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Straight Lines
Straight Lines
Curriculum Ready
Copyright © 2009 3P Learning. All rights reserved.
First edition printed 2009 in Australia.
A catalogue record for this book is available from 3P Learning Ltd.
ISBN 978-1-921861-72-7
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2ISERIES TOPIC
1Straight LinesMathletics Passport © 3P Learning
Q The robot standing on the x-axis at point A needs to get to point B on the y-axis. The solar panels only have enough stored energy to travel the shortest straight line path. Write down the rule of the line the robot needs to follow to get from A to B.
Work through the book for a great way to solve this
Give this a go!
Straight lines all follow a particular pattern or rule and appear in all facets of life.
This curvy optical illusion is made using lots of lines that have different slopes.
7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1x-axis
y-axis
A
B
2 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Straight Lines
x -3 -2 -1 0 1 2 3 4
y -5 -4 -3 -2 -1 0 1 2
(-3,-5) (-2,-4) (-1,-3) (0,-2) (1,-1) (2,0) (3,1) (4,2)
x-axis
y-axis
Write the rule along the line
1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
Remember: The x-value is always written first
0
Join plots with a straight, double arrowed line
Plot tabled values
y
x2
=
-
The table below was completed using the rule: 2y x= -
Summary for graphing from a table of values:
• Plot the coordinates read from the table as small dots• Use a ruler to join the points and put neat arrows on either end of the line• Write the rule used along the line
Graphs using tables of values
Let’s review graphing lines from a completed table of values.
Values in the table are paired together to find the coordinates.
How does it work?
2ISERIES TOPIC
3Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
Graphs using tables of values
Plot each of the completed table of values below:
1 y x= 1y x= -
2y x= + y x3 2= -
x -3 -2 -1 0 1 2 3
y -1 0 1 2 3 4 5
x -3 -2 -1 0 1 2 3 4
y -3 -2 -1 0 1 2 3 4
x -3 -2 -1 0 1 2 3 4
y 4 3 2 1 0 -1 -2 -3
x -2 -1 0 1 2 3 4
y 7 5 3 1 -1 -3 -5
GRAPHS USING TABLES OF
VALUES
�
GRAPHS USI
NG TABLES OF
VALUES
�
..../...../20...
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
3 4
2
4 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight Lines
x -2 -1 0 1 2
y -3 -1 1 3 5
Complete and plot the graph for the table of values using the rule y x2 1= +
Completing and graphing tables of values
Sometimes you are only given a rule and need to complete your own table of values. The table of values often needs to be completed using the given rule before plotting the graph.
( )y 2 2 1
3
= - +
= -
y 2 1 1
1
= - +
= -
^ h y 2 0 1
1
= +
=
^ h y 2 1 1
3
= +
=
^ h 2 1y 2
5
= +
=
^ h
x-axis
y-axis7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
Plot tabled values
Join plots with a straight, double arrowed line
Write the equation along the line
21
yx
=+
Tables of values can also be written vertically. When drawn this way they are often called T-charts.
Coordinates are paired horizontally
x y-1
0
1
1
T-Charts
y x2 1= +
Always include zero, positive and negative values in your table of values
2ISERIES TOPIC
5Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
Completing and graphing tables of values
1 Complete these tables of values for each given rule and then plot the graph.
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
GRAPHING AND COMPLETING TAB
LES OF VAL
UES *
..../...../20...
21
yx
=+
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
a y x 3= - b y x2= -
c y x 2= - + d y x 4= +
6 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight LinesYour Turn
a b 2y x 1= - +
Completing and graphing tables of values
2 Complete these tables of values for each given rule and then plot the graphs.
c y x2 2= - d y x23=
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
x -2 -1 0 1 2
y
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
y x3 1= -
2ISERIES TOPIC
7Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
Completing and graphing tables of values
3 Complete the T-charts below using the given rule and then plot the graphs.
a y x2 3= -
x y
-2
-1
0
1
2
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
b y x 5= - -
x y
-2
-1
0
1
2
c y x25= +
x y
-2
-1
0
1
2
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
8 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight Lines
Pattern of movement
The vertical (up & down) and horizontal (left & right) movement from one point to the next on a straight line follows a pattern. This remains the same (is constant) all along the line.
Describe the constant pattern of movement for this linear graph:
Describe the constant pattern of movement for the linear graph below:
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
Vertical and horizontal construction lines
Across 1 unit
• For vertical movement, up is positive (+ ) and down is negative (- )
• Because we always read left to right, the horizontal movement is always positive
Remember: Linear means straight line
1 2 3 4 5 -5 -4 -3 -2 -1
Up 1 unit
+ or - ?
