(2,0)$. The line has no $y$-intercept.
  • The $x$-values stay constant ($x=2$) while the $y$-values change.
  • The line is parallel to the $y$-axis.

  Let's use the slope formula for two given points of the line, for example $(2,0)$ and $(2,3)$, to confirm that the slope of $x=2$ is undefined. You subtract the $y$-coordinates and divide the resulting difference by the difference of the $x$-coordinates in the appropriate order:

  $m=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{2-2}=\frac30$.

  We cannot divide by zero because, if we could, then we could multiply both sides of the equation $m=\frac{3}{0}$ by zero to get that $0m=3$, which is not true! As any number times zero is zero. So our slope is undefined.

• Identify the corresponding equation to the graphed line.

  Hints

  The equation for any horizontal line, or line parallel to the $x$-axis, is given by $y=c$, where $c$ is a constant.

  The equation of any vertical line, or line parallel to the $y$-axis, is given by $x=c$, where $c$ is a constant.

  Solution

  Remember:

  • A horizontal line (i.e. parallel to the $x$-axis) has the equation $y=c$, where $c$ is a constant.
  • A vertical line (i.e. parallel to the $y$-axis) has the equation $x=c$, where $c$ is a constant.

  Let's start with the horizontal lines:

  • The red one intersects the $y$-axis at $y=2$. This is already the desired equation.
  • The orange one intersects the $y$-axis at $y=1$.
  • The green one intersects the $y$-axis at $y=4$.

  Finally there are still two vertical lines:

  • The violet one intersects the $x$-axis at $x=3$. That's the desired equation.
  • The blue one intersects the $x$-axis at $x=1$.

• Decide if the line is horizontal or vertical.

  Hints

  In general, a linear equation is given by $y=mx+b$, where $m$ is the slope and $b$ the $y$-intercept.

  For any slope $m\neq 0$ the line is neither horizontal or vertical.

  Determine the $y$- or $x$-intercept.

  A horizontal (vertical) line has no $x$- ($y$-) intercept.

  • $y=-2$ is a horizontal line.
  • $x=-2$ is a vertical line.
  • $y=-2x-2$ is a line which is neither horizontal or vertical.

  Solution

  Keep the following in mind:

  • The equation of any horizontal line is given by $y=c$, where $c$ is a constant.
  • The equation of any vertical line is given by $x=c$, where $c$ is a constant.

  The following equations correspond to horizontal lines:

  • $y=0$
  • $y=7$

  Vertical lines are given by the following equations:

  • $x=-2$
  • $x=7$

  The two equations remaining are lines which are neither horizontal or vertical:

  • $y=x+1$
  • $y=7x+7$

• Identify the type of line and their corresponding equations.

  Hints

  Here you see the line given by $y=\frac32x$.

  You can see that the line is neither horizontal or vertical.

  A horizontal line is parallel to the $x$-axis.

  A vertical line is parallel to the $y$-axis.

  The equation of any horizontal line is given by $y=c$ with a constant $c$.

  The equation of any vertical line is given by $x=c$ with a constant $c$.

  Solution

  In the picture you see three lines:

  • The line corresponding to $y=1$ is parallel to the $x$-axis. This line is horizontal.
  • The line corresponding to $x=4$ is parallel to the $y$-axis. This line is vertical.
  • The last line according to $y=-x+2$ passes the $y$-axis in $(0,2)$ and the $x$-axis in $(2,0)$. This line is neither horizontal or vertical.

• Calculate the slope of the line given two points and identify what type of line it is.

  Hints

  Use the slope formula for two given points $(x_1,y_1)$ and $(x_2,y_2)$:

  $m=\frac{y_2-y_1}{x_2-x_1}$.

  Pay attention to the consistent order of subtraction.

  The slope tree represents a vertical line.

  In general a linear equation is given by $y=mx+b$, where $m$ is the slope and $b$ the $y$-intercept.

  • A slope $m=0$ indicates a horizontal line.
  • A slope $m\neq 0$ indicates a line which is neither horizontal or vertical.
  • An undefined slope indicates a vertical line.

  Solution

  To determine if a line corresponding to given set of points on a line is horizontal, vertical, or neither, we determine the slope of the line which the set of points defines. For this we use the slope formula:

  $m=\frac{y_2-y_1}{x_2-x_1}$.

  If the denominator is equal to zero the slope is undefined.

  For $\mathbf{(3,3)}$ and $\mathbf{(4,3)}$, we have that

  $m=\frac{3-3}{4-3}=\frac01=0$.

  So we can conclude that the corresponding line is horizontal.

  For $\mathbf{(3,3)}$ and $\mathbf{(3,4)}$, we have that

  $m=\frac{4-3}{3-3}=\frac10$.

  Division by zero is not defined, so the slope is undefined and the corresponding line must be vertical.

  For $\mathbf{(3,3)}$ and $\mathbf{(4,4)}$, we have

  $m=\frac{4-3}{4-3}=\frac11=1$.

  This time we get a non-zero slope. Thus the line is neither horizontal or vertical.

  With $\mathbf{(0,0)}$ and $\mathbf{(1,3)}$, we get the slope,

  $m=\frac{3-0}{1-0}=\frac31=3$.

  Once again we get a non-zero slope and thus also a line which is neither horizontal or vertical.