03Alineargraphs

= Plotting Linear Graphs = toc

Cartesian Plane
The ** cartesian plane ** has two axes (x-axis and y-axis) meeting at the ** origin ** (0, 0)
 * the x-axis is always horizontal with negative numbers going out to the left and positive numbers going out to the right
 * the y-axis is always vertical with negative numbers going down and positive numbers going up

When drawing a set of axes, you should always:
 * put arrows on the end of each axis (to indicate that they keep going to infinity)
 * label each axis with **X** and **Y**

A ** point ** (or location) is specified by a pair of numbers (x, y) called ** co-ordinates **.
 * They are __always__ in the order (x-coodinate, y-coordinate).
 * The x-coordinate tells how far the point is in the x direction -- to the left or right of the origin.
 * The y-coordinate tells how far the point is in the y direction -- up or down from the origin.

For example, the point (–2, 3) is 2 to the left and 3 up from the origin.

For an interactive exploration of coordinates, go here: MathsIsFun

Quadrants
The axes divide the cartesian plane into four ** quadrants ** (four quarters).
 * Quadrant 1 is the top left quarter (where both x and y are positive)
 * The quadrants are numbered anticlockwise around from quadrant 1

Standard Form
The standard form of a linear function is **y = mx + c**
 * **m** is a constant representing the **gradient** of the line
 * **c** is a constant representing the **y-intercept** {where the line cuts through the y-axis}
 * Any equation that produces a linear graph can be written in standard form.

For example, **y = 2x – 1** has: The ** gradient ** is a measure of how steep the line is.
 * gradient = 2
 * y-intercept = –1
 * A gradient of 0 means the line is horizontal
 * The larger the value of the gradient, the steeper the line is.
 * A __positive__ gradient means the line __slopes up__ as it goes from left to right
 * A __negative__ gradient means the line __slopes down__ as it goes from left to right
 * The gradient can be interpreted as the distance the line goes up (or down) for each step of 1 to the right.

Plotting a Graph
An equation such as ** y = 2x – 1 ** describes an infinite set of points because x can be __any__ real number.

If we choose any particular x-value, the equation tells us the y-value that goes with it. (in this case, the y-value is 2 times the x-value then minus 1)

If we plot all of those infinite points on the cartesian axes, they would all appear to join up and form a single continuous straight line. Fortunately, we only need to plot a few points, then we can rule a straight line to indicate where the rest of those points would be.

When starting off, it is common to select and plot a small number of points (between 3 and 7) so the student can be satisfied that the points will, in fact, produce a straight line.

If we __know__ the equation produces a straight line, we only actually __need__ two points to produce the line.

** Example 1 **
Use the rule, ** y = 2x – 1 ** to fill out a table of values for x = {–1, 0, 1, 2, 3}

Then use those values to plot the graph of ** y = 2x – 1 **.

__**Solution:**__

When x = –1, the rule gives y = –3 etc.



Plot points at (–1, –3) etc. Then join with a straight line. Extend the line out to the edge of the graphing area.

Note: Using the rule ** y = 2x – 1 ** compared to ** y = mx + c **we get
 * m = 2 {gradient}
 * c = –1 {y-intercept}

We can see from the graph shown here, the ** y-intercept ** is the point where the graph cuts through (or intersects) the y-axis. ** (0, –1) **

Gradient = 2 means that for every step of 1 to the right, the graph goes up by 2. This can be verified between any two adjacent red points on the graph shown here. Go to top of page flat

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