1.3 algebra relationships between tables, equations or graphs 4 credits (external)

1.3 ALGEBRA Relationships between tables,

equations or Graphs4 Credits (External)

Note 1: Gradients

• The gradient of a graph tells us how steep the graph is.

• It is given as a whole number or a fraction.• The top number is the change in y, and the

bottom number is the change in x. • Gradient =

)x(run

)y(rise

Note 1: Gradients

Special gradients to remember

A horizontal line has a gradient of 0 (the rise = 0) Gradient = 0

A vertical line has a gradient that is undefined. (the run = 0)

Gradient = undefined

A line that is going up hill is positive. Gradient = positive

A line that is going down hill is negative. Gradient = negative

To draw lines with a certain gradient:

1.) Pick a starting point.2.) Count the number of squares Up (if rise is +)

Down (if rise is -)

3.) Count the number of squares Right (if run is +) Left (if run is -)

If there is no run, (i.e. the gradient is a whole number) it is always 1.

4.) Mark the endpoint.

Note 1: Gradients

e.g. Draw lines with the following gradients.

a = b = 3 23

Note 1: Gradients

To find the gradients of a straight line: Pick 2 points on the line that go through a corner of a grid line.

The points marked here are the only ones that can be used as they are the only ones that go through the corners of the grid.

Count the rise from one point to the other, then count the run.

Simplify where possible.

3

1

9

3=

Note 1: Gradients

e.g. Find the gradients of the following lines.

a b c d e f

2

542

12

11

21

b=

undefined

GAMMA Pg 100 Ex. 8.02

21

=

Note 1: Gradients

Note 2: Graphs of linear functions

• The graph of a linear function is a straight line.• Plot points by:

– Draw a table of x values.– Substitute the x value into the formula and solve

for y.– Plot the pair of co-ordinates onto the axes

In a linear equation, x and y are only raised to the power of 1. (i.e. Not x2, x3 etc)There is a common difference between ordered pairs.

x -2 -1 0 1 2 3

yCo-

ordinate (-2, ) (-1, ) (0, ) (1, ) (2, ) (3, )

y

x1 2 3 4-1-2-3

12345

-1-2-3-4-5

y = 2x - 1

Drawing up a table of x values.Substitute the x value into the formula and solve for y.Plot the pair of co-ordinates onto the axes

eg: Draw the graph of y = 2x – 1

When x = -2 y = 2 -2 – 1 = -5

When x = -1 y = 2 -1 – 1 = -3

-5 -3 -1 1 3 5

Note 2: Graphs of linear functions

Starter Identify if the following relations are linear.

x Y-2 7-1 40 11 -22 -53 -8

Diff. x Y-2 2-1 40 81 162 323 64

Diff.

Common difference = -3Therefore, Linear

No Common difference Therefore, Not Linear

-3

-3-3-3-3

2

481632

y = 2x – 3

xy = -13

y = x(x + 2)

y = 2

Linear y1 = 2x1 - 3

Not Linear x = -13y-1

Not Linear y1 = x2 + 2x1

Linear y1 = 2

Starter Identify if the following relations are linear.

Note 3: Straight Line Graphs

The general formula for a straight line is

y = mx + c

There are a couple of straight lines that do not fit this general formula:

y = number gives a horizontal linex = number gives a vertical line

gradient y-int

e.g. y = 2 x = 4

y

x1 2 3 4-1-2-3-4

1

2

3

4

-1

y

x1 2 3 4 5

1234

-1-2-3-4


To draw straight line graphs using y = mx + c

• Mark the y-intercept (given by c). If there is no c, the y-intercept is 0.

• From the y-intercept, count the gradient (given by m) Put a point and then count the gradient again. If there is no m, it is always 1.

• Rule the straight line between the three points.

• Put arrows on both ends of the line and label it.


e.g. Draw the following straight lines.

a) y =

b) y = 4x – 2

c) y = x + 1

d) y = -2x

22

1x

x

y 22

1xy = y = -2x y = 4x - 2

y = x + 1


Note 4: Straight Line Graphs Using Intercept-Intercept Method

To find the x-intercept, set y = 0.To find the y-intercept, set x = 0.Join the two intercepts to make a straight line. e.g. Find the x and y intercepts of the line

y = -4x + 12

Find y-intercept, set x = 0y = -4 (0) + 12y = 12 therefore the y intercept is (0, 12)Find x-intercept, set y = 00 = -4x + 124x= 12x = 3 therefore the x intercept is (3, 0)

Find x-int, Set y = 02x – 0 = 6 2x = 6 x = 3 therefore the x-intercept is (3, 0)Find y-int, Set x = 02(0) – y = 6 -y = 6 y = -6 therefore the y-intercept is (0, -6)

Draw the graph of 2x – y = 6

Note 4: Straight Line Graphs Using Intercept-Intercept Method

y

x2 4 6 8

2

4

6

8

-2

-4

-6

-8

x-intercept is (3, 0)

y-intercept is (0, -6)

The cover up method – a quick way of graphing a straight line in the form ax + by = c

