y

9

COORDINATE GEOMETRY How do I measure the steepness of the slope?

Am I past the midpoint yet?

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How far is it up this hill?

y

Pr

7 6 5 4 3 2 1

Contents

Pa g

9:04

Number plane review The midpoint of an interval The gradient of an interval Investigation 9:03 Gradients and building The length of an interval GeoGebra activity 9:04 The midpoint, gradient and length of an interval Graphing straight lines Investigation 9:05 x- and y-intercepts GeoGebra activity 9:05 Graphing lines by plotting points

e

9:01 9:02 9:03

9:05

−3 −2 −1 0 −1

x

You are here.

1 2 3 4 5x

Horizontal and vertical lines The gradient–intercept form of a straight line: y = mx + b Investigation 9:07 What does y = mx + b tell us? GeoGebra activity 9:07 The gradient–intercept form of a line Fun spot 9:07 Why did the banana go out with a fig? 9:08 Non-linear graphs GeoGebra activity 9:08 Non-linear graphs Maths terms, Diagnostic test, Assignments 9:06 9:07

Syllabus references (See pages x–xiii for details.)

Number and Algebra Selections from Linear Relationships [Stages 5.1, 5.2] • Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software. (ACMNA294) • Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software. (ACMNA214) • Sketch linear graphs using the coordinates of two points. (ACMNA215) • Interpret and graph linear relationships using the gradient-intercept form of the equation of a straight line. (NSW) Selections from Non-linear Relationships [Stages 5.1, 5.2] • Graph simple non-linear relations, with and without the use of digital technologies. (ACMNA296)

Working Mathematically • Communicating • Problem Solving • Reasoning • Understanding • Fluency

ASM9SB5-2_09.indd 246

9/07/13 9:25 AM

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9:01 Number plane review

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The French mathematician René Descartes first introduced the number plane. He realised that using two sets of lines to form a square grid allowed the position of a point in the plane to be recorded using a pair of numbers or coordinates. The basic ideas behind his system should have been met in Years 7 and 8. Check your knowledge by doing the prep quiz below.

e

PREP QUIZ 9:01

2 Name this axis

7

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Give 7 coords.

4 Give coords.

6 5 4

How long is CD ? 10

D

A

3 Give coords. 2 8 1

−4 −3 −2 −1 0 −1

C

6 Give coords.

−2

Keep doing this test until you get all questions right.

9 How long is AB ?

B

Name this axis 1

1

3

2

3

4

Give 5 coords.

Give name and coordinates.

The x-reading always goes first.

9 Coordinate geometry


247

9/07/13 9:25 AM

Exercise 9:01 1

2

Foundation worksheet 9:01 Coordinates

Write the letter naming the point with these coordinates. a (3, 2) b (-2, 4) c (3, 4) d (2, -1) e (0, 3) f (-2, -2) g (0, 0) h (-3, 0) i (3, -2)

y K

J

3 F

R

U

G

Write the coordinates of these points. a A b B c C d D e E f F g G h H i I j J k K l L

A

2

P

1

E

O

−4 −3 −2 −1 0 1 −1 H S

X

L

M

2

T

4 x

3

D

−2 −3 Q

C

V

I W

−4

Using the number plane above, write the points: a with a y-coordinate of 2 b with an x-coordinate of -3.

4

Plot each set of points on a number plane. What shapes are formed if the points are joined in the given order? a (-3, 1), (3, 1), (3, -1), (-3, -1), (-3, 1) b (-1, -1), (0, 4), (1, -1), (-1, -1) c (2, 3), (6, 3), (5, 0), (1, 0), (2, 3) d (1, 3), (3, 3), (4, 1), (-2, 1), (1, 3)

5

Find the distance between: a (2, 0) and (5, 0) c (-1, 0) and (1, 0) e (−5, 0) and (-1, 0) g (3, 0) and (-1, 0) i (0, 5) and (0, -2) k (-2, -1) and (1, -1) m (3, 1) and (3, 4)

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3

e

Pr

b (0, 1) and (0, 4) d (0, -3) and (0, 1) f (0, −4) and (0, -2) h (2, 0) and (-2, 0) j (1, 2) and (4, 2) l (4, 3) and (-2, 3) n (-2, -3) and (-2, 1)

7

One, two, three units.

y

5

4

(1, 4)

3 2 1

(1, 1)

0

1 2 3

In which quadrant does each point fall? a (3, 2) b (-1, 2) c (-2, -2) d (1, -2) e (-3, -2) f (1, 1) g (-1, -2) h (-3, 1) i (3, -1) j (-1, 1) k (-2, -1) l (1, -1) m (2, 2) n (2, -2) o (-2, 2)

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6

248

4 N

B

What can you say about the coordinates of a point that lies on: a the x-axis b the y-axis?

x y 3 2nd quadrant

2 1

−3 −2 −1 −1 3rd −2 quadrant −3

1st quadrant 1

2

3 x

4th quadrant

Australian Signpost Mathematics New South Wales 9 Stages 5.1–5.2


9/07/13 9:25 AM

8 a Write

the letter naming the points: i (-1⋅2, -1⋅6) ii (3⋅4, -1⋅2) iii (1⋅4, -2⋅6) iv (2⋅4, -1⋅8) b Write the coordinates of: i C ii A iii E iv D

y

F

2

A 1

−1

C

0

1

2

3

4

−1

x

B

H

G

−2

E −3

9

oo fs

D

Write the coordinates of the points A, B, C and D on each of the following lines. a b c y y y A 4 3

−2 −1 0 −1

B

2

C D

−1 0 −1

1

2

3

10 a The

−2

4 x

A −3

1

2

B

3 x

B

0

1

x

Pa g

y

-2

Line C x y

-2

-4

-2

2

3

y

A

4 3

2

1

y

x

D

2

0

−4 −3 −2 −1 0 −1

B

Line D

-1

-1

0

6 x

4

−6

−4

e

-3

2

−4

x-coordinates for some of the points on the lines A to D are given. Find the missing y-coordinates. Line A Line B x

4

−4 −2 0 2 −2 C

C

Pr

1

A

D

1

2

2

3

4 x

D

−2 −3

4

y

1

C

−4

b If line C was extended it would pass through the points

(5, a) and (−4, b). What are the values of a and b?



249

9/07/13 9:25 AM

11 a Write

the coordinates of three points that lie on the line AB between the points A and C. b Write the coordinates of three points on the line ED between F and D. c If the line ED was extended it would pass through the points (2, a), (-2, b) and ( -2 21 , c ) . Find the values of a, b and c.

y 3

E

C 2

1 A −3

−2

−1

0

1

−1

B

2

3x

F

−2

D

−3

points on a line can be described by an equation. Which of the lines A to F are best described by the following equations. a y = 2x b y = x − 2 c 2x + y = 0 d y = x e x + y = 3 f 2x + y = 2 y y y 4 3 2 1

y

4 3 2 1

D

1 2 3x

13 Which

1 2 3x

−3 −2 −1 −1 −2 C −3

y

4 3 2 1

−3 −2 −1 −1 −2 −3

Pa g

−3 −2 −1 −1 −2 −3

−3 −2 −1 −1 −2 B −3

e

1 2 3x

4 3 2 1

Pr

−3 −2 −1 −1 −2 −3

4 3 2 1

A

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12 The

1 2 3 x

y

4 3 2 1 −3 −2 −1 −1 −2 −3

E

1 2 3x

1 2 3 x

F

of the lines A, B, C or D in Question 10 could be described by the following equations? b x + y = 4 c 2x + y + 2 = 0 d x − 2y + 2 = 0

a x − y = 2

Roger has four different pizza toppings. How many different pizzas could be made using: • one topping • two toppings • any number of toppings?

250



9/07/13 9:25 AM

9:02 The midpoint of an

interval

The number plane is the basis of coordinate geometry, which is an important branch of mathematics with many technological applications. Three of the basic tools of coordinate geometry are the midpoint, gradient and length of an interval joining two points.

