Linear Relations

Ontario 10 Applied (MFM2P)

Graph straight lines

Lesson

To graph any liner relationship you only need two points that are on the line. You can use any two points from a table of values, or substitute in any two values of $x$`x` into the equation and solve for corresponding $y$`y`-value to create your own two points. Often, using the intercepts is one of the easiest ways to sketch the line.

x | 1 | 2 | 3 | 4 |

y | 3 | 5 | 7 | 9 |

To sketch from a table of values, we need just any two points from the table. From this table we have 4 coordinates, $\left(1,3\right)$(1,3), $\left(2,5\right)$(2,5), $\left(3,7\right)$(3,7), $\left(4,9\right)$(4,9).

Drag the $2$2 of the points on this interactive to the correct positions and graph this linear relationship.

If we are given the equation of a linear relationship, like $y=3x+5$`y`=3`x`+5, then to sketch it we need two points. We can pick any two points we like.

Start by picking any two $x$`x`-values you like, often the $x$`x`-value of $0$0 is a good one to pick because the calculation for y can be quite simple. For our example, $y=3x+5$`y`=3`x`+5 becomes $y=0+5$`y`=0+5, $y=5$`y`=5. This gives us the point $\left(0,5\right)$(0,5)

Similarly look for other easy values to calculate such as $1$1, $10$10, $2$2. I'll pick $x=1$`x`=1. Then for $y=3x+5$`y`=3`x`+5, we have $y=3\times1+5$`y`=3×1+5, $y=8$`y`=8.This gives us the point $\left(1,8\right)$(1,8)

Now we plot the two points and create a line.

The general form of a line is great for identifying both the x and y intercepts easily.

For example, the line $3y+2x-6=0$3`y`+2`x`−6=0

The x intercept happens when the $y$y value is $0$0. $3y+2x-6=0$3y+2x−6=0 $0+2x-6=0$0+2x−6=0 $2x=6$2x=6 $x=3$x=3 |
The y intercept happens when the $x$x value is $0$0. $3y+2x-6=0$3y+2x−6=0 $3y+0-6=0$3y+0−6=0 $3y=6$3y=6 $y=2$y=2 |

From here it is pretty easy to sketch, we find the $x$`x` intercept $3$3, and the $y$`y` intercept $2$2, and draw the line through both.

Start by plotting the single point that you are given.

Remembering that slope is a measure of change in the rise per change in run, we can step out one measure of the slope from the original point given.

For a slope of $4$4 $1$1 unit across and $4$4 units up. | For a slope of $-3$−3 $1$1 unit across and $3$3 units down. | For a slope of $\frac{1}{2}$12 $1$1 unit across and $\frac{1}{2}$12 unit up. |

The point can be any point $\left(x,y\right)$(`x`,`y`), or it could be an intercept. Either way, **plot **the point, **step **out the slope and **draw **your line!

For example, plot the line with slope $-2$−2 and has $y$`y` intercept of $4$4.

Start with the point, ($y$y intercept of $4$4) |
Step out the slope, (-$2$2 means $2$2 units down) |
Draw the line |

To sketch linear graphs, it's easiest to substitute in values to find coordinates to put it in slope-intercept form.

The Slope-Intercept Form

$y=mx+b$`y`=`m``x`+`b`

where $m$`m` is the slope and $b$`b` is the $y$`y`-intercept

Our graphs may not always be in this form so we may need to rearrange the equation to make $y$`y` the subject (that means $y$`y` is on one side of the equation and everything else is on the other side).

Sometimes, it doesn't matter. We can sketch a straight line on a graph just by knowing a couple of its features such as a point that lies on the line and it's slope. At other times, we may need to generate an equation before we sketch it. So other than the slope-intercept form, we can use:

- Slope-point formula: $y-y_1=m\left(x-x_1\right)$
`y`−`y`1=`m`(`x`−`x`1)

- Two point formula: $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$
`y`−`y`1`x`−`x`1=`y`2−`y`1`x`2−`x`1

Ok let's look at this in action with some examples.

Plot the graph of the line whose slope is $-3$−3 and passes through the point $\left(-2,4\right)$(−2,4).

- Loading Graph...

Graph the linear equation $y=3x-1$`y`=3`x`−1 using the point $Y$`Y` as the $y$`y`-intercept.

- Loading Graph...

Graph the linear equation $-6x+3y+24=0$−6`x`+3`y`+24=0 by finding any two points on the line.

- Loading Graph...

On horizontal lines, the $y$`y` value is always the same for every point on the line.

On vertical lines, the $x$`x` value is always the same for every point on the line.

Graph the line $y=-3$`y`=−3.

- Loading Graph...

Graph lines by hand, using a variety of techniques