-x + 2y = 8Is to cover up the x term to get the y-intercept

-x + 2y = 8And then cover the y term to get the x-intercept

-x + 2y = 8Then plot the 2 points.

i.e. y = 4 (0,4)

i.e. x = -8 (-8,0)

x

y

(0,4)

(-8,0)

-x + 2y = 8

IWB MathematicsEx 13.04 pg 310

HMWK pg 88

Starter – Graph to Equation

y = 2

x + 3

y = -1/2 x + 0

y = 1/3x - 5

y = -4x + 4

2

1

3

4

Note 5: Applications of the Straight Line Graph

Many circumstances in nature and life can be graphed using straight line relationships.

e.g. De Vinces specializes in pizza. The approximate relationship

between x, the # of pizzas they sell daily and y their daily costs, is given by:

y = 10x + 50

Draw a graph showing the number of pizzas they sell on the x-axis, and their daily costs on the y-axis. Use an appropriate scale. (Because the numbers are large, it is best to have the y-axis going up in 20’s).

y = 10x + 50.

x

yCost $

Number of Pizzas sold8642 10 1412

120

100

80

60

40

20

140

What are their costs if they sell 8 pizzas? Either read off the graph or substitute into the equation.

If their costs are $100, how many pizzas did they sell? Either read off graph or substitute into the equation.

Give the y-intercept. What does it represent?

Give the gradient. What does it represent?

$130

5

50If no pizzas are sold, it still costs them $50

10 Represents the fact that each pizza they make, costs $10

y

x 2 4 6 8 10

20

40

60

80

100

120

140

160

Number of Pizzas sold

Cost

$

Note 6: Rate of Change

If two variables (e.g. distance and time) are directly dependent, the graph of this relation is a straight line. We can calculate the rate of change of one variable compared to the other.

rate of change = change in dependent variable change in independent variable

e.g. = change in distance change in time

Time (hrs)

Dist

ance

(km

)

The gradient of the graph is also a measure of the rate of change of the dependent variable

e.g.The graph below shows the value of a section in Auckland. The rate of increase shows two distinct slopes. Find the rate in increase before and after 2004.

Price in $000

Year

Value of section in Auckland

280

320

340

1998

Rate of change = Δ price / Δ time

= 330 – 280 2004 – 1998

= 50 6 = $8 333/yr

300

360

400

380

2000

2002

2004

2006

2008

2010

Before 2004

After 2004 = 400 – 330 2010 – 2004 = 70 6 = $11 667/yr

e.g.An engineer studying the expansion of a bar of metal made the following measurements. At 14°C the length of was 41.4 mm, at 22°C the length was 42.0 mm and at 34°C the length was 42.9 mm. Plot the data on a graph & find the rate of change of length compared to temperature.

mm

°C

Expansion of a metal bar

41

42

43

10 20 30

Plot points (14,41.4), (22,42) & (34,42.9)

Rate of change = Δ length / Δ temperature

= 42.9 – 41.4 34 – 14 = 1.5 20 = 0.075 mm/°C

Starter This graph shows the distance a car has travelled after leaving Dunedin.

a) Find the rate of change for each of the three sections A =

B = C = b.) Give the equation that

represent section A of the journey D =

c.) Using your equation from b) find what time the car is 68 km from Dunedin.

Time (pm)4 5 6 7 8 9

km

20

40

60

80

100

120

140

A

B

C100 km/hr20 km/hr-65 km/hr

100t - 400

4.68 hrs = 4.41 IWB MathematicsEx 13.06 Pg 316 - 321

StarterDraw a distance-time graph to describe the fable of the tortoise and the hare. (assume 3 km race)

Time taken (hours)

Dist

ance

from

star

t (km

)

1

32

2

3

1 4

Note 7: Simultaneous EquationsThe point (x,y) where two linear equations cross, gives a unique solution for x and y that satisfies both equations.

105

5

10

15

15

These 2 linear equations are simultaneously equal at pt (4,7)

(4,7)y = - x + 11

y = 1/2 x + 5

7 = - 4 + 11

7 = 1/2 (4) + 5

Note 7: Simultaneous EquationsTwo boys go out for lunch. The first orders 2 pies and 1 sandwich for $13.50 and the other orders 1 pie and 2 sandwiches for $12.00.

105

5

10

15

15

(5,3.50)

Let p be the cost of pies and s be the cost of sandwiches

pies (p)

snd. (s)

1p + 2s = 12.00 (2)

2p + 1s = 13.50 (1)

Note 7: Simultaneous EquationsAn animal nursery needs 12 cans of food to feed 2 dogs and 6 cats. The next day they have 4 dogs and 4 cats and need 16 cans of food.

105

-5

5

15

10

(3,1)

How many cans does a cat & a dog need?

dogs (d)

Cats (c)

IWB MathematicsEx 13.05 Pg 313-314

2d + 6c = 12 (2)4d + 4c = 16 (1)

Dogs need 3 cans and cats need 1 can of food.

