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Linear Graphs &
Best line fits (Least squares)
Learning objectives
•
After the session you will be able to:
•
•
•
•
•
Compute the salient details of a straight line in
space
Find the solution to two simultaneous equations
using graphical means
Algebraic solutions to systems of two equations.
Computer solutions to systems of equations
Use a computer (and/or calculator) to compute
the best line fit
2
Graphs
 Draw a table of values
 Plot the points accurately
 Draw a straight line or smooth curve, with a
sharp pencil.
 Read off values accurately
3
Recap: Graphs
 Plot the graph y = 2x + 1
x
-3
 2x -6
-4
 +1 1
1
y
-5
 Plot x and y
-2
-2
1
-3
-1
0
1
-1
0
2
1
1
1
4
1
3
2
6
1
5
3
7
4
y = 2x + 1
-3
-2
-1
8
7
6
5
4
3
2
1
0
-1 0
-2
-3
-4 Run
-5
-6
Rise
Run
1
2
Rise
5
3
y = 3x - 2
Plot the graph y = 3x - 2
•x
-3
-2
•y
-11 -8
-1
0
1
2
-5
-2
1
4
2
0
-2
-4
-6
-8
-10
6
-12
-3
-2
-1
0
1
y = -2x + 3
4
2
0
MATLAB Commands:
>>y=‘-2*x+3’
>>fplot(y,[0,6])
OR
>>y=‘-2*x+3’
>>ezplot(y)
-2
-4
-6
-8
-10
0
1
2
3
4
5
6
7
Straight line summary
 The equation for a straight line is always in the form:
y = mx + c


m is the gradient (calculated from the rise over the
run).
And c is simply the intercept on the y axis.
Rise y
NB : m 

Run x
+ve
-ve
8
Fractional coefficients
-1
-1
-1
1
y  x2
2
1
y   x2
3
1
y  1 x  2
4
-1.5
-1.5
-1.5
-2
-2
-2
-2.5
-3
-2.5
-2.5
-3.5
-3
-4
-3.5
-4.5
-5
-4
-2
-2
00
22
4
6
9
3x + 2y = 12
How would we approach this problem
Note for implicit
functions you must
use the ezplot
command
MATLAB Commands:
>>ezplot(‘3*x+2*y=12)
10
2x – 3y = 18
MATLAB Commands:
>>ezplot(‘2*x-3*y=18)
11
Group Discussion
Sketch the graphs of the
following:
a) y  4 x  1
1
b) y  5  x
2
c) x  y  3
12
The Intercept Method
a) x  y  3
b) 2 x  y  4
d )3 x  2 y  6
c)3x  2 y  6
13
Learning Check
 Solve the following using a
graphical method
 y=2x -1 &
 y= 8-x
 Use MATLAB to verify your
results.
12
10
8
6
4
2
Commands:
>> ezplot('8-x‘) 0
>> fplot(y,[0,6]) -20
1
2
3
4
5
6
>> hold on
Read off or use ginput
>> grid on
>> y='2*x-1‘
 Q. Is there a way to solve
>> fplot(y,[0,6])
such problems algebraically?
14
>> fplot(y,[0,6],'r')
Class Examples Time
• Solve the following problems using a
•
•
•
•
•
graphical method and verify the results
using algebra/MATLAB
y=x+3 & y=7-x
y=x-4 & 2x+y=5
x+y =3 & y=1-2x
y=x+4 & y=3x
y=2x-1 & y=3x+2
15
Finding the Line
• Here’s the StarGate bit:
• A point in three dimensions needs six pieces of
information to be fully described.
• A course therefore seven
• Since a line exists essentially in 2D then only two
pieces of information.
•
Two points
•
A gradient and a point
16
Example
 Find the equation of the line given that it
passes though the points (-2,1) and (6,5)
y
1 5
1
Find the Gradient m 


x  2  6 2
1
Find the intercept y  x  c
2
1
1
Any point will do 6  (5)  c or 1  (2)  c
2
2
17
Examples
Find the equation of the line given the
points (-2,4) and (4,1) expressing your
answer in the form ax+by=c.
2. A line has a gradient of –0.75 and passes
through a point (3,-4), state the equation of
the line.
3. Find the equation of a line with a gradient of
unity given that it passes through the point
(-1,-2).
1.
18
Best Line Fit
 Consider the
following data if
we suspect that
these must fall on
a straight line
what is the best
way of finding the
relationship (the
law of the line)
10
5
0
-5
-10
-4
-2
0
2
4
19
Example: Best Line Fit
 Find the best line fit
given the following
data
10
5
 Hence or otherwise
0
find the law of the
line
-5
-10
-4
-2
0
2
4
20
Class Discussion Time
• Plot the graph
given the
following data:
•
•
Find the best
line fit
Hence the law of
the line.
y
x
-4
-3
-2
-1
0
1
2
3
4
-9.81
-5.87
-3.18
-1.81
-2.33
1.50
3.71
5.81
7.66
21
Analytical: Best Line Fit
 MATLAB is a useful tool to
evaluate the line of best fit
for you. The procedure is:
• Enter data
•
•
x=[-4:1:4]
y=[-9.81 -5.87 etc
• Plot graph
• plot(x,y,’x’)
y
x
-4
-3
-2
-1
0
1
2
3
4
-9.81
-5.87
-3.18
-1.81
-2.33
1.50
3.71
5.81
7.66
22
Analytical: Best Line Fit
• Fit data to line
• ployfit(x,y,1)
•
ans=2.0545 -0.5500
• Set up function
•
y1=‘2.0545*x-0.5500’
• Plot the function
•
•
hold on
fplot(y1,[-4:4],’k’)
10
5
0
-5
-10
-4
-2
0
2
4
23
Group Activity:
 The width of keyways for various shaft
diameters are given by:
D
10
20 30
w
3.8 6.3 8.3
40
50 60
11.3 13.8 16.3
(1) Show that the relationship between D and W is
linear and find the law of the line
(2) Use MATLAB or otherwise to find the analytical
law of the line
24
Lesson summary
• Have we met our leaning objectives, specifically
can you now:
•
•
•
•
Find the solution to two simultaneous equations
using graphical means
Algebraic solutions to systems of two equations.
Computer solutions to systems of equations
Use a computer (and/or calculator) to compute the
best line fit
25
Homework
1.
2.
3.
Find the equation of the line given the points (-1,6)
and (4,1) expressing your answer in the form
ax+by=c.
A line has a gradient of 1/2 and passes through a
point (-2,4), state the equation of the line.
Find the equation of a line with a gradient of unity
given that it passes through the point (0,-4).
26
Homework
During an experiment to find the coefficient of friction
between two surfaces the following results were obtained.
 Load W (N)

10
20 30 40 50 60
 Friction Force F (N)

1.5 4.3 7.6 10.4 13.5 15.6
Find the law connecting the variables in the form of:
F=aW+b (You may use a computer if you wish)
27