Ontario 09 Academic (MPM1D)
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Horizontal and vertical lines
Lesson

Straight lines on the $xy$xy-plane can actually be in any direction and pass through any two points. This means that straight lines can be:

   
decreasing     horizontal
       
       
   
increasing     vertical

 

Horizontal and vertical lines

We can quickly identify that a line is horizontal if it is parallel to the $x$x-axis. Similarly, a line is vertical if it's parallel to the $y$y-axis.

 

Horizontal lines

We can draw a set of points that makes a horizontal line in the same way we can draw any set of points as before. Consider the following table of values:

$x$x $-5$5 $-4$4 $-3$3 $-2$2
$y$y $2$2 $2$2 $2$2 $2$2

Plotting each column as points on the $xy$xy-plane gives us the following:

Plotting points from the table of values

 

Clearly the set of points form a line that is parallel to the $x$x-axis. But just to confirm, we can draw a line through these points to show this.

Horizontal line passing through all four points

 

It's also clear from the graph that the distance between any two neighbouring points is exactly $1$1 unit apart. We obtain the distance by referencing the $x$x-axis, and subtracting the $x$x-values of the two points. This means we don't have to plot the points to determine the distance between a pair of points. Consider the table of values earlier.

$x$x $-5$5 $-4$4 $-3$3 $-2$2
$y$y $2$2 $2$2 $2$2 $2$2

The distance between the points $\left(-4,2\right)$(4,2) and $\left(-3,2\right)$(3,2) is:

$-3-\left(-4\right)=1$3(4)=1 unit.

Careful!

You may find yourself with a negative distance depending on what order you subtract the two $x$x-values. Make sure to subtract in the correct order or change your final answer to a positive number.

 

Vertical lines

Just as we did before, we can plot a set of points obtained from a table of values and show that the points fall on a vertical line. Consider the following table of values:

$x$x $2$2 $2$2 $2$2 $2$2
$y$y $-3$3 $-1$1 $1$1 $3$3

Plotting the points in the table of values gives us the following graph. We can also pass a line through the points to show that the points lie on a vertical line.

Plotting points from the table of values

 

By referencing the $y$y-axis we can see that the distance between a pair of neighbouring points is $2$2 units. Alternatively we can refer to the ordered pairs $\left(2,-1\right)$(2,1) and $\left(-1,-3\right)$(1,3) and subtract the two $y$y-values:

$-1-\left(-3\right)=2$1(3)=2 units

We still need to be careful about the order we subtract the two $y$y-values to ensure we obtain a positive distance (or you could change your final answer so that it's positive).

Let's have a look at some examples of questions that might arise when dealing with horizontal or vertical lines.

Examples

Question 1

Plot the points in the table of values.

$x$x $2$2 $3$3 $4$4 $5$5
$y$y $5$5 $5$5 $5$5 $5$5
  1. Loading Graph...

 

Question 2

Consider the points in the plane below.

Which of the following statements is true?

Loading Graph...

  1. The set of points lie on a decreasing line.

    A

    The set of points lie on an increasing line.

    B

    The set of points lie on a vertical line.

    C

    The set of points lie on a horizontal line.

    D

    The set of points lie on a decreasing line.

    A

    The set of points lie on an increasing line.

    B

    The set of points lie on a vertical line.

    C

    The set of points lie on a horizontal line.

    D

 

Question 3

What is the shortest distance between any two of the following points?

Loading Graph...

 

Outcomes

9D.AG1.02

Identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b

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