Starter

(0,2)

4y = 8

y = 2

Subst. y = 2 into either equation

2x + 3(2) = 62x = 0x = 0

ParabolasFound in nature, motion, structure & art

Note 8: Quadratic Functions

Quadratic functions are algebraic expressions where the highest power of x is x2

e.g. y = x2

y = x2 + 3x – 5y = (x-2)(x+2)y = (x+2)2

the graph of these functions are called parabolas.

y = x2

vertex (0,0)

To draw the graph of a parabola• select a range of x values• substitute into the equation to get a corresponding y-term• Plot the points

x y =x2

-3

-2

-1

0

1

2

3

9

4

1

0

1

4

9

y = x2

Transformations of the parabola

y = x2 + 7

y = x2 + 4

y = x2

y = x2 − 3

By adding/ subtracting a constant, the parabola shifts up /down.

1.) Vertical Translation y = x2 ± c

y = x2

By adding/ subtracting a constant to the x variable, the parabola shifts left /right.

2.) Horizontal Translation y = (x+k)2

y = (x−3)2 y = (x+5)2 y = (x+2)2

y = x2

3.) Combined – Horizontal & Vertical Translation y = (x+k)2 + c

y = (x−4)2+2

y = (x+3)2−4

IWB Mathematics Pg 347 Ex. 15.01

To get the y-int:Let x = 0

StarterWrite the equations for each of the parabolas drawn below

y = (x+6)2 - 3y = (x-4)2 - 5

y = (x-5)2 + 4

Note 9: Parabolas (factorised form)

A simple way to graph parabolas when the equation is given in factorised form, is to plot the intercepts.e.g. y = (x - 4)(x + 2)

Y-int – set x = 0 X-int – set y = 0

y = (0 - 4)(0 + 2)

y = -4 x 2y = -8

0 = (x - 4)(x + 2)x - 4 = 0 or x + 2 = 0 x = 4 x = -2

Graph of y = (x – 4)(x + 2)

x-int

y-int

Midpoint of parabola x = 1Subst x = 1 to get vertex y = (1 - 4 )(1 + 2) = -3 x 3 = 9

Step by step for graphing factorised parabolas

1.) Find y-int by setting x = 02.) Find x-int by setting y = 03.) Find the axes of symmetry (midpoint between

both of the x-int)4.) Subst. the x-value that the axes of symmetry

cuts through and solve for y. This is the vertex.

Graph the following parabolas

a.) y = (x – 3)(x +4) b.) y = x(x - 6)

y-int (x = 0) y-int (x = 0)

y = -3 x 4y = 0

= - 12

x-int (y = 0) x-int (y = 0)

0 = (x – 3)(x + 4) 0 = x (x - 6)x – 3 = 0 x + 4 = 0 x = 0 x - 6 = 0 x = 3 x = -4 x = 6

y = (x – 3)(x +4)

y = x(x - 6)

(3, -9)

(-0.5, -12.25)

IWB Mathematics Pg 357-9 Ex. 15.04

HMWK pg 108-109

Note 10: Parabolas – Vertical Scale Factors

If the coefficient of x2 is negative, then the parabola is inverted (upside down)

e.g. y = −x2 + 2y = −(x + 3)2

y = −(x + 3)(x − 2) y = (x + 3)(2 − x)

Different forms of the same equation

y = -x2

y = −x2

y = −2x2

y = −3x2

y = x2

y = −x2

y = 1/2x2

y = x2

y = 1/4 x2

y = 1/10 x2

IWB Mathematics Pg 351 Ex. 15.02

HMWK pg 104-107

Starter

B F ED

C

J I H G

A

Task: Graph the following parabolas

y = -(x – 5)(x – 1) y = 2(x – 1)2 – 2

* Invertedy – intercept = x – intercepts = Midpoint is x = Vertex = (3, )

-55, 13

4

Task: Graph the following parabolas

y = 2(x – 1)2 – 2

* Opens upwardVertex = y – intercept =

(1, -2)0

Over 1, up 1 x2 = 2

Over 1, up 3 x2 = 6

Over 1, up 5 x2 = 10

Graph to Equation in form y=(x+k)2+c

y =- (x+7)2 + 2

2

1

y = x2 - 5

Graph to Equation in form y=(x+a)(x+b)

y = (x+5)(x-1)

1

2

y = −(x-2)(x-4)

Check your intercepts are correct once you have composed an equation

Graph to Equation– Vertical scale factor

y =-1/3 (x+9)(x+3)

1

2

y = 2(x-4)2 - 5

Check your intercepts are correct once you have composed an equation

y =-1/3 (x+6)2 + 3

HMWK pg 112-113

IWB Mathematics Pg 357-359 Ex. 15.04 # 6 -16

Starter

C EA

D BF

Note 11: Linear Patterns

Linear Patterns are sequences of numbers where the difference between successive terms is always the same.

e.g. 4, 7, 10, 13, 16, 19, 22…….

This sequence has a common difference of _______+3

Each Term = 3 x Term # + C

Term # (x) 1 2 3 4 5 6 7 …….