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
Across 3 units
` Constant pattern of movement is: Down one unit (-1) vertically for every three units (+3) to the right horizontally
1 2 3 4 5 6 7 8 9 -5 -4 -3 -2 -1
Down 1 unit
` Constant pattern of movement is: Up one unit (+1) vertically for every one unit (+1) to the right horizontally
2ISERIES TOPIC
9Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
Pattern of movement
1 Describe the constant pattern of movement for each of the following linear graphs.
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
b
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Pattern of movement:
vertically for every across
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
Pattern of movement:
vertically for every across
Pattern of movement:
vertically for every across
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
d
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
Pattern of movement:
vertically for every across
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
Pattern of movement:
vertically for every across
f
Pattern of movement:
vertically for every across
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
a
c
e
..../.....
/20...
+
+
-
*
*
PATTERN OF MOVEMENT PATTE
RN OF MOVE
MENT
10 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight LinesYour Turn
Pattern of movement
2 Draw a linear graph that matches these constant patterns of movement.
1+ vertically for every 2+ horizontally 3+ vertically for every 1+ horizontally
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
2- vertically for every 1+ horizontally 3- vertically for every 2+ horizontally
1- vertically for every 4+ horizontally 0 vertically for every value horizontally
d
fe
a b
c
2ISERIES TOPIC
11Straight LinesMathletics Passport © 3P Learning
How does it work? Straight Lines
Horizontal and vertical lines
These lines are special because the pattern of movement changes in one direction only.
• The rule is: y=a constant number
• Graph is parallel to the x-axis
Horizontal Lines
Write the rule of the graph below:
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
` The y-value remains unchanged with a value of 2, regardless of the x-value` equation is: 2y =
Vertical Lines
• The rule is: x =a constant number
• Graph is parallel to the y-axis
The constant pattern of movement is:
0vertically for every value horizontally
Write the equation of the graph below:
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
` The x-value remains unchanged with a value of 3, regardless of the y-value` equation is: x 3=
The constant pattern of movement is:
0horizontally for every value vertically
Parallel lines never cross
12 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight LinesYour Turn
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Horizontal and vertical lines
1 Write the rule for these graphs.
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
1 2 3 4
1 2 3 4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
Rule:
Rule:
b
d
Rule:
Rule:
a
c
ba
2 Sketch the following graphs:
x 0=.y 1 5= -
..../.
..../2
0...
HORIZONTAL AND VERTICAL LINES
+ HORIZONTAL AND VERTICAL LINES
+
1 2 3 4
2ISERIES TOPIC
13Straight LinesMathletics Passport © 3P Learning
How does it work? Straight Lines
For the equation 2 2y x= + :
Intercepts
The exact point where a graph crosses an axis is called an intercept.
Intercepts are named using the axis they cross.
x-axis
y-axis
7
6
5
4
3
2
1
0
-1
-2
-3
y-intercept: The intercept on the vertical axisx-intercept: The intercept on the horizontal axis
` x-intercept = -1 and y-intercept=2
1 2 3 4 5 -5 -4 -3 -2 -1
x-axis
(i) Write down the intercepts of the graph for 2 2y x= +
Intercept points with axes will have at least one zero coordinate value.
(ii) Write down the coordinates for these intercepts
x-intercept ( , )1 0= -
y-intercept ( , )0 2=
Intercept points can be found by substituting zero into the rule for each variable.
(iii) Use the rule 2 2y x= + to find the intercepts another way
y-intercept is where 0x =
` y-intercept is ( , )0 2
x-intercept is where y 0=
` x-intercept is ( , )1 0-
substitute 0x = into the rule2
y x
y
y
2 2
0 2
2
#
= +
= +
=
20 2 2
y x
x
x
x
2
2 2
0 2 2
1
- -
= +
= +
= +
= -
substitute y 0= into the rule and solve for x
22
yx
=+
14 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight Lines
For the equation x y2 1 0+ - = :
Let’s try another one!
x-axis
5
4
3
2
1
0
-1
-2
-3x-intercept
` x-intercept = 1 and y-intercept21=
1 2 3 4 5 -5 -4 -3 -2 -1
(i) Write down the intercepts of the graph for x y2 1 0+ - =
Intercept points with axes will have at least one zero coordinate value.