PREP QUIZ 9:02 4 + 10 −2 + 4 =? 2 =? 2 2 4 What is the average of -2 and 4? 3 What is the average of 4 and 10? 6 What number is half-way between 5 What number is half-way between 4 and 10? -2 and 4? x −4 −3 −2 −1 0 1 2 3 4 5 6 7 x 1 2 3 4 5 6 7 8 9 10 11 12

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1

8 What number is half-way 7 What number is half-way y 6 between 1 and 5? between -1 and 3?

1+ 5 =? 2

10

e

9

Pr

5 4 3 2 1 0

Pa g

• The midpoint of an interval is the half-way position. If M is the midpoint of AB then it is half-way between A and B.

y

4 3 2 1 0 −1 −2

−1 + 3 =? 2

y

8 7 6 5 4 3 2 1

0

B (10, 7)

If M is the midpoint of AB then AM = MB

M (p, q) A (4, 3) 1 2 3 4 5 6 7 8 9 10 x

Consider the x-coordinates. 7 is half-way between 4 and 10.

The average of 4 and 10 is 7.

4 + 10 =7 2

p=7

The average of 3 and 7 is 5.

3+7 =5 2

q=5

Consider the y-coordinates. 5 is half-way between 3 and 7. Hence, M is the point (7, 5). 9 Coordinate geometry


251

9/07/13 9:25 AM

• Drawing a right-angled triangle with AB as the hypotenuse is helpful when finding the midpoint. By drawing a vertical line from the upper point and a horizontal line from the lower point, a triangle is formed. Finding the middle of the horizontal and vertical sides will give the x- and y-coordinates of the midpoint. When a diagram is not suitable the averaging method gives the same result. −3 + 7 4 + −2  = (2, 1) , Midpoint =   2 2 

A (−3, 4) 3

y 4 2 (2, 1)

−4 3

−2

0

2

4

−2 5

8 x

6

5

B (7, −2)

WORKED EXAMPLES 1 Find the midpoint of the interval joining (2, 6) and (8, 10).

Solutions x1 + x 2 y1 + y 2  , 2 2  2 + 8 6 + 10  = ,  2 2 

x1 + x 2 y1 + y 2  , 2 2  −3 + 4 5 + −2  = ,  2 2 

2 Midpoint =  

= ( 21 , 23 ) or ( 21 , 1 21 )

e

= (5, 8)

2 Find the midpoint of interval AB, if A is the point (-3, 5) and B is (4, -2).

Pr

1 Midpoint =  

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If M is the midpoint of the interval AB, its coordinates are found by taking the averages of the x- and y-coordinates of the points A and B.

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Exercise 9:02 1

Use the graph to find the midpoint of each interval. a y b y 5 4 3 2 1

−1

d

(1, 4)

(5, 2)

1 2 3 4 5x

e

y

−2 −1 −1 (−1, −1) −2 −3 −4 −5 −6

252

Foundation worksheet 9:02 Midpoint

1 2 3 x

(3, −5)

2 1

(−3, 1)

−3 −2 −1 −1 −2 −3

c

1 2 3 x

(1, −3)

(−2, 4) y 4 3 2 1 −2 −1 −1 −2

1 2 3 x (2, −2)

y 3 2 1 −2 −1 −1 −2 (−1, −2)

(4, 2)

1 2 3 4 x



9/07/13 9:25 AM

2

Use the graph to find the midpoints of the intervals: a AB b CD c GH d EF e LM f PQ g RS h TU i VW

y 8

H

C D

6

U B

4 2

T −6

−4

A 0

−2

2

F

−2

G

E −4

R

L

S

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M

−8

V

P

c A f A i A l A o A r A

Find the midpoint of the interval joining: a (-3, -3) and (2, -3) b (8, -1) and (7, -1) d (6, −7) and (−7, 6) e (0, −4) and (−4, 0) g (111, 98) and (63, 42) h (68, -23) and (72, -29) j (−82, 41) and (−51, 63) k (−71, 42) and (23, -17)

c (5, 5) and (5, −5) f (6, −6) and (5, −5) i (400, 52) and (124, 100) l (49, −7) and (19, -22)

Pr

Find the midpoint of each interval AB if: a A is (2, 4), B is (6, 10) b A is (1, 8), B is (5, 6) d A is (0, 0), B is (4, 2) e A is (4, 8), B is (0, 0) g A is (3, 4), B is (8, 1) h A is (4, 0), B is (1, 7) j A is (1, 1), B is (5, 6) k A is (7, 1), B is (2, 5) m A is (0, 0), B is (−4, 2) n A is (−8, −6), B is (0, 0) p A is (-2, −6), B is (4, 2) q A is (-1, 0), B is (5, 4)

5 a i

e

4

Pa g

Find the midpoint of AC. ii Find the midpoint of BD. iii Are the answers for i and ii the same? iv What property of a rectangle does this result demonstrate?

Q

W

−6

3

6 x

4

is (4, 1), B is (8, 7) is (0, 0), B is (6, 6) is (7, 2), B is (4, 7) is (3, 2), B is (7, 5) is (0, 0), B is (0, -10) is (-3, −7), B is (1, 3)

y 4 A(1, 3) B (4, 3) 3 2 1 D (1, 1) C (4, 1) 0 1 2 3 4 5 x

b If (4, 6) and (2, 10) are points at opposite ends of a diameter

y

of a circle, what are the coordinates of the centre? c i Find the midpoint of AC. ii Find the midpoint of BD. iii Are the answers for i and ii the same? iv What property of a parallelogram does this result demonstrate?

7 6 5 4 3 2 1 0 −1

C (7, 7) B (3, 5)

D (5, 1) 1 2 3 4 5 6 7 x

A (1, −1)



253

9/07/13 9:25 AM

6 a

If the midpoint of (3, k) and (13, 6) is (8, 3), find the value of k. b If the midpoint of (c, d) and (10, 2) is (8, 5), find the value of c and d. c If the midpoint of (x, 1) and (7, y) is (9, 6), find the value of x and y. d If the midpoint of (-2, −6) and (b, c) is (−5, 1), find the value of b and c. e The midpoint of AB is (7, -3). Find the value of d and e if A is the point (d, 0) and B is (-1, e). f The midpoint of AB is (−6, 2). If A is the point (4, 4), what are the coordinates of B? g A circle with centre (3, 4) has a diameter of AB. If A is the point (-1, 6) what are the coordinates of B?

PREP QUIZ 9:03 y

0

F

(15, 6)

D

A 1

B (1, 1) 2

C

(4, 2) 3

4

5

6

7

(10, 4)

E

(7, 1) 8

9

10

11

(13, 1)

12 13

Pr

6 5 4 3 2 1

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9:03 The gradient of an interval

14

(17, 1)

15

16

G

17 x

1 Which is steepest, AB or EF?

If I travel from left to right, between which two points am I travelling upwards: for the second time 4 for the last time? 2 for the first time 3

e

If I travel from left to right, between which two points am I travelling downwards: for the second time 7 for the last time? 5 for the first time 6

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Is the hill sloping up, down, or not at all, at these points? 9 G 10 F 8 A

The gradient slope of an interval (or line) is a measure of how steep it is.

Negative gradient

254

Steep (positive gradient)

Steeper



9/07/13 9:25 AM

If we move from left to right: • an interval going down is said to have a negative gradient (or slope) • an interval going up is said to have a positive gradient (or slope) • a horizontal interval (not going up or down) has zero gradient. We find the gradient of an interval by comparing its rise (change in y) with its run (change in x). y 2 • So, a gradient of 21 rise Gradient = means that for every run 1 run of 2 there is a rise change in y of 1 (or for every 2 you = change in x −2 −1 go across, you go up 1). 2

1 1 2

1

−1

2

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Finding the gradient of a line

1

1 Select any two points on the line. 2 Join the points and form a right-angled triangle by drawing a vertical line from the higher point and a horizontal side from the lower point. 3 Find the change in the y-coordinates (rise) and the change in the x-coordinates (run). 4 Use the formula above to find the gradient.