C is found by substituting ANY term and term # into the formula

C = constant term

e.g. 4 = 3 x 1 + CC = 4 - 3C = 1

Rule to find each term = 3x + 1

e.g. Identify if the following relation are linear.

x Y-2 7-1 40 11 -22 -53 -8

Diff. x Y-2 2-1 40 81 162 323 64

Diff.


No Common difference Therefore, Not Linear

-3

-3-3-3-3

2

481632

e.g. Find the differences, and if the result is linear find the rule.

Term # Sequence

1 162 133 104 75 46 1

Diff.


-3-3-3-3

Each Term = −3 x Term # + C16 = −3 x 1 + C

16 = −3 + C19 = C

Each Term = −3x + 19

(X)

-3

Check that your formula works !

Find the differences, and if the result is linear find the rule.

Term # Sequence

1 -82 -63 -44 -25 06 2

Diff.

Common difference = Therefore,

2222

Each Term = 2 x Term # + C -8 = 2 x 1 + C

-8 = 2 + C-10 = C

Each Term = 2x − 10

(X)

2


NOW YOU TRY THIS ONE -8, -6, -4, -2, 0, 2……

+2

HMWK pg 74-82

linear

Find the differences, and if the result is linear find the rule.

Term # Sequence

1 62 113 164 215 266 31

Diff.

Common difference = Therefore,

5555

Each Term = 5 x Term # + C 6 = 5 x 1 + C

6 = 5 + C1 = C

Each Term = 5x + 1

(X)

5


NOW YOU TRY THIS ONE 6, 11, 16, 21, 26, 31……

5linear

HMWK pg 74-82

IWB Mathematics Pg 269-273 Ex. 12.02 # 3 -12

Interpret features of graphs in Context

a) How much does the Mighty Minibus Co charge per kilometre?

b) Charges by Various Vans include a fixed fee for each trip. How much is this fixed fee?

c) For which length of trip do both companies charge the same?

d) Write the equation for the Various Vans graph.

$1.40/km

$30

50 km

y = 0.8 x + 30

Jim gets quotes from the Mighty Minibus Co and Various Vans for a minibus. Jim can use the graph below to work out the cost of a trip for each rental company.

Starter

FA

CE

N K J

H

DP

Q

B

Note 12: Quadratic Patterns

Quadratic patterns have a constant second order of differences (differences of differences).

Term #

Sequence

1 52 113 214 355 536 75

Diff.Diff.

610

141822

444

4

The value of the 2nd order of differences determines our Rule

2nd

1 means a 0.5n2

2 n2

3 1.5n2

4 2n2

In general 2nd Diff n2

2

Now that we know what the constant for n2

we can solve for the rest of the equation by substituting the term # into ____________

Term # (n)

Sequence 2n2 Adjustment

1 52 113 214 355 536 75

2n2

2818325072

333333

Rule: 2n2 + 3

HMWK pg 94-100

IWB Mathematics Pg 277-281Ex. 12.03

NOW YOU TRY 4, 6, 10, 16, 24, 34….

Quadratic patterns have a constant second order of differences (differences of differences).

Term #

Sequence

1 42 63 104 165 246 34

Diff.Diff.

24

6810

222

2

The value of the 2nd order of differences determines our Rule

2nd

1 means a 0.5n2

2 n2

3 1.5n2

4 2n2

In general 2nd Diff n2

2

Now that we know what the constant is for n2

we can solve for the rest of the equation by substituting the term # into _______________

Term # (n)

Sequence n2 Adjustment

1 42 63 104 165 246 34

n2

149162536

3210-1-2

Rule: n2 – n + 4

HMWK pg 94-100


Write the rule for the following patterns

a.) 5, 8, 11, 14, 17 …..

b.) 6, 12, 20, 30, 42,…...

3n + 2

Term #

Seq

1 6

2 12

3 20

4 30

5 42

Diff

6

8

10

12

Diff

2

2

2

2nd1st

Term #

Seq

1 6

2 12

3 20

4 30

5 42

n2

1

4

9

16

25

Diff

5

8

11

14

17

n2 + 3n + 2n2


3

3

3

3

Starter

=

Find the gradient of the line joining the points D(1,5) and E(5,2)

gradient of DE = x

y

12

12

xx

yy

15

52

4

3

=

=

Note 13: Exponential Relationships

• Commonly known as the ‘growth curve’

• General form is y = ax

• Curve always passes through (0, 1) • x-axis is an asymptote (graph approaches, but

never touches)• this year we only explore when is a natural

number and > 1

1 = a0

base

exponent

a a

Graph by plotting points

x value y = 2x

-3-2-10123

0.1250.250.51248

y = 2x

y=2x+2

the exponential function can be vertically translated up or down by adding or subtracting a constant

HMWK pg 119-120


StarterWrite the rule for the following sequences

4, 8, 12, 16, 20, 24, ……

10, 26, 48, 76, 110, 150, ……

4n

1st Difference

2nd Difference16 22 28 34 40 6 6 6 6 3n2

3n2 3, 12, 27, 48, 75, ……. Difference 7 14 21 28 35

+ 7n

Note 14: Exponential Patterns

Exponential patterns do not have a constant difference between terms

Term #

Sequence

1 32 53 94 175 336 65

Diff.Diff.

