# Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

## Coordinate/Cartesian Plane

Most of the two dimensional graphing we do is done on a coordinate plane (often called the Cartesian plane) using a (rectangular/Cartesian) coordinate systemA coordinate plane consists of a pair of perpendicular coordinate lines, called coordinate axes (often labelled as the x and y axes, for the horizontal and vertical axes, respectively), which are placed so that they intersect at the origin

Note: While the labeling of axes with letters x and y is a common convention, any two letters or other labelling may be used. If the letter x and y are used to label the coordinate axes, then the resulting plane is often called the xy-plane.

## Ordered Pairs

An ordered pair of real numbers are two real numbers in an assigned order. Every point on a coordinate plane can be associated with a unique ordered pair of real numbers, usually $$(x,y)$$, also called coordinates. The first value, $$x$$, is commonly called the x-coordinate, and the second value, $$y$$ is called the y-coordinate.

To plot a point associated with a coordinate on the Cartesian plane, start at the origin which has the coordinates (0,0). Then move x units right or left, depending on whether the x-coordinate is positive or negative, and move y units up or down, depending on whether the y-coordinate is positive or negative.

We can label points by writing its "name" before the coordinates, for example the point $$P$$ at the coordinates $$(0,3)$$ can be denoted as $$P(0,3)$$. The following diagram demonstrates what some points look like on the Cartesian/xy-plane.

Plotting multiple points associated with ordered pairs creates the foundation of graphing using a (rectangular/Cartesian) coordinate system.

Tip: Make sure to double check the order in which you write coordinates! The x-coordinate always comes before the y-coordinate. For example, the coordinates $$(3,4)$$ and $$(4,3)$$ correspond to two different points! (Exercise: Plot these two points to see the difference.)

Intercepts

Points where a graph intersects the horizontal or vertical axes (i.e. the point of a graph that is on the x-axis or y-axis) are called intercepts. If a point intersects the horizontal or x-axis, it is called the x-intercept. Likewise, if a point intersects the vertical or y-axis, it is called the y-intercept.

In a rectangular coordinate system, the coordinate axes divide the plane into four regions called quadrants. These are numbered counter clockwise with roman numerals as shown:

• In quadrant I, the x values and y values are both positive.
• In quadrant II, the x values are negative and y values are positive.
• In quadrant III, the x values and y values are both negative.
• In quadrant IV, the x values are positive and y values are negative.

## Graphing Definitions

The graph of an equation in two variables x and y is the set of points in the xy-plane whose coordinates are members of the solution set of that equation. In other words, a graph of an equation is made up of all the points (i.e. x and y values) that satisfy that equation. In the Cartesian system, each point on the graph is defined by a pair of numbers, which are written as an ordered pair in the form $$(x,y)$$. The x value comes from the horizontal axis and the y value comes from the vertical axis.

Graphing is useful because a graph can often provide information about the relationship between the independent and dependent variables (usually denoted x and y). For example, given a linear function, we can sketch its graph to find that there's a linear relationship between the variables, i.e. the graph is a line (hence its name linear). See more on graphing linear functions on the Graphing Linear Functions page.

## Examples

Example: Find the coordinates of point $$q$$ and express it as an ordered pair.

Solution:

First, we determine the x-coordinate by following a vertical straight line down from point $$q$$ to the x-axis. In this example, the x-coordinate is 3.

Then, determine the y-coordinate by following a horizontal straight line left from point $$q$$ to the y-axis. In this example, the y-coordinate is 2.

Finally, we can write the coordinates in the form of an ordered pair since we now know the x-coordinate is 3 and y-coordinate is 2. So, the coordinates of the point $$q$$ is $$(3,2)$$.

Example: Determine the quadrant that $$M(3,-4)$$ is located in by:
 a) First plotting the point on the Cartesian plane, then determining the quadrant it's located in. b) Analyzing the x and y coordinate values (without plotting the point).

Solution:

a) First we plot $$M$$ on the Cartesian plane. We do so by drawing a vertical line from the x-axis at $$x=3$$ and horizontal line from the y-axis at $$y=-4$$. Then we plot the point $$M(3,-4)$$ where these two lines intersect:

$$\Rightarrow$$        $$\Rightarrow$$

Finally, now we can see that the point $$M(3,-4)$$ is in quadrant IV.

b) We take a look at the x and y coordinates to determine which quadrant point $$M$$ lies in. Since the x-coordinate is positive ($$x=3$$) and the y-coordinate is negative ($$y=-4$$), we have that the point $$M(3,-4)$$ is in quadrant IV.