LEARNING OBJECTIVE:
Understanding:
(1) (a) Cartesian coordinate (x,y),
(b) y = mx +c
m = the gradient
c = y-intercept
(2) Using the equation y = mx + c with given data
(1) A Cartesian coordinate graph: two axes ("axes" plural of "axis"):
- Horizontal axis is called the x-axis.
- Vertical one is the y-axis.
(2) Coordinates are points on a graph
Format: (x, y)
(3) The centre point of the graph is called the origin (0, 0).
It is the zero point of both the x-axis and the y-axis.
Plotting a coordinate:
Step1: locate on x-axis
Step2: locate on y-axis
Step3: "Move"/plot for both to "meet"
Example:
Practice on same graph:
Plot coordinates (-5, 5), (10,0), (0,5)
Graph of Equation of a Straight Line
A linear (straight line) equation has a standard form:
y = mx + c
Example:
y = 2x +c
y = -x + 5
y = 4
x = 3 are all linear equations.
On the graph, the equation of the line y = mx+c = the LINE drawn.
Equation of the Line and its Coordinates
Coordinates are points on the line that satisfies/can be substituted into the equation.
Examples
To test if the following coordinates are on the line y = 2x + 8.
a. (-4,0), b. (-3,2) c. (5, 5)
Equation of the line is y = 2x + 8
(a) Substitute (-4,0) into y = 2x + 8
=> 0 = 2(-4) +8
(-0, -4) is on the line.
(b) Substitute (-3,2) into y = 2x + 8
=> 2 = 2(-3) +8
(-3, 2) is on the line.
(c) Substitute (5,5) into y = 2x + 8
=> 5 ≠ 2(5) + 8
(5, 5) is a coordinate on the graph but not on the line.
Graph Below : y = 2x + 8, and the coordinates (-3,2), (-4,0) and (5,5).
Name the other coordinate on the line (satisfy the equation).
=> (1,10)
If the line passes through a coordinate (x,y), => can be used for the (x,y) of the equation : y = mx + c
Practice: What are the coordinates of (x1, y1), (x2, y2), (x3, y3)?
What is c (the y-intercept)?
c = y-intercept
=> where the equation(line) intercept/‘cuts’ the y-axis.
Finding the value of c from the equation
Equation of a straight line:
y = mx + c
c= the y-intercept; coordinate (0,c)
Example:
(1) y = 3x + 5 => c = 5, (0,5)
(2) y = -x – 3 => c = -3 (0,-3)
(3) 2y = 5x + 6 (must change equation of format: y = mx + c)
y = 5x + 6
2
= 5x + 3, => c = 3 (0, 3)
2
(4) -3y = -2x + 9 (must change equation format to: y = mx + c)
2x - 9 = 3y
3y = 2x - 9
3
= 2/3 x - 3, => c = -3 (0, -3)
What is m?
y = mx + c
Gradient(m) is the ‘steepness’ or slope of a graph.
Formula:
Gradient = y2 – y1
x2 – x1
We need 2 sets of coordinates (x, y) on the line to find the gradient(m).
Example:
Find the gradients of the following graphs.
(x1, y1) = (0, -2), (x2, y2) = (5,8) (x1, y1) = (-5, 7), (x2, y2) = (1,-5)
Gradient = -2 – 8 Gradient = 7 – (-5)
0 - 5 -5 - 1
= -10/-5 =12/-6
= 2 = -2
Observation:
1. The gradient/"slope" is positive (2) for (0, -2), (5,8) as it is "ascending" from left to right.
Slope Upward => positive gradient(+m)
2. The gradient/"slope" is -2 for (-5,7), (1-5) as it is "descending" from left to right.
Slope downward => -m
3. When calculating the value of m, the (x,y) must be of the same "set order" =>
y1 - y2 or y2 - y1 NOT y1 - y2
x1 - x2 x2 - x1 x2 - x1
To find the Value of m from the equation
Equation of a straight line:
y = mx + c
Example:
(1) y = 3x + 5 , m = 3
(2) y = -x – 3, => y = (-1)x - 3, m = -1
(3) 2y = 5x + 6 (must change equation of format: y = mx + c)
y = 5x + 6
2
= 5x + 3, m = 5/2
2
(4) -3y = 2x + 9 (must change equation of format: y = mx + c)
-2x - 9 = 3y
3y = -2x - 9
3
= -2/3 x - 3, m = -2/3
Equation Of Vertical And Horizontal Lines
(1) x = numeric value
-> a vertical line graph
Example:
x=5
Equation of the line : x = 5
(2) y = numerical value
=> a horizontal line graph.
Example:
y=5
Equation of the line : y = 5
Level 2
To find intercept of 2 lines
(1) Use simultaneous equation or
(2) the point of interception of the 2 lines on the graph
2 linear equations: y = 2x – 2
y = -2x + 4
From the graph, the intersection coordinate is (3/2, 1)
=> x = 3/2 and y = 1
Practice
1. Draw a simple graph. Plot and label the point A(-2, 1) and B (3, 5).
2. Find the gradient of the line AB [17/I/92,2/T]
3. Find the gradient of the line joining the points A (2, 6) and B(7,3)
4. Find the gradient of the line joining the points A(2, 1) and B(4, 6)