A

2 4 x

3

B

rise

Pr

run

WORKED EXAMPLES

Use the points A and B to find the gradient of the interval AB in each case. 2 y 3 y 1 y A (1, 6) 6 B (4, 5) 5 4 3 4 2 A (2, 1) 2 1

Pa g

e

6 5 4 3 2 1

0 −1

1 2 3 4 5 x

B (5, 3)

A (2, 2)

1 3

0 −1

1 2 3 4 5 x

6 5 4 3 5 2 1

B (3, 1) 2

1 2 3 4 5 x

0 −1

Solutions

1 Gradient

change in y change in x up 4 = across 2 4 = 2 =2

change in y change in x up 1 = across 3 1 = 3

2 m =

=

m is used for ‘gradient’

3 m =

change in y change in x

down 5 across 2 −5 = 2 = −2 21 =



255

9/07/13 9:25 AM

Exercise 9:03 1

Foundation worksheet 9:03 Gradients

State whether each interval below has a positive or negative gradient. a b c y y y

d

y

x

e

f

y

g

y

d

3 3 A (1, 2) 1 2 3 4 5 x

A (1, 3)

−1

3

6 5 4 3 2 1

4

1 2 3 4 5 x

−1 0

Find the gradient of the interval AB. a y b B (5, 6) 6 5 4 3 2 1

−1

256

y

B (5, 1)

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3 2 2 1

A (1, 4)

1 2 3 4 5 x

B (3, 6)

c

5

1

1 2 3 4 5 x

A (2, 6)

e

6 5 4

−1 0

e

y

y 6 5 4 3 2 A (2, 1) 1

Pr

−1 0

B (4, 5)

x

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Find the gradient of the interval AB. a y b 6 5 4 3 2 1

y

x

x

2

x

x

x

−1 0

f

4

B (4, 2) 2

1 2 3 4 5 x

3 2 1

−3 −2 −1 −1 A (−2, −3) −2 −3

y 6 5 4 3 2 1 −1 0

c

y

y 6 5 B (5, 4) 4 1 4 A (0, 3) 5 2 1

B (2, 3) 1 2 3 x

1 2 3 4 5 x

A (2, 6)

5

B (3, 1) 1 1 2 3 4 5 x

y

1 A (1, 0)

−3 −2 −1 −1 −2 B (−3, −2) −3 −4 −5

1 2 3 x



9/07/13 9:25 AM

d

e A (−2, 2)

y 6 5 4 A (−2, 3) 3 2 1

1 2 3 4 x

−2 −1

−3 −2 −1 −1 −2 −3

B (1, −4)

d

6 5 C 4 3 2 1

D

1 2 3 4 5 x

−1 0

e

y 6 5 4 3 2 1

C

−3 −2 −1 0

c

6 5 4 C3 2 1

D

1 2 3 4 5 x

−1 0

f

y

−1 0

C −1

1 2 3 x

−2 −3 −4

1 2 3 4 5 x

y

3 2 1

1 2 3 4 5 x

D

1

1

2

3

4 x

Pa g

−2 −1 0 −1

e

2

−3 −2 −1 0 −1 −2 C −3

−2

1 2 3 x

y 4

4

2

2

−4

D

6

−4

−2

6

D

In each of the following, use any two points on the line to find its slope. a b c y y 3

B (3, −2)

y

2 1

D

1 2 3 x

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−1 0

5

1 2 3 x

Use the points C and D to find the slope of the line. a y b y 6 5 C 4 3 2 1

y A (−3, 2) 3 2 1

Pr

4

−3 −2 −1 −1 −2 −3 −4

B (2, 1)

f

y 2 1

0 −2

2 x

−2

0

2 x

−2 −4

Find the slope of the line that passes through each pair of points. a (1, 0) and (3, 4) b (-2, 1) and (0, 3) c (1, -3) and (3, -2) d (-1, 4) and (1, 2) e (-2, -2) and (0, −5) f (0, 4) and (2, -2)



257

9/07/13 9:25 AM

INVESTIGATION 9:03

GRADIENTS AND BUILDING

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• Engineers, architects and builders need to understand and calculate gradients when designing and constructing ramps, driveways and roads. • They usually refer to a gradient in ratio form. e.g. A driveway can have a maximum gradient of 1 in 4 (1 : 4). This means a run of 4 units will produce a rise (or fall) of 1 unit. We would say the gradient is 41 .

1 The graph below shows three slopes. Which of the slopes A, B or C is: a 1 in 4 b 1 in 6 c 1 in 12?

Pa g

e

3

0

2

4

Slope C

2

Slope B 1

Slope A

6

8

10

12

2 A slope is to be 1 in 4 (1 : 4). What would the run have to be if the rise is: a 2 m b 1⋅4 m c 3⋅2 m?

3 Use the graph above (or ratio) to find the rise for a: a 1 in 6 slope, if the run is 9 m b 1 in 12 slope, if the run is 6 m? 4 A builder has to construct a wheelchair ramp from the front of the house to the front of the property, a run of 10 m. He calculates the fall to be 0⋅8 m. Building regulations state that the maximum gradient of a ramp is 1 in 14. Is it possible for the builder to build a straight ramp? If not, how can he build it and satisfy the regulations?

258



9/07/13 9:25 AM

9:04 The length of an interval PREP QUIZ 9:04 1 Which of the following is the correct statement of Pythagoras’ theorem for the triangle shown? A a2 = b2 + c2 B b2 = a2 + c2 C c2 = a2 + b2

c

For Questions 2 to 4 use Pythagoras’ theorem to find the value of d. 3 4 2 12 cm d cm

3 cm

a dm

2m

5 cm

b

d cm

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4m

4 cm

6 Distance AB = … units. 7 Distance AB = … units. 5 Distance AB = … units. y A B y −1 0 1 2 3 4 5 6 7 8

x

4 3 A 2 1

2 B 1 0 −1 A −2 −3

−1 0 −1

B

1 2 3 4 5 x

1

4

B

3 A

4

A 3

B

e

1 2 3 4 5 x A

B −1

4 x

2

Pa g

−1 0 −1 −2 −3 −4

Pr

8 Distance AB = … units. 9 Find the distance AB. 10 Find the distance AB. y y y

3

x

Pythagoras’ theorem can be used to find the distance between two points on the number plane. By drawing a right-angled triangle, we can use Pythagoras’ theorem to find the distance between any two points on the number plane. y y y B

B

A

A x

B

BC A

C x

C AC

x

c 2 = a 2 + b2 AB 2 = AC 2 + BC 2



259

9/07/13 9:25 AM

WORKED EXAMPLES 1 Find the distance between the points (1, 2) and (4, 6).

2 If A is (4, 5) and B is (-2, 2) find the length of AB.

Solutions 1

2

y 7 A (4, 6) 6 5 4 4 3 3 2 B (1, 2) 1

B (−2, 2)

1 2 3 4 5 6 7x

Exercise 9:04

6 1 2 3 4 5x

c2 = a2 + b2 AB 2 = AC 2 + BC 2 = 32 + 62 = 9 + 36 = 45 ∴ AB = 45 ∴ Length of AB is 45 units.

Foundation worksheet 9:04 Distance between points

e

Use Pythagoras’ theorem to find the length of each of the following. (Leave your answer as a surd, where necessary.) a y b y c y B (7, 9) 9 8 7 6 5 4 3 2 1

7 6 5 4 3 2 1

8

3

C (3, 3) 4

0

6

C (7, 1) 1 2 3 4 5 6 7 8 x

4 3 2 1

A (3, 6)

B (4, 3)

5 −1 0 1 2 3 4 5 x −1 3 −2 C A (1, −2) −3

B (7, 3)

1 2 3 4 5 6 7 8 x

A (1, 1)

0

d

e

y B (−3, 3) 2

C

5

5 4 3 2 1

−4 −3 −2 −1 0 −1 −2

260

3

−3 −2 −1 0 −1

Pa g

1

A (4, 5)

oo fs

c2 = a2 + b2 AB 2 = AC 2 + BC 2 = 42 + 32 = 16 + 9 = 25 ∴ AB = 25 ∴ Length of AB is 5 units.

7 6 5 4 3 2 1

45 is a surd. We simplify surds if they are perfect squares.

