Straight+line+graphing

Online notes: [|here] for how to graph a straight line.
 * Linear patterns (Download here [[file:Finding patterns and graphing straight lines.docx]]) ** Revision:

[|Graphing straight lines] The difference between each term is 5. (13 - 8 = 5, 18 - 13 = 5, etc)

The difference (5) is the multiplier, so the rule must be w = 5b + ?.

To work out the ?, substitute in b = 1 and w = 8 from the first line of the table

8 = 5 x 1 + ?.

The constant term must be 3.

Giving the rule as w = 5b + 3.

Online revision:[| here] for finding a pattern / creating an equation


 * Straight line graphs**

In a sequence of numbers, each number is called a term.
 * Linear patterns**

For the sequence 2, 5, 8, 11, …

The first term is 2, the second term is 5, etc. The gap between the numbers is called the difference.

In this example the difference is 3, because 5 – 2 = 8 – 5 = 11 - 8 = 3. Linear patterns have the same difference between the terms.

To find the rule, look for a relationship between the number of the term and the term itself. The rule is the same for each term.

In this example: 2 = 3 x 1 – 1, 5 = 3 x 2 – 1 and 8 = 3 x 3 – 1,

so the rule is t = 3 x n – 1 or t = 3n - 1 For the sequence of diagrams below, find the rule for the number of white tiles (w) in terms of the number of black tiles (b). The difference between each term is 5. (13 - 8 = 5, 18 - 13 = 5, etc)
 * Problem**
 * Answer**
 * **Number of black tiles (b)** || **Number of white tiles (w)** || **Difference** ||
 * 1 || 8 ||  ||
 * 3 || 13 || 5 ||
 * 4 || 18 || 5 ||
 * 4 ||  ||   ||

The difference (5) is the multiplier, so the rule must be w = 5b + ?.

To work out the ?, substitute in b = 1 and w = 8 from the first line of the table

8 = 5 x 1 + ?.

The constant term must be 3.

Giving the rule as w = 5b + 3. Lines in the form y = **m**x + **c** If the equation is written in this form: i.e. the distance up, over the distance across Positive slope so m is positive Negative slope so m is negative For more information, see Straight Lines and Slope To draw a line written in y = mx + c form:
 * Straight line graphs**
 * **m** (the number in front of x) is the gradient of the line
 * **c** (the number by itself) is where the graph crosses on the y-axis
 * remember the sign belongs to the term directly following it.
 * The value of m is the gradient.
 * Write this number as a fraction if it is a whole number by putting over 1.
 * Mark a point at c on the y-axis.
 * From here step out the gradient. Count up the top number and along the bottom number.
 * If the gradient is negative count down.
 * Join the points with a ruler.
 * If the gradient is negative count **down** the top number.

Jan obtains two quotes for printing booklets.
 * Example of a problem in context**

Firm A quotes: "$8 plus $4 per booklet".

Firm B quotes: "$12 plus $3 per booklet".

For how many copies will the cost be the same with both firms?

Draw a graph for each firm on the same axes. This can be done by writing an equation for both lines first. If n is the number of booklets and C is the total cost: Use the equations to draw both lines. The point where the lines intersect (4,24) will give number of booklets that can be printed for the same cost. So the printing of 4 booklets will cost the same at both firms and the cost will be $24.
 * Answer**
 * the equation for Firm A is C = 4n + 8
 * the equation for Firm B is C = 3n +12.

A horizontal line has equation y = c
 * Special cases**

– the gradient is 0 and c is where the line crosses on the y-axis. A vertical line has equation x = k

– the gradient is undefined and k is where the line crosses on the x-axis. For more information on horizontal and vertical lines see:

Horizontal and Vertical Lines

To graph lines which have equations that are written in a form other than y = mx + c, for instance 2x + 3y – 6, use one of the following methods:
 * Linear equations in other forms**
 * gradient/intercept method (first rearrange into y = mx + c form)
 * two-intercept method (put y = 0 to find x-intercept, and x = 0 to find the y-intercept)
 * plotting points.