1. REFRESHER
Linear Graphs

2. 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.

3. Standard linear equation
y = mx + c Where ‘m’ is the gradient and ‘c’ is the y-intercept.

4. 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).
400 m
500 m
Gradient = m = Rise
Run
rise
200 m
400 m
rise
run
run
m =
m =

5. Gradient The graph of y = 2x - 4 m = 8 4 = 2 8 4 Gradient = m = Rise
Run
rise
m = 8 4
= 2 8
run 4

6. Gradient The graph of y = 2x - 4 4 m = 4 2 = 2 2
The gradient is the same at any point along a straight line
Gradient = m = Rise
Run
rise 4
m = 4 2
= 2
run 2

7. If a line goes up from left to right,
Gradient: Positive or negative?
If a line goes up from left to right,
then the slope has to be positive
If a line goes down from left to right,
then the slope has to be negative
Lines that are horizontal have zero slope.
Vertical lines have no slope, or undefined
slope.

8. Gradient: Positive or Negative?
Positive gradient
x- value increases
y- value increases
Negative gradient
x- value increases
y- value decreases

9. 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 the change between two points on a line.
In 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

10. Gradient Exercise 6.2 Q1, Q2, Q3, Q4, Q5, Q6 RHS
The graph of y = 2x - 4
(4,4)
1. Choose two points on the
graph
2. Substitute the values and
calculate
(0,-4) (4,4)
x1, y1 x2, y2
m = 4- -4
4-0
= 8 4
= 2
(0,-4)

11. Gradient Exercise 6.2 Q1, Q2, Q3, Q4, Q5, Q6 RHS
The graph of y = -2x + 4
(0, 4)
1. Choose two points on the
graph
2. Substitute the values and
calculate
(0,4) (2,0)
x1, y1 x2, y2
(2,0)
m = 0 - 4
2 - 0
= -4 2
= -2

12. Intercepts The x-intercept occurs where the line cuts the x- axis.
At the x- intercept the y-value always equals to 0.
The y- intercept occurs where the line cuts the y- axis.
At the y-intercept the x-value is
always equal to 0.
(2,0)
x-intercept
(0,-4)
y-intercept

13. Brain Storm! Explain how you could find the y-intercept by:
Looking at a graph
b) Looking at an equation
y = 2x +3
c) Looking at a table of values
Find the coordinates for the point where the line passes through the x-axis (the x-intercept), and for the point where the line passes through the y axis (the y-intercept)
Substitute y = 0 into the equation to find the 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 and look at the graph. OR... Work out the rule used and use the method listed above under looking at an equation X 2 4 Y 8 12
Rule: y = 2x +4

14. Find the Intercepts:
Find the x and y intercepts of the following graphs:
a b.
c d.
(2,0)
(2,0)
x-intercept
x-intercept
(0, 4)
y-intercept
(0,-4)
y-intercept
(0,0)
(4,0)
x-intercept
x-intercept
(0, 4)
(0,0)
y-intercept
y-intercept

15. Find the intercepts
Example 1. Find the x and y intercepts of the following equation: y = 3 x + 2
x int: y = 0
0 = 3 x + 2
0-2 = 3 x
-2 = 3 x
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)

16. 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 = -2x
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)

17. 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
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 3
y = 2
y-intercept = 2 i.e. the line cuts the y axis at (0,2)

18. REMEMBER!! The x-intercept occurs where the line cuts the x- axis.
At the x- intercept the y-value always equals to 0.
The y- intercept occurs where the line cuts the y- axis.
At the y-intercept the x-value is
always equal to 0.
(2,0)
x-intercept
(0,-4)
y-intercept

19. Sketching linear graphs using the x- and 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.

20. Sketch the linear equation
Example 1. Find the x and y intercepts of the following equation and sketch the line. y = 2 x + 4
STEP 1: FIND THE INTERCEPTS...
x int: y = 0
0 = 2 x + 4
0-4 = 2 x
-4 = 2 x
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)

21. Sketch the linear equation
Example 1. Find the x and y intercepts of the following equation and sketch the line. y = 2 x + 4
X int
(-2,0)
Y int
(0,4)
STEP 2: PLOT THE COORDINATES AND RULE A STRAIGHT LINE BETWEEN THEM... y
y = 2 x + 4 x