24

81632

4÷2 = 28÷4 = 2

16÷8 = 2

32÷16 = 2

The ratio of the 1st order of differences determines our Rule

Ratio of 1st

2 means a base 2 2x

3 …… 3x

4 …… 4x

Now that we know our rule is of the form 2x

we can solve for the rest by substituting the term # into _____

Term # (n)

Sequence 2x Adjustment

1 32 53 94 175 336 65

2x

248163264

111111

Rule: 2x + 1

What will the value of term 9 be?

29 + 1= 513

HMWK pg 116-118IWB Mathematics Pg 286-290 Ex. 12.04

Applications to ParabolasA courier company is investigating leasing an old hangar as storage for its vehicles at night. The hangar has a length of 18.5 m and a parabolic cross-section with an equation y = -0.2x(x – 13.5). Each van needs a space 6 m in length, 2.8 m in width and 2.5 m high to allow for access. Height

(m)

y = -0.2x(x – 13.5)

Width (m)

a.) What is the maximum height of the hangar?

What are the two x-intercepts

18.5 m(0,0) and (13.5, 0)

The maximum height occurs at the midpoint (6.75,0)

Solve for y given that x = 6.75 y = -0.2(6.75)(6.75 – 13.5) = (-1.35)(-6.75) = 9.1125 m is the maximum height

Applications to ParabolasA courier company is investigating leasing an old hangar as storage for its vehicles at night. The hangar has a length of 18.5 m and a parabolic cross-section with an equation y = -0.2x(x – 13.5). Each van needs a space 6 m in length, 2.8 m in width and 2.5 m high to allow for access. Height

(m)

y = -0.2x(x – 13.5)

Width (m)

b.) Is there room to fit 12 courier vans?

Required width = 2.8 x 4 = 11.2 m

18.5 m

Required length = 3 x 6 = 18 m

Is there sufficient height?

13.5 m – 11.2 m = 2.3 m spare

Subst. x = 1.15 to find the height of the hangar at that point

y = -0.2(1.15)(1.15 – 13.5) = (-0.23)(-12.35) = 2.8405 m

< 13.5 ok

< 18.5 ok

= 1.15 on each side

< 2.5 okYes, there is sufficient room for 12 courier vans

y-intercept = (0,-4.5)

Find an equation for the following parabola in the form y = k (x – c)(x – d)

Applications to ParabolasJacqui plans to hang shade cloth over a paved area. The shade cloth will hang between two posts 4 metres apart & will fall in the shape of a parabola. At its lowest point it will hang 2.2 m above the paving. The shade cloth is attached at the top of the 3 metre posts.

Post Post

Height (m)3

2

1

Supports

1 2-2 -1 0Width (m)

a.) Find the equation that represents the fall of the shade cloth

b.) To support the shade cloth, two supports are attached. Find the equation that models either of the two supports.

Applications of Parabolas

a.) Parabola is of the form y = kx2 + c

‘At its lowest point it will hang 2.2 metres above the paving.’ Therefore c = 2.2

y = kx2 + 2.2

Parabola is of the form y = kx2 + 2.2

A point that satisfies this equation is (2,3) (-2,3)

3 = k(2)2 + 2.20.8 = 4k

k = 0.2 y = 0.2x2 + 2.2

When x = 1 y = 0.2 (1)2 + 2.2 y = 2.4

(1, 2.4)

Applications of Parabolas

b.) Line is of the form y = mx ± c

2 points that satisfies this equation are (2,2)(1,2.4)

This gradient is true for any 2 points on the line

y = - 0.4x + 2.8

Gradient isxrun

yrise

12

12

xx

yy

12

4.22

= − 0.4

2

2

x

y = − 0.4

Multiply both sides by (x-2) y – 2 = - 0.4 (x − 2) y – 2 = - 0.4x + 0.8

HMWK pg 114-115


Find an equation for the following parabola in the form y = k (x – c)2 + d

y = k (x +6)2 - 8

Subst (1,4) into eq’n

4 = k (1 +6)2 - 84 = k (7)2 - 8

12 = 49k

k = 0.245

y = 0.245 (x +6)2 - 8

Find an equation for the following parabola in the form y = k (x – c)2 + dand then find the y-intercept

y = k (x – 3)2 + 13

Subst (4.22,3.57) into eq’n

3.57 = k (4.22 – 3)2 + 13

k = -6.34

y = -6.34 (x – 3)2 + 13

3.57 = k (1.22)2 + 13-9.43 = 1.4884k

To find the y-intercept, set x = 0

y = -6.34 (0 – 3)2 + 13y = -6.34 (9) + 13y = -6.34 (9) + 13y = -44.0

(0,-44)

The following parabola has been translated 4 units up and 6 units to the left. Give the equation of the parabola in its new position and find its new y-intercept.

