the straight line. reminder: gradient = change in vertical distance change in horizontal distance...
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Dis
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Reminder: Gradient = Change in vertical distance Change in horizontal distance
You can start and end anywhere on the line.
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5m
rise run
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What happens when the line slopes down?
In this case the gradient is negative.
3
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12
8m
Reminder: Gradient Formula12
12
xx
yym
x
y
13
303
3612
12
xx
yym
22
4
)1()3(
3712
12
xx
yym
(0, 3)
(3, 6)(-3, 7)
(-1, 3)
What are the gradients of these two lines?
Positive gradient1
Negative gradient-2
Now check the gradients using the formula.
Reminder: Sketching Lines
Given the equation of a line, we can sketch it by making a table and finding points which lie on the line.
We usually find three points.
Example: Sketch the line .23 xy
x 0 2 4
y -2 4 10
(0, -2) (2, 4) (4, 10)We plot the points
-4
-3
-2
-1
0
1
2
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
(0, -2)
(2, 4)
(4, 10)
.23 xy•Plot the points
•Draw line through the points
x
y
xy 2 2
x
y
xy 2 1
x
y
xy 2 3
x
y
xy 2 2
x
y
xy 2 4
x
y
xy 2 5
x
y
xy 2
x
y
xy 2 1
What do you notice?
What is the gradient of each line?
Where does each line cut the y axis?
We will now look at more lines and their equations
x
y
- 6
x
y
- 2
x
y
+ 1xy 3
Here are some more lines
What do you notice?
What is the gradient of each line?
Where does each line cut the y axis?
So far all the lines we have looked have been of the form
y= mx + c
gradienty-
intercept
We will look at this more closely using Autograph
Autograph.lnk
23 xy
cmxy m = 3 c = -2
What do you think the gradient of the line is?23 xy
-4
-3
-2
-1
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
.23 xy
Check this using the graph.
Where do you think it cuts the y axis?
Again check this using the graph.
What about ?3 xy
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
.3 xy
The equation of the line shown is
62 xy(a)
321 xy(b)
321 xy(c)
62 xy(d)
-4
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-1
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
-4
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-1
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
-4
-3
-2
-1
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
-4
-3
-2
-1
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
-4
-3
-2
-1
0
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-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
The graph of y = 3x + 1 is
A B
DC
You join a video shop for a membership fee of £3 and then charge £2 for each video you hire.
We can draw a graph of Cost against Number of Videos by making a table.
No of Videos (N) 0 1 2 3 4 5 6Cost of Videos(£) (C)
3 5 7 9 11 13 15
Now draw a graph of the table above.
Consider the following problem
0123456789
10111213141516171819202122232425
0 1 2 3 4 5 6 7 8 9 10
Number of Videos
Co
st o
f V
ideo
s
N
C
We can use y = mx + c to find a formula for the cost of hiring any number of videos.
•Instead of x we have N.
•Instead of y we have C.
•What is the gradient of this line?
•What is the y-intercept?
What is the equation of this line?Answer:C = 2N + 3
21
2m
3c
32
32
3
NC
xy
mxy
You always have to pay £3.
For every square that you move to the right you go two squares up because the cost of each video is £2.
Were you correct?
What does the y-intercept tell us?
What does the gradient of 2 tell us?
Another problem: Find the equation of the line below.
• Write down the coordinates of 2 points on the line.
• Use the gradient formula to find m.
• Read off c.
• Write down the equation in the form y = mx + c.
• Write down the equation in terms of s and P.
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-2
-1
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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
s
P
Answer: 32
1 sP
Method:
Special Cases
Lines parallel to the x and y axes.
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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12
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
Using the gradient formula with (0, 3) and (6, 3) gives
03
036
3312
12
xx
yym
3
3)(0
y
xy
cmxy
Notice: the y coordinate of every point on this line is 3We say the equation
of the line is y = 3Also c = 3
3
0
c
mUsing y = mx + c
1. Parallel to the x-axis
-4-3-2-10123456789
101112
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
x
y
Special Cases
Lines parallel to the x and y axes.
Using the gradient formula with (4, 2) and (4, 5) gives
undefined
xx
yym
0
344
2512
12
Notice: the x coordinate of every point on this line is 4
We say the equation of the line is x = 4
This is because you cannot divide by zero!This means you cannot use y = mx + c
2. Parallel to the y-axis
However……..