(ii) Write down the coordinates for these intercepts
x-intercept (1 , 0)=
y-intercept (0 , )21=
Intercept points can be found by substituting zero into the rule for each variable.
(iii) Use the rule 2 1x y 0+ - = to find the intercepts another way
y-intercept is where 0x =
` y-intercept is (0 , )21
x-intercept is where y 0=
` x-intercept is (1 , 0)
22 1
y
y
y
y
2
0 2 1 0
2 1
21
' '
+ - =
=
=
=
11 0
x
x
x
x
1
2 0 1 0
1 0
1
#
+ +
+ - =
- =
- =
=
y-intercept
Remember:• For any point on the y-axis, x 0= ` y-intercept is found by substituting x 0= into the rule• For any point on the x-axis, y 0= ` x-intercept is found by substituting y 0= into the rule
y-axis
substitute 0x = into the rule
substitute y 0= into the rule
xy210
+-=
This way is called general form
2ISERIES TOPIC
15Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Intercepts
1 Write down the coordinates of the x and y intercepts for the following graphs.
(,)x-intercept =
b
d f
a c
e
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Horizontal & vertical lines
l Horizontal lines only intercept the y-axis
l Vertical lines only intercept the x-axis
y
x
y
x
y-intercept
x-intercept
(,)x-intercept =
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
16 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
How does it work? Straight LinesYour Turn
Intercepts
2 Write down the coordinates of the x and y intercepts for the graphs of the following straight line rules.
b
d
f
a
c
e
(,)x-intercept =
(,)y-intercept =
1y x= + 2 6y x= + (,)x-intercept =
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
2y x41= + 2 4y x= - (,)x-intercept =
(,)y-intercept =
(,)x-intercept =
(,)y-intercept =
y x3 2 6+ = 2x y5 20- = (,)x-intercept =
(,)y-intercept =
hint: look at part (iii) of the examples for the method
2ISERIES TOPIC
17Straight LinesMathletics Passport © 3P Learning
How does it work? Straight LinesYour Turn
Combining intercepts with the pattern of movement
Write down the coordinates of the other intercept point using the given information.hint: On the number plane, start from the given point and use the pattern of movement to help draw the line first
1 x-intercept is (1,0) , y-intercept is? Constant pattern of movement is 2+ vertically for every 1+ horizontally.
y-intercept is (0,1),x-intercept is? Constant pattern of movement is 1+ vertically for every 3+ horizontally.
3 y-intercept is (-2,0),x-intercept is? Constant pattern of movement is 2- vertically for every 4+ horizontally.
2
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
COMBO TIME * COMBO TIME *
COMBO T
IME *
..../...../20...
18 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Straight Lines
Draw the graph of 3 6y x- = by finding and plotting the intercepts first
Graphing using the intercepts
A linear graph can be drawn from a rule by plotting and joining the intercepts.
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1: Plot the intercepts
1 2 3 4 5 -5 -4 -3 -2 -1
For 3 6y x- = :
Where does it work?
7
6
5
4
3
2
1
0
-1
-2
-3
Step 2: Draw a double-arrowed line through both points
Step 3: Write the rule along the line
1 2 3 4 5 -5 -4 -3 -2 -1
3 0 6
y x
y
y
3 6
6
#
- =
- =
=
y-intercept is where x 0=
6 3
y x
x
x
x
3 6
0 3 6
2
'
- =
- =
= -
= -
x-intercept is where 0y =
(0 , )6
( , )2 0-
x-axis
y-axis
x-axis
y-axis
` y-intercept is (0 , )6
` x-intercept is ( , )2 0-
substitute 0x = into the rule
substitute y 0= into the rule
36
yx
-=
2ISERIES TOPIC
19Straight LinesMathletics Passport © 3P Learning
Where does it work? Straight Lines
Draw the graph of y x2 4= - + by finding and plotting the intercepts first
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1: Plot the intercepts
1 2 3 4 5 -5 -4 -3 -2 -1
For y x2 4= - + :
7
6
5
4
3
2
1
0
-1
-2
-3
Step 2: Draw a double-arrowed line through both points
Step 3: Write the rule along the line
1 2 3 4 5 -5 -4 -3 -2 -1
y x
y
y
2 4
2 0 4
4
#
= - +
= - +
=
y-intercept is where x 0=
y x
x
x
x
2 4
0 2 4
2 4
2
= - +
= - +
=
=
x-intercept is where 0y =
(2 , 0)
(0 , )4
x-axis
y-axis
x-axis
y-axis
y
x24
=-+
` y-intercept is (0 , )4
` x-intercept is (2 , 0)
substitute 0x = into the rule
substitute y 0= into the rule
20 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Where does it work? Straight LinesYour Turn
Graphing using the intercepts
Graph each of the following rules using the intercepts method.
