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Gradient and Intercept April 19, 2023
• Lesson Objective: • To Plot the graphs of
simple linear functions, and also find the equation of a straight line - L5
• Lesson Outcomes• Must: plot graphs
given equations of the line – D
• Should: find equation of line –D/C
• Could: Find the equation given the graph– C
Gradients of straight-line graphs
The gradient of a line is a measure of how steep the line is.
y
x
a horizontal line
Zero gradient
y
x
a downwards slope
Negative gradient
y
x
an upwards slope
Positive gradient
The gradient of a line can be positive, negative or zero if, moving from left to right, we have
If a line is vertical its gradient cannot by specified.
The equation of a straight line
y = mx + cm = gradient of the line. It tells us how steep the line is or the SLOPE of the line
c = intercept on the y axis tells us where the line crosses the y-axis.
y
x3
5
1
4-5 -4 -3 -2 -1
4
3
2
-2
-1
-4
-3
-6
-5
021
6
= Gradient is 1
1= 1
Intercept is 2
y = 1x + 2
= Gradient is 3
1= 3
Intercept is 3
y = 3x + 3
= Gradient is 0.5
1= 0.5
Intercept is 1
y = 0.5x + 1
The gradient and the y-intercept
equation gradient y-intercept
y = 3x + 4
y = – 5
y = 2 – 3x
1
–2
3 (0, 4)
(0, –5)
–3 (0, 2)
y = x
y = –2x – 7
x2
12
(0, 0)
(0, –7)
Finding the gradient from two given points
If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows,
the gradient =change in ychange in x
the gradient =y2 – y1
x2 – x1
x
y
x2 – x1
(x1, y1)
(x2, y2)
y2 – y1
Draw a right-angled triangle between the two points on the line as follows,
Draw a Graph Plot the points and write the equation
Level 7/8 Pack 1
Sketching Graphs
Y = x + 3Y = -x + 2
The sign in front of the x tells you if the gradient is positive or negative.
Positive gradient
Negative gradient
This tells you where the line cuts the vertical line
3
cuts the vertical line at 2
2
1. y = x + 5
2. y = x + 1
3. y = x + 3
4. y = x - 6
5. y = x - 5
6. y = x - 1
7. y = -x + 7
8. y = -x + 3
9. y = -x - 4
10. y = 2x + 5
Starter: sketch the graph of each of the equation.
51.
1
2. 3
3.
-6
4.
-5
5.
-1
6.
3
8.
-4
9.
77.
510.
Linear graphs with positive gradients
Investigating straight-line graphs
Rearranging equations into the form y = mx + c
Sometimes the equation of a straight line graph is not given in the form y = mx + c.
The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.
We can rearrange the equation by transforming both sides in the same way
2y + x = 4
2y = –x + 4
y =–x + 4
2
y = – x + 212
Rearranging equations into the form y = mx + c
Sometimes the equation of a straight line graph is not given in the form y = mx + c.
The equation of a straight line is 2y + x = 4.Find the gradient and the y-intercept of the line.
Once the equation is in the form y = mx + c we can determine the value of the gradient and the y-intercept.
So the gradient of the line is 12
– and the y-intercept is 2.
y = – x + 212
What is the equation?
Look at this diagram:
C
A
B
E
G H
F
D
0 5
5
10-5
10
What is the equation of the line passing through the points
a) A and E
b) A and F
c) B and E
d) C and D
e) E and G
f) A and C?
x = 2
y = 10 – x
y = x – 2
y = 2
y = 2 – x
y = x + 6
x
y
Substituting values into equations
What is the value of m?
To solve this problem we can substitute x = 3 and y = 11 into the equation y = mx + 5.
This gives us: 11 = 3m + 5
6 = 3mSubtracting 5:
2 = mDividing by 3:
m = 2
The equation of the line is therefore y = 2x + 5.
A line with the equation y = mx + 5 passes through the point (3, 11).
Pairs
Matching statements
Exploring gradients
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