Pr

−1 0 −1

y

y A (−6, 2)

A (2, 1) 1 2 3 x

3 2 1

−6 −5 −4 −3 −2 −1 0 −1 5 −2 −3 C 12 −4

1 2 3 4 5 6 x

B (6, −3)



9/07/13 9:25 AM

Find the lengths BC and AC and use these to find the lengths of AB. (Leave your answers in surd form.) a y b y c B (−7, 4) y B (5, 4) B (4, 4)

−1 0 −1

d

A (0, 0)

C 1 2 3 4 5 x

0

1 2 3 x

−4 −3 −2 −1 0 −1

1 2 3 4 5 x

1 2 3 x A (2, −1)

C

A (2, −3)

i

5 4 3 2 1

B (−4, 1) B (2, 3)

y −4 −3 −2 −1 0 −1 −2 −3 −4 B (−4, −4) −5

y

C

1x

1

C

h

A (4, 7)

7 6 5 4 3 2 1

f

4 3 2 1

−3 −2 −1 0 −1 −2 −3

C

−7 −6 −5 −4 −3 −2 −1 0 −1

y B (−2, 3)

1 2 3 4x

−3 −2 −1 0 −1 −2 B (−3, −2)

g y

1 2 3 4 5 x

e A (4, 4)

4 3 2 A (−1, 2) 1

C

C

A (1, 1)

−1 0 −1

y 5 4 3 2 1

y

5 4 A (−2, 3) 3 2 1

A (4, 4)

C

1 2 3 4 x

−2

C

1 2 3 4 5x

−1 0 −1

B (5, −1)

Use Pythagoras’ theorem to find the length of interval AB in each of the following. (Leave your answers in surd form.) a y b y c y

e

3

4 3 2 1

oo fs

4 3 2 1

Pr

2

A (3, 6)

Pa g

6 5 4 3 2 B (1, 2) 1

−1 0 −1

d

1 2 3 4 5 x

−1 0 −1 −2

y

e y

5 4 3 2 1

5 4 3 2 1

−1 0 −1

B (4, 4)

A(0, 0) 1 2 3 4 5 x

A (4, 5)

5 4 3 2 1

0 −1

5 4 3 2 B (−2, 1) 1

1 2 3 4 5 x

f

1 2 3 4 5 6 x B (6, −1)

1 2 3 4 5 x

−2 −1 0 −1 −2

B (1, −2)

A(1, 4)

A (5, 3)

y 3 A(−3, 1) 2 1 −3 −2 −1 0 −1 −2

1 2 3 x

B (2, −2)



261

9/07/13 9:25 AM

g

−2 −1 0 −1 −2

i

y

B (3, 1)

−3 −2 −1 0 −1 −2 B (−3, −2)

1 2 3 4 x

y 3 2 1

5 A(1, 3) 4 3 2 1 1 2 x

0 −1 −2 −3

A(8, 1) 1 2 3 4 5 6 7 8x B (2, −1)

Find the distance AB for each of the following. (Leave your answers in surd form.) a A(1, 2), B(3, 8) b A(-2, 0), B(2, 4) c A(0, 0), B(4, 3) d A(-2, -3), B(6, 5) e A(4, 5), B(5, 6) f A(1, 4), B(4, 7) g A(-1, 4), B(2, -2) h A(4, 3), B(2, −4) i A(0, 0), B(-2, -2) j A(−6, 4), B(6, -1) k A(-1, -1), B(−4, −5) l A(0, 6), B(4, 0)

GEOGEBRA ACTIVITY 9:04

oo fs

4

h

y

5 A(−2, 3) 4 3 2 1

The midpoint, gradient and length of an interval

Pa g

e

Pr

In this activity you can click and drag points to change the length and gradient of an interval. Calculate the midpoint, gradient and length of the interval, and then use a checkbox to see the solution.

Architectural design often requires an understanding of gradients (slopes).

262



9/07/13 9:25 AM

9:05 Graphing straight lines PREP QUIZ 9:05 If y = 2x + 1 find y when x is: If x + y = 3 find y when x is: If x − y = 3 find y when x is:

1 0 4 1 6 1

3 -2

2 1 5 -2 7 -2

The tables below give the coordinates of three points on a line. Which of the graphs corresponds to each table? y y y 3

3

2

2

2

1

1

oo fs

3

1 −2 −1 0 −1

1

2x

−2 −1 0 −1

−2

1

2x

−2

1

-1

Pr

3

y

0

2x

1

2

1

3

−2

Graph A Graph B Graph C 9 10 8 x -1 0 1 x -1 0 1 x 0 y

1

−2 −1 0 −1

1

2

To graph a straight line we need: • an equation to allow us to calculate the xand y-coordinates for each point on the line • a table to store at least two sets of coordinates • a number plane on which to plot the points

y

-1

y

e

x-intercept

Pa g

Is THAT all? Hey, no problem! I can do that!

Two important points on a line are: • the x-intercept (where the line crosses the x-axis) • the y-intercept (where the line crosses the y-axis)

x y-intercept

WORKED EXAMPLES

Draw the graph of each straight line. From the graph write the line’s x- and y-intercepts. 2 y = 3x − 2 3 4x + y = 2 1 x + y = 5

Solutions 1 x + y = 5 2 y = 3x − 2 3 4x + y = 2 x 0 1 2 x 0 1 2 x 0 1 2 y

5

4

-3

y

-2

1

4

y

-1 -2 -6

Note: We may choose any three values for x, but it is usually best to use 0, 1 and 2.



263

9/07/13 9:25 AM

Only two points are required to graph a line, however, it is usual to use three. This is done to check that no errors have been made with the calculation of the points. If a mistake is made with one of the calculations it is usually impossible to draw a straight line through the three points. When x = 0, y = 3 × 0 − 2 = -2

When x = 0, 4 × 0 + y = 2 0 + y = 2 ∴ y = 2

When x = 1, 1 + y = 5 ∴ y = 4

When x = 1, y = 3 × 1 − 2 ∴ y = 1

When x = 1, 4 × 1 + y = 2 4 + y = 2 ∴ y = -2

When x = 2, 2 + y = 5 ∴ y = 3

When x = 2, y = 3 × 2 − 2 ∴ y = 4

When x = 2, 4 × 2 + y = 2 8 + y = 2 ∴ y = −6

−2 −1 −1 −2 −3

y

y x+y=5

1 2 3 4 5 6x

x-intercept is 5 y-intercept is 5

y

5 4 3 2 1

5 4 3 4x + y = 2 2 1

y = 3x − 2

1 2 3 4x

−2 −1 −1 −2 −3

Pr

5 4 3 2 1

oo fs

When x = 0, 0 + y = 5 ∴ y = 5

Pa g

e

x-intercept is 23 y-intercept is −2

−2 −1 −1 −2 −3 −4 −5 −6

1

x-intercept is 2 y-intercept is 2

Exercise 9:05 1

1 2 3 4 x

Foundation worksheet 9:05 Graphing lines

Complete the table for each equation. a y = 3x b y = x + 1 c y = 5 − x x 0 1 2 x 0 1 2 x 0 1 2 y

y

y

d y = 6x - 7

e y = 18 - 5x f y = 7x + 2 x 2 3 4 x 2 3 4 x 2 3 4

y g y = x - 7

y

y

h y = 1 - x

x 0 1 2 x 0 1 2 y

264

y



9/07/13 9:25 AM

2

Draw the graph of each of the following lines. Use axes numbered from -3 to 7. In each case complete the table first. a y = x + 1 b y = 2x c y = −x + 1 x 0 1 2 x 0 1 2 x 0 1 2 y

y

y

d y = 3x + 1

e y = 6 - 2x f y = x + 3 x 0 1 2 x 0 1 2 x 0 1 2

y

y

Graph the lines described by these equations. a x + y = 3 b x + y = 5 d y = x + 1 e y = x + 4 g y = 2x − 1 h y = 3x − 1

c x + y = 2 f y = x + 2 i y = 2x − 2

oo fs

3

y

On the same number plane, draw the graphs of these equations. a y = 2x − 1 b y = 2x c y = 2x + 3 How are these lines alike?

5

Graph these equations on one number plane. a y = x b y = 2x What do these lines have in common?

c y = 3x

On one number plane, graph these equations. a y = 2x − 1 b x + y = 2 What do these lines have in common?

c y = x

A point lies on a line if its coordinates satisfy the equation of the line.

e

6

Pr

4

Does the point lie on the line?

Pa g

E xample Does the point (6, 7) lie on the line y = 2x − 5?

Solution Substitute x = 6 and y = 7 into y = 2x − 5.

0

7

0·2

0·4

0·6

0·8

1

y = 2x − 5 7=2×6−5 7=7 which is true. ∴ (6, 7) lies on the line y = 2x − 5.