(,)x-intercept =
(,)y-intercept =
y x 3+ =1
(,)x-intercept =
(,)y-intercept =
y x3 6+ =2
(,)x-intercept =
(,)y-intercept =
y x4 4- =3
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
..../...../20...
* GRAP
HING USING THE INTERCEPTS
* GRAPHING USING T
HE I
NTERCEPTS
2ISERIES TOPIC
21Straight LinesMathletics Passport © 3P Learning
Where does it work? Straight LinesYour Turn
Graphing using the intercepts
Graph each of the following rules using the intercepts method.
(,)x-intercept =
(,)y-intercept =
y x2 4 8- =4
(,)x-intercept =
(,)y-intercept =
y x2 1= -5
(,)x-intercept =
(,)y-intercept =
y x4 5 10+ =6
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
7
6
5
4
3
2
1
x-axis
y-axis
22 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Where does it work? Straight Lines
Slope
The slope of a line is the special name given to the constant pattern of movement written as a fraction.
Slope =Vertical movement Also called slope or Rise Horizontal movement Run
If the line slopes up the slope is positive.
Positive
What is the slope of the line graphed below?
Pattern of movement is: 1+ (up) vertically for every 1+ across
` Slope 11 1= =
Negative
What is the slope of the line graphed below?
Pattern of movement is: 1- (down) vertically for every 3+ across
` Slope 131
3= - = -
If the the line slopes down the slope is negative.
x-axis
y-axis
0
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1Across 1 unit
Up 1 unit
sloping up =positive slope
x-axis
y-axis
0
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1Across 3 units
Down 1 unit
sloping down =negative slope
-1
-2
-1
-2
2ISERIES TOPIC
23Straight LinesMathletics Passport © 3P Learning
Where does it work? Straight Lines
Using the slope formula on horizontal and vertical lines gives special results.
Horizontal lines have a slope of 0.Zero
What is the slope of the horizontal line graphed below?
Pattern of movement is: 0 vertically for every horizontal value
Undefined
What is the slope of the vertical line graphed below?
The slope for vertical lines cannot be defined. Let’s see why with this example.
` Slope =0 =
any x-value
0
Pattern of movement is: every vertical value for 0 horizontally
Slope =any y-value =
0
undefined
because you cannot divide by 0
No vertical change
x-axis
y-axis
0
7
6
5
4
3
2
1
1 2 3 4 5 -5 -4 -3 -2 -1
any values across
-1
-2
Any values vertically
0
7
6
5
4
3
2
1
-5 -4 -3 -2 -1x-axis
y-axis
1 2 3 4 5
-1
-2
-3
No horizontal change
As the value of the slope gets bigger, the graph gets steeper.
`
24 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Where does it work? Straight LinesYour Turn
Slope
1 (i) What is the slope of the lines graphed below? (ii) Circle the graph for each pair that has the steeper slope.
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
b
d
a
c
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Slope =
Slope =
Slope =
Slope =
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Slope =
Slope =
Slope =
Slope =
SLOPE * SLOPE * SLOPE * SLOPE
* SLOPE * S
LOPE
*
..../...../20...
* GR
ADIEN
T * GRADIENT *
GRADIENT *
GRADIENT *
GRADIENT
RARR
DAA Rise (+)
Run
2ISERIES TOPIC
25Straight LinesMathletics Passport © 3P Learning
Where does it work? Straight LinesYour Turn
Slope
2 What is the slope of these lines?
ba
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
d
f
c
e
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
x-axis
y-axis
Slope =
0
26 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Where does it work? Straight LinesYour Turn
Slope
3 Sketch a line that has the given slope on the number planes below.
a
c
e
g
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope21= b
d
f
h
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope 3=
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope 2= - 4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope32= -
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope 0= 4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope = undefined
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope41= -
4
3
2
1
1 2 3 4-4 -3 -2 -1
-1
-2
0
x-axis
y-axisSlope52=
2ISERIES TOPIC
27Straight LinesMathletics Passport © 3P Learning
Straight Lines
For the rule 2 1:y x= +
Graphing straight lines using the slope-intercept rule
The slope and y-intercept can be found easily from a straight line rule when written a special way.
