Cartesian coordinates use a horizontal x-axis and vertical y-axis intersecting at (0,0) to locate points on a graph. The point (x,y) is located x units from the y-axis along the x-axis and y units from the x-axis along the y-axis. The slope or gradient of a line is calculated by rise/run or the change in y over the change in x. The equation of a line is y=mx+b, where m is the slope and b is the y-intercept. Perpendicular lines intersect at 90 degrees, while parallel lines never intersect and remain the same distance apart.

1. Cartesian Coordinates
Cartesian coordinates can be used to pinpoint where you are on a map or
graph.
Cartesian Coordinates
Using Cartesian Coordinates you mark a point on a graph by how far
along and how far up it is:
The point (12,5) is 12 units along, and 5 units up.
X and Y Axis
The left-right (horizontal) direction is commonly
called X.
The up-down (vertical) direction is commonly
called Y.

2. Put them together on a graph ...
... and you are ready to go
Where they cross over is the "0" point,
you measure everything from there.


The X Axis runs horizontally through zero
The Y Axis runs vertically through zero
Axis: The reference line from which distances are measured.
The plural of Axis is Axes, and is pronounced ax-eez
Example:
Point (6,4) is
6 units across (in the x direction), and
4 units up (in the y direction)
So (6,4) means:
Go along 6 and then go up 4 then "plot the dot".

3. Gradient (Slope) of a Straight Line
The Gradient (also called Slope) of a straight line shows how steep a straight
line is.
Calculate
The method to calculate the Gradient is:
Divide the change in height by the change in horizontal distance
Gradient
=
Change in
Y
Change in
X
Examples:
3
The Gradient of this line =
=
3
1
So the Gradient is equal to 1

4. 4
Gradient =
2
=
2
(The line is steeper, and so the Gradient is larger)
3
Gradient =
5
=
0.6
(The line is less steep, and so the Gradient is smaller)
Positive or Negative?
Important:

Starting from the left end of the line and going across to the
right is positive
(but going across to the left is negative).

Up is positive, and down is negative

5. -4
Gradient =
= –2
2
That line goes down as you move along, so it has a negative Gradient.
Straight Across
0
Gradient =
=
5
0
A line that goes straight across (Horizontal) has a Gradient of zero.
Straight Up and Down
Gradient =
3
0
=
undefined
That last one is a bit tricky ... you can't divide by zero,
so a "straight up and down" (Vertical) line's Gradient is "undefined".

6. Rise and Run
Sometimes the horizontal change is called "run", and the vertical change is
called "rise" or "fall":
They are just different words, none of the calculations change.
Equation of a Straight Line
The equation of a straight line is usually written this way:
y = mx + b
(or "y = mx + c" in the UK see below)

7. What does it stand for?
Slope (or Gradient)
y = how far up
x = how far along
m = Slope or Gradient (how steep the line is)
b = the Y Intercept (where the line crosses the Y axis)
How do you find "m" and "b"?


b is easy: just see where the line crosses the Y axis.
m (the Slope) needs some calculation:
Change in Y
m =
Change in X
Y Intercept

8. Knowing this we can work out the equation of a straight line:
Example 1
2
m =
= 2
1
b=1
(where the line crosses the Y-Axis)
So: y = 2x + 1
With that equation you can now ...
... choose any value for
For example, when
x and find the matching value for y
x is 1:
y = 2×1 + 1 = 3
Check for yourself that
x=1 and y=3 is actually on the line.
Or we could choose another value for
x, such as 7:
y = 2×7 + 1 = 15

9. And so when
x=7 you will have y=15
Example 2
3
m =
= –3
-1
b=0
This gives us y = –3x + 0
We do not need the zero!
So: y = –3x

10. Example 3: Vertical Line
What is the equation for a vertical line?
The slope is undefined ... and where does it cross the Y-Axis?
In fact, this is a special case, and you use a different equation, not "y=...",
but instead you use "x=...".
Like this:
x = 1.5
Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5
Rise and Run
Sometimes the words "rise" and "run" are
used.


Rise is how far up
Run is how far along
rise
And so the slope "m" is:
You might find that easier to
remember
m =
Run

11. Perpendicular and Parallel
Perpendicular
It just means at right angles (90°) to.
The red line is perpendicular to the blue line in both these cases:
(The little box drawn in the corner, means "at right angles", so we didn't really
need to also show that it was 90°, but we just wanted to!)
Parallel
Lines are parallel if they are always the same distance apart (called
"equidistant"), and will never meet. (They also point in the same direction). Just
remember: Always the same distance apart and never touching.
The red line is parallel to the blue line in both these cases:
Example 1
Example 2

12. Perpendicular to Parallel
Question: What is the difference between perpendicular and parallel?
Answer: 90 degrees (a right angle)
That's right, if you rotate a perpendicular line by 90° it will become parallel (but
not if it touches!), and the other way around.
Perpendicular ...
Rotate One Line 90°
... Parallel !
Parallel Curves
Curves can also be parallel when they are always the same distance apart
(called "equidistant"), and never meet. Just like railroad tracks.The red curve is
parallel to the blue curve in both these cases:

13. Parallel Surfaces
Surfaces can also be parallel, so long as the rule
still holds: always the same distance apart and never touching.