y = k (x-5)2 + 2y = -2.2 (x-5)2 + 2y = -2.2 (x+1)2 + 6

y-intercept Set x = 0

y = -2.2 (0+1)2 + 6

y = 3.8

(0, 3.8)


Applications to ParabolasThe occupancy rate, as a percentage, over a period of 12 months for a new hotel, is expected to be represented by the graph drawn below. The graph compromises a straight line (which gives the occupancy rate for the first 3 months) and a parabola (which gives the occupancy rate for the remaining 9 months)

Write an equation for each part of the graph

1 2 3 4 5 6 7 8 9 10 11 12months

100

90

80

70

60

50

40

30

20

10

occupancy %

First 3 months (linear)gradient = y-intercept =

-30/3 = -10= 70 y = -10x + 70

Last 9 months (quadratic)vertex = (8, 20)

y = k(x-8)2 + 20 Subst (3, 40) to solve for k

40 = k(3-8)2 + 2020 = k(-5)2 k = 0.8

y = 0.8(x-8)2 + 20


STARTER Applications of Parabolas A civil engineer designs a model of a 6 metre suspension bridge using the equation , where h is the height, in metres, of the hanging cables above the road and d is the distance from the first pillar.

a.) What are possible values for d?

h = (d – 3)2 + 7

when d = 0, h = (0 – 3)2 + 7

6 m

1st pillar

b.) Sketch the graph of for appropriate values of d

h = (d – 3)2 + 7

c.) What is the height , h, of the pillars above the road?

h

= 16when d = 6, h = (6 – 3)2 + 7

= 16 The pillars are 16 m tall.

0 ≤ d ≤ 6

Applications of Parabolas - MERIT A civil engineer designs a model of a 6 metre suspension bridge using the equation , where h is the height, in metres, of the hanging cables above the road and d is the distance from the first pillar.

d.) If vertical cables are hung every 0.5 m along the bridge, determine the length of the cable needed:

h = (d – 3)2 + 7

i.) 2 m from the pillar

6 m

1st pillar

h

when d = 2, h = (2 – 3)2 + 7 = 8 metres

ii.) 3.5 m from the pillar

when d = 3.5, h = (3.5 – 3)2 + 7 = 7.25 metres

Applications of Parabolas - MERIT

The height of a parachutist above the ground is modelled by the equation

where t is the time in minutes after jumping from the airplane.

h = -40t2 + 4t + 10 000

a.) Sketch the graph of h vs. t

b.) How long does it take the parachutist to reach the ground?

c.) At what height did the parachutist jump from the plane?

set h = 0 (factorise the quadratic)

set t = 0 h = 10 000 m

t = 15.86 min or -15.76 min


A wood turner hones out a bowl according to the formula

where d is the depth of the bowl and x is the distance in cm from the centre

d = 1/3 x2 – 27

a.) Sketch the graph for appropriate values of x

b.) What is the width of the bowl?

from -9 to +9 - width is 18 cm

-2.5 = 1/3 x2 – 27 24.5 = 1/3 x2

c.) What is the distance away from the centre when d = -2.5?

73.5 = x2

x = ±8.57 cm (3 sf)


A wood turner hones out a bowl according to the formula

where d is the depth of the bowl and x is the distance in cm from the centre

d = 1/3 x2 – 27

d.) What is the depth d, 7 cm away from the centre.

d = 1/3 (7)2 – 27 d = -10.7 cm (3 sf)

d = 1/3 x2 – 27


The cross section of a curved skateboard ramp can be modeled by a parabola, given the equation The distance between B and C (the top points on either side of the ramp) is 6 m.

y = 0.2 x2

a.) Calculate the distance from the ground at the highest point (B or C)

y = 0.2(3)2 = 1.8 m

x = 3 or -3

b.) When it rains the ramp partially fills with a puddle of water. The puddle is 0.3125 m deep.

How wide is the puddle?


The cross section of a curved skateboard ramp can be modeled by a parabola, given the equation The distance between B and C (the top points on either side of the ramp) is 6 m.

y = 0.2 x2

0.3125 = 0.2(x)2 x2 = 1.5625

y = 0.3125

b.) When it rains the ramp partially fills with a puddle of water. The puddle is 0.3125 m deep.

How wide is the puddle?

The width is 2 x 1.25 = 2.5 m (2 sf)

x = 1.25

Applications of Parabolas (last question in assignment)Andrew takes part in a competition to see which of the players could throw a ball over a 5 m fence from the greatest distance. The ball must be on its way down as it clears the fence. He has a couple of trial and the flight of the ball is shown below. If he stood 43 m from the wall would the ball still clear it?