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1: Plot the y-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
(i) Write the slope and y-intercept for the graph of 2 1y x= +
7
6
5
4
3
2
1
0
-1
-2
-3
Step 3: Draw a double arrowed line through the points
1 2 3 4 5 -5 -4 -3 -2 -1x-axis
y-axis
x-axis
y-axis
What else can you do?
2 1y x= +
Slope 2m12= =++^ h y-intercept 1b = +^ h
` Graph of the rule moves 2+ vertically for every 1+ horizontally and passes through the y-axis at (0,1)
(ii) Graph 2 1y x= +
2+
Step 2: Use the slope to plot a second point
Step 4: Write the rule along the line
y mx b= +
the number in front of x (m) is the slope the constant term (b) is the y-intercept
1+
All whole numbers can be written as fractions
21
yx
=+
The number in front of the x in the rule is called the coefficient of x
28 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
What else can you do? Straight Lines
For the rule 4:y x32= - +
Here is another example with a fraction written in the rule:
7
6
5
4
3
2
1
0
-1
-2
-3
Step 1: Plot the y-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
(i) Write the slope and y-intercept for the graph of 4y x32= - +
7
6
5
4
3
2
1
0
-1
-2
-3
Step 3: Draw a double arrowed line through the points
1 2 3 4 5 6 7
-5 -4 -3 -2 -1
( , )2 0-
x-axis
y-axis
x-axis
y-axis
Slope m32
32= - =+-^ h y-intercept b 4= +^ h
` Graph of the rule moves 2- vertically for every 3+ horizontally and passes through the y-axis at (0,4)
(ii) Graph 4y x32= - +
2-
Step 2: Use the slope to plot a second point
Step 4: Write the rule along the line
4y x32= - +
Rules like 4y x32= - + , can also be written as y x
32 4= - +
x x32
32- = -
3+
2x
y
34
= -
+
Sometimes we need to extend an axis to find an intercept
2ISERIES TOPIC
29Straight LinesMathletics Passport © 3P Learning
What else can you do? Straight LinesYour Turn
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Graphing straight lines using the slope-intercept rule
1 (i) Write the slope and y-intercept for each straight line rule. (ii) Sketch each graph.
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Slope = b
d
a
c
fe
y-intercept =
Slope =
y-intercept =
Slope =
y-intercept =
Slope =
y-intercept =
2 1y x= -
y x4 2= -
2y x 0= - +
0 1 2 3 4 5 -5 -4 -3 -2 -1x-axis
y-axis7
6
5
4
3
2
1
-1
-2
-3
0 1 2 3 4 5 -5 -4 -3 -2 -1x-axis
y-axis7
6
5
4
3
2
1
-1
-2
-3
Slope =
y-intercept =
Slope =
y-intercept =or 2y x= -
y x 2= +
y x 3= - +
4y x 2= +
..../...../20...
* GRAP
HING STRAIGHT LINES USING THE GRADIENT-INT
ERC
EPT RULES
y = m x + b
GRAPHING STRAIGHT LINES USING TH
E SL
OPE-INTERCEPT
RULE *
30 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
What else can you do? Straight LinesYour Turn
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Graphing straight lines using the slope-intercept rule
2 (i) Write the slope and y-intercept for the graph of each straight line rule. (ii) Sketch each graph.
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
Slope = b
d
a
c
fe
y-intercept =
Slope =
y-intercept =
Slope =
y-intercept =
Slope =
y-intercept =
1y x21= +
1y x41= -
y x52= + Slope =
y-intercept =
Slope =
y-intercept =
or y x41= -
y x31 3= - -
y x32 1= - +
.y x31 1 5= +
Be careful finding the second point
x-axis
y-axis
0
4
3
2
1
1 2 3 4 5 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
-4 -3 -2 -1
-1
-2
-3
-4
1 2 3 4
2ISERIES TOPIC
31Straight LinesMathletics Passport © 3P Learning
What else can you do? Straight Lines
Finding the slope-intercept rule from the graph
We reverse the method of graphing to find the rule of a linear graph in slope-intercept form.
