Slides: GCSE Straight Line Equations

Report
GCSE: Straight Line Equations
Dr J Frost ([email protected])
Last modified: 3rd September 2014
GCSE specification:
 Understand that an equation of the form y = mx + c corresponds to a straight line graph
 Plot straight line graphs from their equations
 Plot and draw a graph of an equation in the form y = mx + c
 Find the gradient of a straight line graph
 Find the gradient of a straight line graph from its equation
 Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept
of c (ie. crosses the y axis at c)
 Understand how the gradient of a real life graph relates to the relationship between the two
variables
 Understand how the gradients of parallel lines are related
 Understand how the gradients of perpendicular lines are related
 Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line
perpendicular to it will be -1/m
 Generate equations of a line parallel or perpendicular to a straight line graph
y
What is the equation of
this line?
And more importantly,
why is it that?
4
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
? 2
=
The line -3represents
all points which
satisfies -4the
equation.
□ “Understand that
an equation
corresponds to a
line graph.”
6
y
4
Starter
A
D
3
F
C
2
B
1
E
G
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
H
-3
-4
What is the equation of
each line?
Equation of a line
 Understand that an equation of the form  =  + 
corresponds to a straight line graph
The equation of a straight line is  =  + 
gradient
y-intercept
Gradient using two points

Given two points on a line, the gradient is:
ℎ  
=
ℎ  
1, 4
5, 7
2, 2
(3, 10)
 = 3?
(8, 1)
?
 = −2
(−1, 10)
8
 = −?
3
Gradient from an Equation
 Find the gradient of a straight line graph from its equation.
 = 1 − 2
Putting in form  =
 + :
 = − + 
Gradient is -2
?
2 + 3 = 4
Putting in form  =
 + :
 = − + 
 ? 
=− +


Gradient is −


Test Your Understanding
Find the gradient of the line with equation  −
2 = 1.
 =  − 


= −
 ? 

=

Exercise 1
1
2
Determine the gradient of the lines
which go through the following
points.
a
3,5 , 5,11
b
−1,0 , 4,3
c
2,6 , 5, −3
d
4,7 , 8,10
e
f
g
h
1,1 , −2,4
3,3 , 4,3
4, −2 , 2, −4
−3,4 , 4,3
 = ?

= ?

 = −
?

= ?

 = −
?
 = ?
 = ?

 = −?

Determine the gradient of the lines
with the following equations:
a  = 5 − 1
 = ?
 = −
b +=2
?
=
c  − 2 = 3
?
d  − 3 = 5
e 2 + 4 = 5
f
2 −  = 1
g 2 = 3 − 7
3

= ?


 = −?


= ?


= ?

A line 1 goes through the points
(2,3) and 4,6 . Line 2 has the
equation 4 − 5 = 1. Which
has the greater gradient:


 =
 =

? 
So  has greater gradient.
Drawing Straight Lines
y
4
Sketch the line with equation:
 + 2 = 4
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
Bro Tip: To sketch a line, just work out
any two points on the line. Then join up.
Using  = 0 for one point and  = 0 for
the other makes things easy.
-3
-4
□ “Plot and draw a
graph of an
equation in the
form y = mx + c”
6
y
Test Your Understanding 4
Sketch the line with equation:
 − 3 = 3
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
□ “Plot and draw a
graph of an
equation in the
form y = mx + c”
6
Finding intersection with the axis


When a line crosses the -axis:
=
?
When a line crosses the -axis:
=
?
The point where the line crosses the:
Equation
-axis
-axis
 = 3 + 1
0,1
1
− ?, 0
3
 = 4 − 2
0, −2
?
1
?, 0
2
1
 = −1
2
0, −1
2,0
?
?
?
Equation given a gradient and point
The gradient of a line is 3. It goes through the point (4, 10). What
is the equation of the line?
 =  − 
? determined)
Start with  =  +  (where  is to be
Substituting:  =  ×  + 
Therefore  = −
The gradient of a line is -2. It goes through the point (5, 10). What
is the equation of the line?
 = − + 
?
Test Your Understanding
1
2
Determine the equation of the line which has gradient 5 and goes through
the point 7,10 .
 = 
? − 
Determine the equation of the line which has gradient −2 and goes through
the point 3, −2 .
 = −
? +
1
3
Find the equation of the line which is parallel to  = − 2  + 3 and goes
through the point 6,1

 = −?  + 

Equation given two points
A straight line goes through the points (3, 6) and (5, 12). Determine
the full equation of the line.
Gradient:
3
Equation:
 = 
? −
(5,12)
?
(3,6)
A straight line goes through the points (5, -2) and (1, 0). Determine
the full equation of the line.
(5, -2)
Gradient:
-0.5
Equation:
 = − ? +
?
(1,0)




Exercise 2
1 Determine the points where the
following lines cross the  and  axis.
1
 = 2 + 1
0,1 , − , 0
2
2
 = 3 − 2
0, −2 , , 0
3
5
2 +  = 5
0, , 5,0
2
?
?
?
2
Using suitable axis, draw the line with
equation 2 +  = 5.