Parabola is of the form y = kx2 + d

0 = k(25)2 + 15-15 = 625k

k = -0.024

y = kx2 + 15

(25, 0) is on the parabola

y = -0.024x2 + 15

y = -0.024(x – 25)(x + 25)

Applications of Parabolas (last question in assignment)Andrew takes part in a competition to see which of the players could throw a ball over a 5 m fence from the greatest distance. The ball must be on its way down as it clears the fence. He has a couple of trial and the flight of the ball is shown below. If he stood 43 m from the wall would the ball still clear it?

y = -0.024(0 +18)2 + 15y = 7.224 m

he has shifted back 18 metres y = -0.024x2 + 15

The new equation for the shifted parabola is:

y = -0.024(x + 18)2 + 15

y = -0.024(x + 43)(x – 7)

18 m

Find y when x = 0

Mere buys a small tent for her little sister for Christmas.The top part is modelled by a parabola.There are three zips: BE, DE, and EF.The equation of the frame of the top part of the tent is where y is the height in cm and x is the distance from the centre line in cm.

a) What is the height of the top of the tent?

b) What is the length of the horizontal zip, EF?

c) The length of AC is 32 cm. Find the length of zip BE.

At x=0, y = 80 cm

At y =0, x = 40 cm, EF = 40 cm

At x = 16, y = 67.2 cm

Excellenced) Two poles, PR and QS help keep the tent upright.They are 25 cm from the centre-line of the tent.The bottom part of the tent is also modelled by a parabola.It cuts the vertical axis at y = –20.The length FG = 10 cm.Find the length of the pole PR.

Subst x = -25 into

To get y = 48.75 cm

Subst (40, -10) to solve for k in y = kx2 – 20 (from part a)k = 1/160

- 48.75

-25 40

Subst x=-25 into y = x2 – 20 160y= -16.09 cm The length of the pole PR is 48.75 + 16.09 = 64.84 cm

-16.09


StarterWrite equations for the lines shown below.

a.) ___________

b.) ___________

c.) ____________

y = 5

y = -2x – 7

y = x + 6

MeritIn decorating the hall for the dance streamers are hung across the hall as shown below. The equation for the height, in metres, of the streamer above the floor is:

y = (x – 5)2 + 3.25

1

Diagram NOT

to scale

(a)(i) What is the least distance between the floor and the streamer ?

Vertex - 3 metres

(ii) What is the height of the ceiling above the floor of the hall ?

Set x = 0 or x = 10

y = (0 – 5)2 + 325

1

y = 4 metres

(iii) What is the width of the hall ?

10 metres

ExcellenceAs part of the entrance to the dance the organizers are going to make an archway as shown in the plan. The archway will be 250 cm wide and 250 cm above the floor at its highest point. The curved part of the archway can be modeled by a parabola. The top curve has its lowest point 200 cm above the floor. The lower curve is going to be joined to the upper curve by connecting rods AB that are placed 75 cm from either side of the highest point and position the lower curve 200 cm above the floor at point B.

Calculate the length of the rods AB. Find an equation that models the curves used to form the archway.

B

250 cm

250 cm

200 cm

75 cm A

B

A

Diagram NOT to scale

y = k (x – 125)2 + 250 subst. (0, 200) to solve for k

200 = k (0 – 125)2 + 250 -50 = 15625k

k = -0.0032

y = -0.0032 (x – 125)2 + 250

ExcellenceCalculate the length of the rods AB.

Find an equation that models the curves used to form the archway.

B B

250 cm

250 cm

200 cm

75 cm AA

y = -0.0032 (x – 125)2 + 250

subst. x = 50 to solve for y

y= -0.0032 (50 – 125)2 + 250

50

Pt B has the coordinates ______

200 = -0.0032 (50 – 125)2 + c

y = 232 pt is (50, 232)

(50, 200)

c = 218

y = -0.0032 (x – 125)2 + 218

The differences between y – values at x = 50 is the length of rod AB

232 – 200 = 32 cm 250 – 218 = 32

Applications to ParabolasIn a science experiment, groups of students were firing small rockets which burnt for about 5 seconds and then fell back to earth.They decided to model the path of the rocket by a straight line and half a parabola.

Write an equation for each part of the graph

10 20 30 40 50 60distance (m)

50

45

40

35

30

25

20

15

10

5

height (m)

First 45 metres (linear)gradient = y-intercept =

30/45 = 0.6720

y = .67x + 20

Last 20 metres (quadratic)vertex = (45, 50)

y = k(x-45)2 + 50 Subst (65, 0) to solve for k

0 = k(65-45)2 + 50-50 = k(20)2 k = -0.125

y = -0.125(x-45)2 + 50

Applications to ParabolasCalculate the height of the rocket after it has travelled a distance of 55 m

10 20 30 40 50 60distance (m)

50

45

40

35

30

25

20

15

10

5

height (m)

= 37.5 m

35 = -0.125(x – 45)2 + 50Subst y = 35

-15 = -0.125(x – 45)2

x = 55.95 m

y = -0.125(55 – 45)2 + 50

After what distance travelled is the rocket at a height of 35 metres

120 = (x – 45)2 10.95 = x – 45

y = -0.125(x-45)2 + 50

Applications to ParabolasIn a science experiment, groups of students were firing small rockets which burnt for about 5 seconds and then fell back to earth.They decided to model the path of the rocket by a straight line and half a parabola.