Find the rule of these graphed lines
` y x2 6= +
You might need to look a little more closely to find another point with clear coordinates to use for slope.
x-axis
y-axis7
6
5
4
3
2
1
0
-1
-2
-3
Step 1: Read the y-intercept
1 2 3 4 5 -5 -4 -3 -2 -1
x-axis
y-axis
7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1
6+ Step 2: Determine the slope in simplest form
y mx b= +
y-intercept b 3= -^ h Slope 5m25
2=+- = -^ h
y-intercept b 6= +^ h Slope m36
12 2=
++ =
++ =^ h
` y x25 3= - -
y mx b= +
2+
(-2,2) is a point with clear coordinates
y-intercept 3= - Step 1: Read the y-intercept
Step 2: Determine the slope in simplest form
5-
3+
32 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
What else can you do? Straight LinesYour Turn
Finding the slope-intercept rule from the graph
1 Find the rule of the line for each graph below:
b
d
f
a
c
e
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
..../...../20...
* FIND
ING T
HE GRADIENT-INTERCEPT RULE FROM THE GR
APH
y = x - l
FINDING THE SLOPE-INTERCEPT RUL
E
FROM THE
GRAPH
*
2ISERIES TOPIC
33Straight LinesMathletics Passport © 3P Learning
What else can you do? Straight LinesYour Turn
Finding the slope-intercept rule from the graph
2 The y-intercept, slope or both for each of these trickier questions are not whole numbers.
b
d
f
a
c
e
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 -4 -3 -2 -1
-1
-2
-3
-4
x-axis
y-axis
0
4
3
2
1
1 2 3 4 5 6 7 8 -4 -3 -2 -1
-1
-2
-3
-4
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
y-intercept =
Slope =
` Rule:
34 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
What else can you do? Straight LinesYour Turn
Remember me?
Graphing straight lines using the slope-intercept rule
3 The robot standing on the x-axis at point A needs to get to point B on the y-axis. The solar panels only have enough stored energy to travel the shortest straight line path. Write down the rule of the line the robot needs to follow to get from A to B.
hint: determine the y-intercept and slope and go from there.
7
6
5
4
3
2
1
0
-1
-2
-3
1 2 3 4 5 -5 -4 -3 -2 -1x-axis
y-axis
Rule of the line for the robot to follow:
4 Two keys are needed to unlock a treasure chest at Canary Cove. Birdy Town and Flutterton have one key each. Two homing pigeons, one from each town, are sent to Canary Cove. Use the map below to find the straight line rule each pigeon needs to fly along to get to Canary Cove.
hint: draw the flight paths of each pigeon first
Rule of the line for the pigeon from Birdy Town:
Rule of the line for the pigeon from Flutterton:
A
B
x-axis
y-axis
1 2 3 4 5 6 7 8-8 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
7
6
5
4
3
2
1
Canary Cove
Flutterton
Birdy Town
2ISERIES TOPIC
35Straight LinesMathletics Passport © 3P Learning
What else can you do? Straight LinesYour Turn
Reflection Time
Reflecting on the work covered within this booklet:
1 What useful skills have you gained by learning about straight lines?
2 Write about one or two ways you think you could apply straight lines to a real life situation.
If you discovered or learnt about any shortcuts to help with straight lines or some other cool facts, jot them down here:
3
36 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Cheat Sheet Straight Lines
Here is a summary of the important things to remember for straight lines
Graphing from a table of values
The x-value is always written first in coordinates: (x-value , y-value) = (horizontal value , vertical value)
Pattern of movement
The pattern of vertical and horizontal movement for a straight line does not change (is constant).
Horizontal and vertical lines
Horizontal: rule: y =a constant value Vertical: rule: x= a constant value
Intercepts
To graph a rule using the intercepts, plot the intercept points first and then join with a straight line.
Slope Slope = Vertical movement =rise
Horizontal movement run
= positive slope, = negative slope, = zero slope, = slope undefined The larger the value of the slope, the steeper the line: = large slope value = small slope value
Slope-intercept rule
coefficient of x (m) is the slope the constant term (b) is the y-intercept
To draw a graph using this rule: • plot the y-intercept • use the slope to plot a second point • join the two points with a straight line (putting arrows on each end).
To find the rule from a graph: • write down the y-intercept • find another point on the line with clear coordinates • find the slope between this point and the y-intercept • put the y-intercept and slope values into the slope-intercept rule.
y mx b= +
x-intercept (where y =0) y-intercept (where x =0)
x-axis
y-axis
2ISERIES TOPIC
37Straight LinesMathletics Passport © 3P Learning
Straight Lines Notes
38 Straight LinesMathletics Passport © 3P Learning
2ISERIES TOPIC
Straight Lines Notes
www.mathletics.com
Straight Lines