5
?
5
2
3
A line has gradient 8 and goes
through the point 2,10 . Determine
its equation.
 =  − 
A line has gradient −3 and goes
through the point 2,10 . Determine
its equation.
 = − + 
?
4

?
5 Determine the equation of the line parallel
to  = 6 − 3 and goes through the point
3,10 .
 =  − 
?
6 Determine1 the equation of the line parallel
to  = −  + 1 and goes through the
3
point −9,5 .

=− +

?
7 Determine the equation of the lines which
go through the following pairs of points:
3,5 , 4,7
 =  − 
4,1 , 6,7
 =  − 
−2,3 , 4, −3  = − + 

0,3 , 3,5
= +


4, −1 , 2,4  = −  + 

?
?
?
?
?
y
4
m = -1/3
?
3
m = 1/2
?
2
1
m=3?
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
m = -2?
-2
-3
-4
Find the gradients of
each pair of
perpendicular lines.
What do you notice?
6
Perpendicular Lines

If two lines are perpendicular, then the gradient of one is the
negative reciprocal of the other.
1
1 = −
2
Or:
1 2 = −1
Gradient
Gradient of Perpendicular Line
1
−?
2
1
?3
2
−3
1
4
5
2
7
7
5
−
-4
?
1
−?
5
7
?2
5
−?
7
Example Problems
Q1
A line is goes through the point (9,10) and is perpendicular to another line with
equation  = 3 + 2. What is the equation of the line?

 −  = −?  − 

Q2
A line 1 goes through the points  1,3 and  3, −1 . A second line 2 is
perpendicular to 1 and passes through point B. Where does 2 cross the x-axis?
, 
?
Q3
Are the following lines parallel, perpendicular, or neither?
1
= 
2
2 −  + 4 = 0


Neither. Gradients are and . But ×? = , not -1, so not perpendicular.


Exercise 3
1 A line 1 goes through the indicated point and
is perpendicular to another line 2 . Determine
the equation of 1 in each case.

2,5
2 :  = 2 + 1  :  = −  + 


−6,3 2 :  = 3
 :  = −  + 

1
0,6
2 :  = −  − 1  :  =  + 
2
1
−9,0 2 :  = −  + 1  :  =  + 
3

10,10 2 :  = −5 + 5  :  =  + 

4

?
?
?
?
?
2
 2,5  4,9
Find the equation of the line which passes through B,
and is perpendicular to the line passing through both
A and B.

 = −  + 

?
3
Line 1 has the equation 2 + 3 = 4. Line 2 goes
through the points (2,5) and (5,7). Are the lines
parallel, perpendicular, or neither?


 = −
 =


  = − so perpendicular.
?

Determine the equation of the line .

=− +

?
5



Determine the equation of the line .
Known point on :
, 
So equation of :

= −

?
GCSE specification:
 Understand that an equation of the form y = mx + c corresponds to a straight line graph
 Plot straight line graphs from their equations
 Plot and draw a graph of an equation in the form y = mx + c
 Find the gradient of a straight line graph
 Find the gradient of a straight line graph from its equation
 Understand that a graph of an equation in the form y = mx + c has gradient of m and a y intercept
of c (ie. crosses the y axis at c)
 Understand how the gradient of a real life graph relates to the relationship between the two
variables
 Understand how the gradients of parallel lines are related
 Understand how the gradients of perpendicular lines are related
 Understand that if the gradient of a graph in the form y = mx + c is m, then the gradient of a line
perpendicular to it will be -1/m
 Generate equations of a line parallel or perpendicular to a straight line graph
Two last things…
Distance between two points
Midpoint of two points
?5
5,9
(3,6)
3
(, .
?)
(3,6)
Just find the average of  and
the average of .
(7,9)
4
Find  change and  change to
form right-angled triangle.
Then use Pythagoras.
Past Exam Questions
See GCSEPastPaper_Solutions.pptx
GCSERevision_StraightLineEquations.docx

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