10 20 30 40 50 60distance (m)

50

45

40

35

30

25

20

15

10

5

height (m)

and x = 55.95 m


After what distance travelled is the rocket at a height of 35 metres

y = .67x + 2035 = .67x + 2015 = .67x

x = 22.5

y = .6

7x + 20

Practice Assessment – Q2A school team is investigating selling sausages or discount cards for a fundraiser.

Discount card scheme is:- set up costs $10

Projected Earnings E – S + 10 = 0

Sausage scheme – how much would they earn/lose if the y spent $10 on sausages.

How much would they have to spend on sausages if they want to earn $50?lose $20

Which is more profitable?Sausages (if over $30 is spent) Steeper gradient.

E

SE = S – 10

The sausage scheme is shown on the graph E = 2S - 4

0

spend $45

E = S - 10

Practice Assessment – Q3The avg. length of local telephone calls (in min) made in Emily’s household over a period time can be modeled by the equation y = -1.5x(x – 16)

duration of calls x min

a.) Max. occurs at x =

y = -1.5(8)(8 – 16)= 96

8 min is the most common duration of call, its occurs 96 times.

b.) Longest length of call 16 min

8

c.) Write in VERTEX form of the equation, then shift 2 units to the right and 4 units up. y = -1.5(x – 8)2 + 96

y = -1.5(x – 10)2 + 100

subst. y = 0 and solve for x to find length of shortest call(find roots of the equation using GDC)

x = 1.84 min or 18.2 min

number of calls made y

Practice Assessment – Q4 (excellence)A telecommunication company undertook an ad campaign to gain clients. The number of new customers signed up daily over 1 month (30 days) is modeled by the parabola shown.

Day (x)

Day is 7 most successful, 115 new customers

(7, 115)

Aim is to sign up at least 65 customers per day. Were they successful ?

y = k(x – 7)2 + 115

subst. x = 30 to find the number of new customers on the last day of the campaign

y = 66.9, therefore Acme Net was successful in signing up at least 65 customers per day

New customers y (18, 104)

104 = k(18 – 7)2 + 115k =

11

1

y = (x – 7)2 + 11511

1

finish d (cm)

Applications of Parabolas A flea can jump up about 120 times its own body height. A flea jump that goes 10 cm sideways and reaches a height of 15 cm can be modeled by a parabola. The horizontal axis, d, the sideways distance (in cm) and the vertical axis shows the height, h, reached (in cm)

What is the equation of the parabola?

0 = k(10 – 5)2 + 15-15 = 25k

k = -0.6

(10, 0) is on the parabola

y = -0.6x2 + 15

h = k(d – 5)2 + 15

start

h(cm)

How tall was the flea? 15 / 120 = 0.125 cm

= 1.25 mm

0 = 0.1(x – 25)2

250 m

Applications of Parabolas The ‘Tower of Terror’ is a thrill ride. It starts along a horizontal track, reaching a top speed of 160 km/h in 7 seconds, then goes up a curve that can be modeled by a parabola and continues vertically up to the top of a 115 m tower. Then the ride drops in reverse back to the starting point.

The equation of the parabolic part of the graph is

What are the equations for the vertical and horizontal components of the ride?

y = 0.1(x – 25)2

start

Top

= 62.5 m

115

m

What are the coordinates of D, the y – intercept of the parabola?

x = 0 y = 0.1(0 – 25)2

D

What are the coordinates of B, the x – intercept of the parabola?y = 0 x = 25

(0, 62.5)

(25, 0)B

x = 0, y = 0

250 m

Applications of Parabolas The ‘Tower of Terror’ is a thrill ride. It starts along a horizontal track, reaching a top speed of 160 km/h in 7 seconds, then goes up a curve that can be modeled by a parabola and continues vertically up to the top of a 115 m tower. Then the ride drops in reverse back to the starting point.

Horizontal = ______________Vertical = _________________Curve = ______________

How long is the track approximately?

y = 0.1(x – 25)2

start

Top

= 67.3 m 1

15 m250 – 25 = 225 m

√(25)2 + (62.5)2D

What is the total distance you would travel while on this ride?Length 344 m ≤ l ≤ 365 m

x ≈ 690 m

(0, 62.5)

(25, 0)B

115 – 62.5 = 52.5 m

Applications of Parabolas A rugby player places the ball on the ground 46 metres from the goal posts. He steps back, runs towards the ball and kicks it towards the goal posts.Will he score a goal? The ball must pass over a bar 3 metres above the ground between the goal posts. The ball’s path through the air can be modeled with a parabola.

-36/242 = k0 = k(0 – 24)2 + 36

y = -0.0625(x – 24)2 + 36

y = k(x – 24)2 + 36

Subst. x = 46 to see if ball will clear the goal

(0,0) is on the parabola

= 5.75 m > 3.00 m yes it will clear

24 m46 m

Maximum height 36 m Goal

3 m high

y = -0.0625(46 – 24)2 + 36

Alternative equation y = -0.0625(x – 24)(x + 24)

Subst. point (0, 36)

1.3 algebra relationships between tables, equations or graphs 4 credits (external)

Documents

x y cost

x intercept

line y

linear x

linear y

graph of y

x term

equation y