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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


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


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




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= +


2ISERIES TOPIC


a b 2y x 1= - +


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




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


2ISERIES TOPIC


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



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





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



x-axis

y-axis

0

4

3

2

1

-4 -3 -2 -1

-1

-2

-3

-4



f



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


2ISERIES TOPIC


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



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


2ISERIES TOPIC


x-axis

y-axis

0

4

3

2

1

1 2 3 4 -4 -3 -2 -1

-1

-2

-3

-4


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



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

=+


2ISERIES TOPIC


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



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 =


2ISERIES TOPIC


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



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...


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


` x-intercept is ( , )2 0-



36

yx

-=

2ISERIES TOPIC


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


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

=-+


` x-intercept is (2 , 0)




2ISERIES TOPIC

Where does it work? Straight LinesYour Turn


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




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


2ISERIES TOPIC


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



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.

`


2ISERIES TOPIC


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



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


2ISERIES TOPIC


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


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


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


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


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


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


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 *


2ISERIES TOPIC


x-axis

y-axis

0

4

3

2

1

1 2 3 4 -4 -3 -2 -1

-1

-2

-3

-4


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


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+


2ISERIES TOPIC



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




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:


2ISERIES TOPIC


Remember me?


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



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


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: 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


Straight Lines Notes


2ISERIES TOPIC

Straight Lines Notes

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Straight Lines

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