On which of the following lines does the point (6, 7) lie? a y = x + 1 b x + 2y = 20 d 4x − 3y = 3 e x − y = 1

c y = 3x − 4 f y = x − 1

8 a

Does (4, 0) lie on the line 4x + 3y = 16? b Does the line y = 2x − 3 pass through the point (7, 11)? c Which of the points (7, 2) and (7, -2) lie on the line y = x − 9? d Which of the points (5, 0) and (0, 5) lie on the line x + 2y = 5? 9 Coordinate geometry


265

9/07/13 9:25 AM

9

Draw the graph of each line and use it to find the x- and y-intercepts. a 2x + y = 2 b 3x + y = 6 c 2x + y = 4 d 2x − y = 4 e 3x − y = 3 f 4x − y = 2

10 Draw

the graph of each equation. 3x x +1 a y = b y = 2 2 d 3x + 2y = 7 e 5x − 2y − 6 = 0

INVESTIGATION 9:05

x -1 2 f 2x − 3y − 5 = 0 c y =

x- AND y-INTERCEPTS

x

e

Pr

oo fs

1 From the points A, B, C, D and E choose the x-intercept of: y 12 a line 1 b line 2 c line 3 2 From the points A, B, C, D and E choose the y-intercept of: 8 a line 1 b line 2 c line 3 Line 1 4 3 Write the coordinates of the x-intercepts. A E D What do you notice? 0 B 4 −8 −4 8 4 Write the coordinates of the y-intercepts. −4 C What do you notice? Line 2 5 What can you say about the y-coordinate of every −8 Line 3 point on the x-axis. −12 6 What can you say about the x-coordinate of every point on the y-axis. 7 How can you find the x-intercept from the line’s equation? 8 How can you find the y-intercept from the line’s equation? 9 Find the x- and y-intercepts of the lines in Questions 3 and 10 of Exercise 9:05.


GRAPHING LINES BY PLOTTING POINTS

Pa g

Choose one of the three sets of equations and work out the coordinates of two points on the line. Then click and drag two given points to these positions. If you have calculated correctly, the given line will appear.You can then repeat the process with the other two lines. The degree of difficulty of calculating the points varies from line to line. New sets of three equations can be generated randomly.

266



9/07/13 9:25 AM

9:06 Horizontal and vertical

lines

The line on the right is vertical. • The following table of values shows the points marked on the line. x

2

2

2

2

2

2

y

-2 -1

0

1

2

3

This line is x = 3.

y

3 2 1

They cut the x-axis at –1 and 3.

The line on the right is horizontal. • The following table of values shows the points marked on the line.

y

2

1

2

3

2

2

2

2

Pa g

2

0

e

-2 -1

• There seems to be no connection between x and y. However, y is always 2. So the equation is y = 2. Horizontal lines have equations of the form y = b, where b is the y-intercept.

1 2 3 x

1 2 3 x

Pr

−3 −2 −1 0 −1 −2 −3

x

x=2

oo fs

This line is

x = −1.

3 2 1

−3 −2 −1 0 −1 −2 −3

• There seems to be no connection between x and y. However, x is always 2. So the equation is x = 2. Vertical lines have equations of the form x = a, where a is the x-intercept.

y

y

This line is 3 y = 1. 2 1

y 3 2 1 −3 −2 −1 0 −1 −2 −3

1 2 3 x

They cut the y-axis at –3 and 1.

−3 −2 −1 0 1 2 3 x −1 −2 −3 This line is y = −3.



267

9/07/13 9:25 AM

Exercise 9:06 Write the equation of each line. a y

b

−2 −1 0 −1 −2

2

y

3 2 1

3 2 1 1 2 3x

For each number plane write the equations of the lines A to F. a b A B C A y

3 a What

D E 1 2 3 4 x

F

C D

4 3 2 1

−5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5

E

1 2 3 4 x

F

Using values from −5 to 5 on each axis, draw the graphs of the following straight lines. Use a new diagram for each part. a y = 4, x = 5, y = -1, x = 0 b x = 1, y = 0, x = 2, y = 3 c y = 4, x = 2, y = -2, x = −4 d x = 5, y = −5, x = 2, y = 2 e y = -2, y = 0, x = 0, x = 3

y = 0 is the x-axis.

Pa g

e

4

B

Pr

is the equation of the y-axis? b What is the equation of the x-axis?

y

oo fs

4 3 2 1 −5 −4 −3 −2 −1 0 −1 −2 −3 −4 −5

1 x

−4 −3 −2 −1 0 −1 −2

x = 0 is the y-axis. y x=0

1

Foundation worksheet 9:06 Horizontal and vertical lines

y=0 x

Which of these encloses a square region?

5

268

Give the point where each pair of lines cross. (You can use the graphs in Question 2.) a x = 3 and y = 1 b x = −4 and y = 3 c x = -1 and y = −4 d x = 2 and y = -2 e x = -3 and y = 2 f x = 4 and y = 4 g x = 0 and y = 3 h x = 2 and y = 0 i x = 0 and y = 0



9/07/13 9:25 AM

9:07 The gradient–intercept

form of a straight line: y = mx + b

PREP QUIZ 9:07 If x = 0, what is the value of: 2 mx + b? 1 2x

y

What is the gradient of: 6 line B? 5 line A

A

oo fs

If x = 0, what is the value of y when: 4 y = 4x − 1? 3 y = 3x + 2

4 3 2 1

−3 −2 −1 −1

B

1 2 3x

−2 −3 −4

What are the coordinates of the y-intercept of: 8 line B? 7 line A

Pr

9 Does every point on the y-axis have an x-coordinate of 0? 10 Can the y-intercept of a line be found by substituting x = 0?

e

• The equation of a line can be written in several ways. For example: x − y − 4 = 0, y = x − 4 and x − y = 4 are different ways of writing the same equation. • x − y − 4 = 0 is said to be in general form. • y = x − 4 allows us to get information about the line directly from the equation.

Pa g

INVESTIGATION 9:07

WHAT DOES y = mx + b TELL US?

y y y y = 2x + 2 4 3 2 1

−3 −2 −1 −1 −2 A −3 −4 B

y=x+2

y=x−2 1 2 3 x

C

4 3 2 1

−3 −2 −1 −1 −2 −3 D −4

4 3 2 1

y = 12 x + 1 y= 1 2 3 x

1 2x

−1

−3 −2 −1 −1 −2 E −3 −4 F

y = 2x − 3

1 2 3 x

1 Use the graphs above to complete the table on the following page.



269

9/07/13 9:25 AM

Line

Equation

A

y = x + 2

B

y = x − 2

C

y = 21 x + 1

D

y = 21 x − 1

E

y = 2x + 2

F

y = 2x − 3

Gradient

y-intercept

2 Complete the table below. Equation y = x + 2

B

y = x − 2

C

y = 21 x + 1

D

y = 21 x − 1

E

y = 2x + 2

F

y = 2x − 3

Coefficient of x

Constant

Pa g

e

A

y = 3x − 5

Pr

Line

3 is the coefficient of x.

oo fs

In an equation like y = 3x − 5: • the number 3, in front of x, tells us how many of x we have and is called the coefficient of x • similarly, the number in front of y is called the coefficient of y. In this case the coefficient of y is 1 • the number −5 is called the constant.

3 From the tables in Questions 1 and 2, state how the gradient and y-intercept of a line are related to its equation. 4 The graphs of equations 2x + y − 3 = 0 and y = -2x + 3 are the same line. The gradient is -2 and the y-intercept is 3. Which form of the equation gives this information directly?

5 What does the form y = mx + b tell us about a line? • When an equation of a line is written in the form y = mx + b: m gives the gradient b gives the y-intercept • Clearly, lines with the same gradient are parallel. (See the pairs of lines in Investigation 9:07.)

270



9/07/13 9:25 AM

WORKED EXAMPLE 1 Write the gradient and y-intercept of these lines. a y = 3x − 5 b y = -2x

Here, m = 3, b = −5. The gradient is 3 and the y-intercept is −5.

c y = 4 − 3x

Here, m = -2, b = 0. The gradient is -2 and the y-intercept is 0.

Here m = -3, b = 4. The gradient is -3 and the y-intercept is 4.

WORKED EXAMPLE 2 Find the gradient and y-intercept from the graph and write the equation of the line.

oo fs

y

3 2 1

−2 −1 −1 −2 −3

1 2 3 4 x

Pr

From the graph: For every run of 2 there is a fall of 1, gradient = - 21 y-intercept = -1 ∴ Equation of the line is y = - 21 x − 1.

This line is 'falling', so, the gradient is negative.

