Linear Rules and Graphs
An algebraic rule relates one number (x) to another number (y). Any x number can be SUBSITUTED
into the rule to find a matching y number and produce a number pair that “SATISFIES” the rule.
Linear Rules are those whose number pairs lie on a STRAIGHT LINE when graphed.
The general form of a linear rule is y = mx + c The value of m = the rate of change of the rule (and
the gradient of the graph). The value of c= constant term this also tells you where the graph crosses
the y axis (Vertical Intercept), this will be the point (0, c)
For each x value in the first column use the given rule to find the corresponding y value. Also
identify the rate of change value (m) and the vertical inetrecpt ( 0 ,C) of the graph of each rule.
Question 1: Positive x values only. Note 2x means 2 times x
x
y= x +2
y=2x - 3
y= - x+ 3
m=
m=
m=
y=0.5x + 1
0
1
2
3
4
5
Rate of change
Vertical
Intercept
( 0,
)
(0,
)
( 0,
m=
)
( 0,
)
Question 2 Using both positive and negative integers as x values
x
y= x - 2
y=2x + 3
m=
m=
y= - 2x- 3
y=0.5x - 1
-2
-1
0
1
2
3
Rate of change
Vertical
Intercept
( 0,
)
(0,
m=
)
( 0,
m=
)
( 0,
)
Finding Linear Rules from a graph
Finding a linear rule involves finding the RATE of CHANGE (m) and the value of any CONSTANT
term (c)
Finding a rule from a graph
First find the gradient of the graph by making a
rise/run triangle. Choose any triangle that is easy
to calculate with whole numbers.
rise=3
For this example rise/run=3/1= 3
So the value of
m=3
Next find where the graph crosses the y axis
run = 1
Vertical
Intercept (0, -3)
For this example the vertical intercept is ( 0, -3 )
So the value of c
= -3
So the rule for this graph is
y = 3x - 3
Find the rules for the following graphs
y=
y=
y=
y=
y =
y=
Sketching a graph from a rule in y = mx + c form. The gradient –intercept method
Provided the rule is written in y = mx + c form a graph can be sketched using the values of m and c
First indentify the value of c and Eg given y = 3x - 2 ( m = 3/1 and c = -2 )
plot the vertical intercept as the
point (0, c)
Finally draw a line through
both points
Next find another point using the
rate of change ratio m.
Finally draw a line through both
points
1st plot Vertical
intercept (0, -2)
2nd point found using
rise/run=3/1
Sketch the graphs of the following rules on the grids provided
y = x+ 2
y = -x -1
y = 2x -2
y=½x+1
y = -2x +3
y = 3x - 4