REFRESHER
Linear
Graphs
INTERPRETING STRAIGHT-LINE
GRAPHS
Equations and graphs are used to study the
relationship between two variables such as
distance and speed.
 If a relationship exists between the variables, one
can be said to be a function of the other. A
function can be described by a table, a rule or a
graph.
 We are going to analyze linear functions.

STANDARD LINEAR EQUATION
y = mx + c
Where ‘m’ is the gradient and
‘c’ is the y-intercept.
GRADIENT





The gradient (or slope) is the steepness of a line.
The gradient gives us information about how much
one variable changes compared to another.
The steeper the line the greater the gradient
Gradient is simply the change in the vertical distance
(rise) over the change in the horizontal distance (run).
Gradient = m = Rise
Run
rise
500 m
rise
200 m
400 m run
200
m=
 0.5
400
400 m run
m = 500  1.25
400
GRADIENT
The graph of y = 2x - 4

Gradient = m = Rise
Run
rise
m= 8
4
= 2
8
run
4
GRADIENT
The graph of y = 2x - 4
The gradient is the same at
any point along a straight
line

Gradient = m = Rise
Run
m= 4
2
= 2
rise
4
run
2
GRADIENT: POSITIVE OR NEGATIVE?
o
If a line goes up from left to right,
then the slope has to be positive
o
If a line goes down from left to right,
then the slope has to be negative
o
Lines that are horizontal have zero slope.
o
Vertical lines have no slope, or undefined
slope.
GRADIENT: POSITIVE OR NEGATIVE?
Positive gradient
o x- value increases
o y- value increases
Negative gradient
o x- value increases
o y- value decreases
GRADIENT: DEFINITION
Since the rise is simply the change in the vertical
distance and the run is the change in the horizontal
distance. The rise and run can be found by calculating
o
the change between two points on a line.
rise y y2  y1
slope  m 


run x x2  x1
oIn
order to use this formula we need to know, or be
able to find 2 points on the line. Eg (0,-4) & (2,0)
x1, y1
x2, y2
GRADIENT
Exercise 6.2 Q1, Q2, Q3,
Q4, Q5, Q6 RHS
The graph of y = 2x - 4
y2  y1
m
x2  x1
(4,4)
1. Choose two points on the
graph
(0,-4) (4,4)
x1, y1 x2, y2
2. Substitute the values and
calculate
m = 4- -4
4-0
=8
4
=2
(0,-4)
GRADIENT
Exercise 6.2 Q1, Q2, Q3,
Q4, Q5, Q6 RHS
The graph of y = -2x + 4
y2  y1
m
x2  x1
1. Choose two points on the
graph
(0,4) (2,0)
x1, y1 x2, y2
2. Substitute the values and
calculate
m=0-4
2-0
= -4
2
= -2
(0, 4)
(2,0)
INTERCEPTS

The x-intercept occurs where the line cuts the xaxis.


At the x- intercept the y-value always equals to 0.
The y- intercept occurs where the line cuts the yaxis.
(2,0)

At the y-intercept the x-value is
always equal to 0.
(0,-4)
y-intercept
x-intercept
BRAIN STORM!
1.
Explain how you could find the y-intercept by:
a)
Looking at a graph
Find the coordinates for the point where the
line passes through the x-axis (the xintercept), and for the point where the line
passes through the y axis (the y-intercept)
b) Looking at an equation
Substitute y = 0 into the equation to find the
y = 2x +3
c)
X
Y
x value for the x-intercept.
Substitute x = 0 into the equation to find the
y-value for the y-intercept.
Carefully plot the points
Looking at a table of values
and look at the graph.
OR... Work out the rule
0
2
4
used and use the method
4
8
12
listed above under looking
at an equation
Rule: y = 2x +4
FIND THE INTERCEPTS:
1.
Find the x and y intercepts of the following
(2,0)
(2,0)
graphs:
x-intercept
x-intercept
a.
b.
(0, 4)
y-intercept
(0,-4)
y-intercept
c.
(0,0)
y-intercept
(0,0)
x-intercept
d.
(0, 4)
y-intercept
(4,0)
x-intercept
FIND THE INTERCEPTS
Example 1. Find the x and y intercepts of the
following equation: y = 3 x + 2
x int: y = 0
0=3x+2
0-2 = 3 x +2 - 2
-2 = 3 x
3 3
X=-2
3
x – intercept = -2/3 i.e. The line cuts the x –axis at (-2/3, 0)
y int: x = 0
y = 3 (0) + 2
y=0+2
y=2
y-intercept = 2 i.e. the line cuts the y axis at (0,2)
FIND THE INTERCEPTS
Example 2. Find the x and y intercepts of the
following equation: y = -2x + 1
x int: y = 0
0 = -2x + 1
0-1 = -2x + 1 - 1
-1 = -2x
-2 -2
X=1
2
x – intercept = ½ i.e. The line cuts the x –axis at ( ½ , 0)
y int: x = 0
y = -2(0) + 1
y=0+1
y=1
y-intercept = 1 i.e. the line cuts the y axis at (0,1)
FIND THE INTERCEPTS
Example 1. Find the x and y intercepts of the
following equation: 2x + 3y = 6
x int: y = 0
2x + 3(0) = 6
2x + 0 = 6
2x = 6
2 2
x=6
2
x=3
x – intercept = 3 i.e. The line cuts the x –axis at (3, 0)
y int: x = 0
2 (0) + 3y = 6
0 + 3y = 6
3y = 6
33
y=2
y-intercept = 2 i.e. the line cuts the y axis at (0,2)
REMEMBER!!

The x-intercept occurs where the line cuts the xaxis.


At the x- intercept the y-value always equals to 0.
The y- intercept occurs where the line cuts the yaxis.
(2,0)

At the y-intercept the x-value is
always equal to 0.
(0,-4)
y-intercept
x-intercept
SKETCHING LINEAR GRAPHS USING THE XAND Y-INTERCEPTS
State the equation
 Find the x- intercept (substitute 0 for y and solve
for x)
 Find the y- intercept (substitute 0 for x and solve
for y)
 Mark the x intercept and y intercept and rule a
straight line through them.

SKETCH THE LINEAR EQUATION
Example 1. Find the x and y intercepts of the following
equation and sketch the line.
y=2x+4
STEP 1: FIND THE INTERCEPTS...
x int: y = 0
0=2x+4
0-4 = 2 x +4 - 4
-4 = 2 x
2 2
X=-4
2
X = -2
x – intercept = -2 i.e. The line cuts the x –axis at (-2, 0)
y int: x = 0
y = 2 (0) + 4
y=0+4
y=4
y-intercept = 4 i.e. the line cuts the y axis at (0,4)
SKETCH THE LINEAR EQUATION
Example 1. Find the x and y intercepts of the following
equation and sketch the line.
y=2x+4
STEP 2: PLOT THE COORDINATES AND RULE A STRAIGHT
LINE BETWEEN THEM...
y
X int
(-2,0)
y=2x+4
Y int
(0,4)
x