WORKED EXAMPLE 3

Change the equation into the form y = mx + b to find the gradient and y-intercept of the following.

e

b 3x + y = 1 − 3x − 3x y = 1 − 3x

Pa g

a 2y = 4x + 3 ÷ 2 ÷ 2 ÷ 2 y = 2x + 1 21

∴ gradient = 2 y-intercept = 1 21

∴ gradient = -3 y-intercept = 1

WORKED EXAMPLE 4

Use the y-intercept and gradient to graph the line y = 3x − 2. Start at the y-intercept of -2. rise 3 Now gradient = = 3 = run 1 ∴ For every run of 1 there is a rise of 3.

c 2x − 3y + 6 = 0 + 3y + 3y 2x + 6 = 3y ÷ 3 ÷ 3 ÷ 3 23 x + 2 = y

∴ gradient = 23 y-intercept = 2 y 5 4 3 2 1

y = 3x − 2 3 1

−1 1 2 3 x 3 −1 −2 1 −3



271

9/07/13 9:25 AM

WORKED EXAMPLE 5 Use the y-intercept and gradient to graph the line 4x + y = 2. First, rearrange the equation.

y

2 4x + y = 2 1

4x + y = 2 − 4x − 4x y = −4x + 2

−1 −1 −2 −3

x

1

oo fs

Exercise 9:07

Foundation worksheet 9:07 Gradient–intercept form

Pr

What are the gradient and y-intercept of each of the following lines? a y = 2x + 3 b y = 5x + 1 c y = 3x + 2 e y = 4x + 0 f y = x g y = x -2 i y = 6x − 4 j y = -2x + 3 k y = -x − 2 x m y = 21 x + 4 n y = 43 x − 2 o y = + 5 3 q y = -2 + 4x r y = 3 − 21 x

d y = 1x + 6 h y = 5x − 1 l y = -3x + 1

p y = 4 − 3x

Find the equation of the line that has: a a gradient of 4 and a y-intercept of 9 b a gradient of -2 and a y-intercept of 3 c a gradient of 7 and a y-intercept of -1 d a gradient of −5 and a y-intercept of -2 e a gradient of 21 and a y-intercept of 5

y = mx + b

Pa g

e

2

1 2

−4 4 −5 −6

Then graph the line using the y-intercept of 2 and gradient of −4.

1

Remember! A negative gradient always slopes down to the right.

f a g a h a i a j a

3

gradient of 23 and a y-intercept of −4 y-intercept of 1 and a gradient of 3 y-intercept of -3 and a gradient of 2 y-intercept of 21 and a gradient of 5 gradient of 1 and a y-intercept of 1 21

For each diagram find the gradient and the y-intercept and use these to write the equation of each line. y a b c d y y y

−2

272

1 x + 5 = –x + 5 – 2 2

6

6

6

6

4

4

4

4

2

2

2

2

0

2

4x

−2

0

2

4x

−2

0

2

4x

−2

0

2

4x



9/07/13 9:25 AM

e

f

y

0

2

x

−2

−2

y 2

2 2

x

0

−2

−2

2

x

−2

0

x

−2

−2

Match the graphs A to G with the equations below. a y = x b y = 6 − 2x c y = x + 2 d y = -2 e y = 21 x − 3 f y = -2x + 2 g y = 2x + 2

2

B y 6

A

C

D

E

4 2 F –6

–4

–2

0

2

4

oo fs

4

0

h

y

2

2 −2

g

y

–2

6 x G

–4 –6

By first changing the equation into the form y = mx + b, find the gradient and y-intercept of the following. y a 2y = 4x + 10 b y + 3 = 3x c = x + 2 2 ÷ 2 ÷ 2 − 3 − 3 × 2 × 2 y = …… y = …… y = …… ∴ gradient is …… ∴ gradient is …… ∴ gradient is …… y-intercept is …… y-intercept is …… y-intercept is …… y d 3y = 6x − 3 e y − 8 = 5x f = 2x − 4 3 g 4y = 12x − 8 h y + 4 = -2x i y − 2x = 7 j x + y = 4 k 2y = x + 3 l 3x + y = 5

Pa g

e

Pr

5

m 6x + 2y = 4

− 6x − 6x 2y = …… ÷ 2 ÷ 2 y = …… ∴ g radient is …… y-intercept is ……

6

n 3x + 2y = 6

o 2x - 5y + 1= 0

− 3x − 3x 2y = …… ÷ 2 ÷ 2 y = …… ∴ gradient is …… y-intercept is ……

+ 5y + 5y …… = 5y ÷ 5 ÷ 5 …… = y ∴ gradient is …… y-intercept is ……

Lines with the same gradient are parallel. Are the following pairs of lines parallel or not? a y = 3x + 2 and y = 3x − 1 b y = 5x − 2 and y = 2x − 5 c y = x + 7 and y = x + 1 d y = x − 3 and y = 1x + 2

7 a Which

of the following lines are parallel to y = 2x + 3? y = 3x + 2 2x − y + 6 = 0 2y = x + 3 y = 2x − 3 b Two of the following lines are parallel. Which are they? y = x − 3 x + y = 3 y = 3x 3y = x y = -x + 8

If two lines are parallel, they have the same gradient.



273

9/07/13 9:25 AM

8

Draw the graph of each line on a separate number plane by following these steps. (Look at Worked Example 4.) • Mark the y-intercept (the value of b) on the y-axis. • ‘Count’ the gradient (the value of m) from this point, and mark a second point. • Draw a line through this second point and the y-intercept. a y = 2x + 1 b y = x + 2 c y = 3x − 1 d y = -2x − 1 e y = -x + 1 f y = -3x − 3 1 1 g y = 2 x + 2 h y = 3 x − 1 i y = 23 x j y = - 21 x + 1 k y = - 13 x l y = - 23 x − 1 Graph lines with a y-intercept of 1 and a gradient of 0, 1, 2 and -1 on the same number plane. b Graph lines with a slope of -1 and y-intercepts of -2, 0, 1 and 2 on the same number plane.

9 a

equation of each line has been given in general form. Find the gradient of each line by rearranging the equation into gradient–intercept form. a 2x + y + 6 = 0 b 4x − 2y + 5 = 0 c x + 2y + 1 = 0 d 4x − 3y + 6 = 0 e 4x − y + 3 = 0 f x + 3y − 6 = 0


oo fs

10 The

THE GRADIENT–INTERCEPT FORM OF A LINE

Pr

1 Finding the equation of a line In this activity a line is generated randomly. Find the gradient and y-intercept of the line and use these to write its equation in the form y = mx + b. Tick a checkbox to see if the equation is correct.

Pa g

e

2 Equations of the form y = mx + b Use sliders to vary the values of m and b on a blue line. As these values are varied you can analyse the effect that each has on the shape of the graph.You can then find the equation of a randomly produced red line. Use the sliders to change the values of m and b on the blue line until it coincides with the red line.

3 Graphing lines using the gradient and y-intercept In this activity you are given an equation and asked to graph the line by first dragging a point to the y-intercept. A gradient triangle and a line appears when this is done correctly. The gradient triangle is used to find another point on the line. When this is successful, the line changes colour. 4 Matching equations of lines to their graphs The graphs and equations of four different lines are given for you to match each equation to its graph.You can then check the solutions and generate a new set of lines and equations.

274



9/07/13 9:25 AM

FUN SPOT 9:07

WHY DID THE BANANA GO OUT WITH A FIG?

Answer each question and write the letter for that question in the box above the correct answer. For the points A(2, 5), B(-1, 3), C(4, 2), D(0, 6), find: E distance AC E distance CD A slope of AB A slope of AD T midpoint of DC N midpoint of AC

E distance BD T slope of BC I midpoint of AB

C y = 2x − 1

What is the gradient of the following lines? U 2y = x − 5

G 2x + y + 1 = 0

What is the y-intercept of the following lines? D y = 2x − 1 T 2y = x − 5

L 2x + y − 1 = 0

Find the equation of the line with:

Write in the form y = mx + b. B 2x − y − 5 = 0 E 2x − y + 5 = 0

y

A x = 21 y − 2

U coordinates of C.

C

y = 2x + 4 y = 4 − 2x √32

2 3 4 3

1 −1 (3, 3 12 ) − 15 −2 √10 −2 12

x

Pa g

e

B(4, 5)

A(1, 1)

Pr

y = 4x − 2 √13 1 ( 2 , 4) (2, 4) y = 2x − 1 y = 3x + 5 (2 12 , 3)

1 2

y = 2x − 5 y = 2x + 5 2 − 12

In the diagram, find: D the slope of AB

O slope of 3 and a y-intercept of 5 S y-intercept of -2 and a slope of 4.

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C gradient of 2 and a y-intercept of -1 T y-intercept of 4 and a slope of -2

Not all graphs are straight lines. A parabola is one type of curve that is non-linear.



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9/07/13 9:25 AM

9:08 Non-linear graphs Up to this point, all the graphs have been straight lines. There are many types of mathematical curves and in this section we will look at two of them, the parabola and the exponential curve.

Parabolas • The equations of parabolas can be identified because they have x2 as the highest power of x. The simplest equation of a parabola is y = x2. • As with the straight line, the equation is used to find the points on the curve. Some of these are shown in the table.

y = x2 x -3 -2 -1 0 y

9

4

1

0

1

2

3

1

4

9

1

2

3x

1

2

3x

y

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For an accurate graph, many points would have to be plotted. • From the graph, we can see that the parabola has a turning point, or vertex, which is the minimum value of y on y = x2. • The y-axis is an axis of symmetry of the curve, so the right side of the curve is a reflection of the left side. This can be seen when points on either side of the axis are compared. • The parabolas that we will consider have two basic shapes. The one on the left is said to be concave up, whereas the one on the right is said to be concave down.

9 8 7 6 5

Pr

4

Parabolas can be happy (up) or sad (down). concave up

3 2

e

1 concave down

–2

–1

Pa g

–3

y

Exponential graphs

• The simplest exponential equation is y = 2x. Notice that in this equation x is a power, and hence there is a connection with the topic of Indices (see Chapter 6). Again, a table of values is used to find points that lie on the curve.

8 7 6 5

y = 2x

4

x

-3

-2

-1

0

1

2

3

y

0.125

0.25

0.5

1

2

4

8

• When plotting points it is often necessary to round the y-values to one decimal place. • Notice what happens to the y-values of the points as the x-values increase and decrease.

276

0

3 2 1 –3

–2

–1

0



9/07/13 9:25 AM

Exercise 9:08 1

Copy and complete the following tables. By comparing the values obtained with the given graphs, find which of the graphs A to C would result from plotting each set of points. a y = x2 + 1 b y = x2 - 1 x

-3 -2 -1

0

1

2

3

y

0

1

2

3

-3 -2 -1

0

1

2

3

y

A

B

y 12 10 8 6 4 2 1

2

3 x

y

12

12

10

10

8

8

6

6

4

4

2

2

−3 −2 −1 0

1

2

3 x

−3 −2 −1 0 −2

Pr

−3 −2 −1 0

C

y

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1

2

3 x

Complete a table of values ranging from x = -3 to x = 3 for the parabola y = x2 − 2 and use the values obtained to graph the equation.

3 a

What are the coordinates of the vertex for each of the parabolas in question 1?

e

2

-3 -2 -1

y

c y = x2 + 2

x

x

Pa g

b Is the vertex also the y-intercept of the parabola? c Are the parabolas symmetrical? d Does the vertex lie on the axis of symmetry?

4

Complete a table of values ranging from x = -3 to x = 3 for each of the following equations. State which of the graphs E, F or G would be obtained by plotting those points. a y = -x2 E

y

−3 −2 −1 0 −2

1

2

3 x

b y = -x2 + 1

c y = -x2 - 2

F

G

y −3 −2 −1 0 −2

1

2

3 x

y 2

−4

−4

−3 −2 −1 0 −2

−6

−6

−4

−8

−8

−6

−10

−10

−8

−12

−12

−10

1

2

3 x

−12



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9/07/13 9:25 AM

5 a

Write the coordinates of the vertex for each of the parabolas in question 4. b Is the vertex also the y-intercept of the parabola? c Are the parabolas symmetrical? d Does the vertex lie on the axis of symmetry?

7

8

For each of the following, complete a table of values and use it to state the graph (A, B, C or D) that matches the equation. a y = x2 + 3 b y = 1 − x2 c y = x2 − 4 d y = −x2 + 4

A

y

B

8 7 6 5 4 3

Using the results from Questions 1 to 6, how can you use the equation of a parabola to tell if the parabola is concave up or concave down?

2 1

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6

Using a table of values ranging from −3 to 3, graph each of the following equations. a y = 3 − x2 b y = x2 + 4 c y = –x2 − 1

−4 −3 −2 −1 0 −1

3

4 x

−3

−4

−5

−6

Pr

D

Copy and complete each of the following table of values. Which of the exponential graphs E, F G or H would result from plotting that set of points? a y = 2x b y = 2-x x

-3 -2 -1

y

0

1

2

Pa g y

x

-3 -2 -1

0

1

2

3

0

1

2

3

y d y = -2-x

c y = -2x

x

3

e

-3 -2 -1 E

0

1

2

y

3

x

-3 -2 -1

y y

F

8

2

7 6

1

5

−3 −2 −1 0 −1

4

−2

3

−3

2

−4

1

−5

−3 −2 −1 0

1

2

1

2

3 x

−6

3 x

−7 G

278

2

−2

C

9

1

−8

H



9/07/13 9:25 AM

10 By

finding suitable points, match each equation with one of the exponential curves A to D. a y = -1 + 2x b y = 4 + 2-x c y = -2x + 2 d y = 2 − 2-x

A

y

B

10 9 8 7 6 5

Hint: Use the equations to find the y-intercept for each curve. If these are the same, find another point.

4 D

3

C

2 1

−4 −3 −2 −1 0 −1

1

2

3

4 x

GEOGEBRA ACTIVITy 9:08

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−2

NON-LINEAR GRAPHS

Pr

1 Graphing parabolas The equations of two parabolas that are given can be varied by using sliders. After completing a table of values the points can be plotted via the Input Bar. Tick the checkbox to reveal the graph and show whether the points have been plotted accurately.

Pa g

e

2 Exponential curves This activity is similar to Activity 1 but uses exponential curves rather than parabolas. 3 Matching parabolas to their equations

The equations of five parabolas and their graphs are given and you are asked to match each parabola to its equation. Tick a checkbox to show the correct answer. A new set of equations and parabolas can be randomly generated.



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9/07/13 9:25 AM

MATHS TERMS 9

y

number plane 4 • a rectangular 3 2 A (3, 2) C (−3, 2) grid that 2 1 3 allows the 1 position of 1 2 3 x −3 −2 −1 0 points in a −1 E (0, −1) plane to be −2 B (3, −2) identified by −3 D (−2, −3) an ordered pair of numbers origin • the point where the x-axis and y-axis intersect, (0, 0); see 3 in number plane plot • to mark the position of a point on the number plane y quadrants 3 • the four 2nd 1st 2 quarters that quadrant quadrant 1 the number −3 −2 −1 1 2 3 x plane is divided −1 into by the 3rd 4th −2 quadrant quadrant x- and y-axes

e

Pr

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coefficient • the number that multiplies a pronumeral in an equation, e.g. 3x − 5y = 6 coefficient of x is 3 coefficient of y is −5 coordinates • a pair of numbers that gives the position of a point in a number plane relative to the origin; see (3, 2), (3, -2), (-3, 2), (-2, -3) and (0, -1) in number plane constant • the number part of an equation or expression, e.g. 3x − 5y = 6 The constant is 6 gradient y • the slope of a line rise or interval; it can be run measured using x the formula: rise Gradient = run gradient-intercept form • a way of writing the equation of a line, e.g. y = 2x - 5, y = 21 x + 2

Pa g

when an equation is rearranged and written in the form y = mx + b then m is the gradient and b is the y-intercept

graph (noun) • a diagram showing the relationship between two variable quantities, usually shown on a number plane graph (verb) • to plot the points that lie on a line or curve

280

−3

x-axis • the horizontal number line in a number plane; see 1 in number plane x-coordinate • the first number in a pair of coordinates; it tells how far right (or left) the point is from the origin x-intercept • the point where a line or curve crosses the x-axis y-axis • the vertical number line in a number plane; see 2 in number plane y-coordinate • the second number in a pair of coordinates; it tells how far the point is above (or below) the origin y-intercept • the point where a line or curve crosses the y-axis



9/07/13 9:25 AM

DIAGNOSTIC TEST 9

Coordinate geometry

These questions reflect the important skills introduced in this chapter. Errors made will indicate areas of weakness. Each weakness should be treated by going back to the section listed. 9:02

1 Find the midpoint of the interval AB in each of the following. a b c y

y

4

4

3

3

3

2

2

2

1

1

1

−3 −2 −1 0 −1

B(−2, −2)

A(−4, 4)

A(2, 4)

4

1

2

3 x

−4 −3 −2 −1 0 −1 −2

−2

1

2 x

A(1, 4)

−3 −2 −1 0 −1

1

2

3 x

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y

B(2, −2)

B(−2, −2)

−2

2 For each of the following find the midpoint of the interval AB. a A(1, 2), B(7, 10) b A(3, 0), B(5, 3) c A(-3, -2), B(1, -3)

9:02

3 For each diagram in Question 1, find the gradient of the interval AB.

9:03

4 Find the gradient of the line that passes through: a (1, 3), (2, 7) b (-2, 8), (4, 5)

9:03

Pr

c (0, 3), (3, 5)

5 For each diagram in Question 1, find the length of the interval AB. (Leave answers in surd form).

9:04

6 Find the distance between the points A and B if: a A is (1, 2), B is (7, 10) b A is (3, 0), B is (5, 3)

9:04

e

c A is (-3, -2), B is (1, -3) 9:05

7 Graph the lines with the following equations: a y = 2x + 1 b y = 2 − x

Pa g

c y = 2x

8 a Does the point (3, 2) lie on the line x + y = 5? b Does the point (-1, 3) lie on the line y = x + 2? c Does the point (2, -2) lie on the line y = x − 4?

9:05

9 State the x-intercept of each line: a y = x − 3 b x + y = 4

9:05 c x + 2y = 4 9:05

10 State the y-intercept of each line: a y = x + 4 b x + y = -1

c x − y = 5

11 Write the equation of each line in this diagram.

b c

y

4 3 2 1

−3 −2 −1 −1 −2 −3

d

9:06

a

1 2 3x



281

9/07/13 9:25 AM

12 Write the equation of the line that has: a a gradient of 3 and a y-intercept of 2 b a gradient of 21 and a y-intercept of -3 c a y-intercept of 3 and a gradient of -1 d a y-intercept of - 21 and a gradient of 13

9:07

13 State the gradient and y-intercept of each line: a y = 2x + 3 b y = 21 x − 3 c y = 3 − 2x

9:07 d y = 4 − x 9:07

15 Rearrange these equations into gradient–intercept form. a 4x − y + 6 = 0 b 2x + 3y − 3 = 0 c 5x + 2y + 1 = 0 d x − 2y + 5 = 0

9:07

16 Graph the lines in Question 12.

9:07

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14 Write the equation of each of the lines AB in Question 1.

17 Match each equation with the corresponding

parabola in the diagram. a y = -x2 + 3 b y = x2 + 1 c y = x2 − 3

A

y

9:08

4 3 2 1

Pr

−3 −2 −1 0 −1

C

18 Match each equation with the corresponding

e

Pa g

1

3 x

2

−2 −3 −4

9:08

y

exponential curve in the diagram. a y = 2x b y = 2-x c y = 2 + 2x

282

B

8 B

7 6 5 4 A

3 2 1

−3 −2 −1 0

C 1

2

3 x



9/07/13 9:25 AM

ASSIGNMENT 9A

Chapter review

1 Find: a the length AB as a surd b the slope of AB c the midpoint of AB.

5 A right-angled triangle OAB is shown.

y 6 5 4 3 2 1 −1 −1 −2

A (6, 5)

y 4

B E

1 2 3 4 5 6 x

A

O

6

x

a Find the coordinates of E, the midpoint

B (2, −2)

of AB. b Find the length of OE. c Find the length of EA. d What can you say about the distance of E from O, A and B?

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2 A is the point (2, 5) and B is the point (7, 17). a What is the length AB (as a surd)? b What is the slope of AB? c What is the midpoint of AB?

6 Complete the table of values below for each of the following equations and draw the graphs on the same number plane. x

-3 -2 -1

1

2

3

a y = 2x

b y = x2

c y = 2x

Pa g

e

4 a A line has an x-intercept of 3 and a gradient of 1. Find where the line crosses the y-axis, and hence write its equation. b A line has a slope of - 21 and a y-intercept of 6. What is its equation? What is its x-intercept?

0

y

Pr

3 The equation of line A is y = 2x − 4 and the equation of line B is y = 4 − 2x. a Which line has a negative gradient? b Which line has a y-intercept of 4? c Which line is parallel to the line 2x + y + 4 = 0? d Find the point of intersection of the lines.

I have 20 Australian banknotes totalling $720. Show how this could be done with: • 2 types of notes • 3 types of notes • 4 types of notes



283

9/07/13 9:25 AM

ASSIGNMENT 9B

Working mathematically

1 The diagram shows a 4-minute timer. a If this timer was started 0 with the pointer on 3 1 zero, what number would it be pointing 2 to after 17 minutes? b At what times between 30 minutes and 1 hour will the pointer be pointing at number 3?

4 Four friends decide to play tennis. Find out how many different: a singles matches can be played. (A singles match is one player against another player.) b doubles matches can be played. (A doubles match is two players against two players.)

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2 The faces of a cube are divided into 4 squares. If each square on each face is to be painted, what is the minimum number of colours needed if no squares that share an edge can be the same colour?

3 John is given $200 each month for a year. Anna is given $1 for the first month, $2 for the second, $4 for the third, and so on until the twelfth month. Who is given the most in the year and by how much is the amount bigger?

Pr

5 The table below has been prepared from information supplied by the Federal Office of Road Safety, for side-impact accidents in which at least one occupant was killed. It gives the number of car occupants in various seating positions and the number killed as a direct result of an impact to the side of the car. The figures exclude 38 occupants as there was insufficient information to determine their exact seating position at the time of the crash.

e

Seating position

Pa g

Left side (passenger’s side)

Centre seat

Right side (driver’s side)

All occupants

Dead

Total

Dead

Total

Dead

Total

Dead

Total

Front seat

114

187

0

1

199

316

313

504

Rear seat

29

63

2

19

20

44

51

126

All occupants

143

250

2

20

219

360

364

630

a How many occupants were in: i the driver’s seat ii the front left side seat iii a front seat iv a passenger’s side seat? b What percentage of drivers were killed? c What percentage of all the occupants who were killed were drivers? d For each of the six seating positions, calculate the number of deaths as a percentage of the

total number of people in that seat. Can you infer anything about the relative safety of the various seating positions from this information? Give reasons for your answer.

284



9/07/13 9:25 AM

ASSIGNMENT 9C

Cumulative revision

1 Solve: 3x x 3x − 5 x 3x − 5 x − 3 − 5 = − 3 b = a = − 3 c 4 4 2 4 2 2

7:07

2 Simplify each of the following. Give your answer in scientific notation correct to

6:05

two significant figures. a (1⋅65 × 103) × (7⋅2 × 10–6) c (6⋅03 × 10–3)2 ÷ (4⋅7 × 10–10)

b

−5

1·06 × 10 5·1 × 10 −9

8:02

4 a Use the net of the triangular prism to calculate its surface area. (All measurements are in cm.) b Calculate the surface area and volume of a cylinder that has a diameter of 6 cm and a height of 6 cm.

5:03

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3 a Jim earns $18 an hour. Calculate his wages for a week in which he works 35 hours at the normal rate , 6 hours overtime at ‘time-and-a-half ’ and two hours overtime at ‘double-time’. b Ralph is a tailor. He charges $15 for a minor alteration and $25 for a major alteration. How much will he earn in a week in which he does 32 minor alterations and 23 major alterations? What is his hourly rate if it takes him a total of 38 hours to do the work? 10

Pr

6

8

e

6

4:03

6 The 2009–10 Census produced the following figures on the involvement of people aged 15 years and over in organised and non-organised sporting and physical recreational activities. If a person over 15 is chosen at random, what is the chance that they: a do not participate in sport or physical recreation NonOrganised organised b participate only in organised activities activities activities c participate in only non-organised activities 15% d participate in an organised activity 11% 38% e participate in both organised and 36% non-organised activites?

4:02

Pa g

5 A card is chosen at random from a standard pack of cards. What is the chance that it is: a a King b red c red or a King d red and a King e not an Ace, King, Queen or Jack?



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