s4 unit 2 relationships.notebook - weebly

S4 Unit 2 Relationships.notebook

1

December 12, 2013

Sep 2513:32

Today we are going to

learn about equations.

Sep 2609:41

Examples:

1. 7x ‐ 5 = 3x + 23

2. 2(3h ‐ 4) = 3(h + 1) ‐ 5

3. 11 ‐ x = 2 ‐ x4

4. x + 2 + x ‐ 12 5 = 10

1

Oct 117:49

Daily Practice 2.10.2013

Q1. Simplify 3k2 x 10k5

Q2. 3x/y ‐ 5/xy

Q3. Solve 3(x ‐ 1) + 2x(x + 5) = 30

6k

Oct 115:23

Oct 210:45


Q1. Multiply out and simplify (3k - 1)(2k + 7)

Q2. 12/5 x 31/3

Q3. Solve the inequality 3y - 2 > 7

Q4. Calculate the volume of a hemishere with radius 7cm

Q5. State the equation of the line with gradient 3 and passes through (4, 5)

Oct 210:47

Today we are going to revise over inequalities.

Homework 6 online Equations Due 23.10.2013


2

December 12, 2013

Oct 210:48

Inequalities 3.10.2013Notation:

> means greater than

< means less than

< means less than or equal to

> means greater than or equal to

Treat inequalities in the same way that you treat equations except when you are multiplying or dividing both sides by a negative number. Then the inequality is reversed.

Oct 210:53

Inequalities 3.10.2013Examples:

1. 4x ‐ 1 > 2x + 9 2. ‐5f > 4f ‐ 9 3. 2(q ‐ 3) < 5 + 7q

Page 142 Q3 & Q8

Oct 1108:08

Daily Practice 22.10.2013Q1. Calculate the volume of a cone with radius 7cm and height 18cm

Q2. Round 8125.39 to 2 significant figures

Q3. Karen put £800 in the bank at 4.2% interest compounded annually for 2 years. How much was in her account at the end of 2 years?

Q4. m5(2m2 + 3m)

Q5. Solve 3(x - 2) < 5x - 4

Oct 1108:09

Today we are going to learn how to solve simultaneous equations.

Oct 1108:10

Simultaneous Equations 22.10.2013Simultaneous Equations are equations that involve two unknowns.

They can be solved by:

1. Sketching the graph and stating the point of intersection.

2. Elimination ‐ eliminating one variable and then finding the other.

3. Substitution ‐ Substituting one equation into the other.

Oct 1108:14

Simultaneous Equations 22.10.2013

Solving by Elimination:

To solve a set of simultaneous equations by elimination add the equations to eliminate one of the variables (you may have to multiply one of the equations by a constant first)

Examples:

(i) 7m - 3y = 2 4m + 3y = 20


3

December 12, 2013

Oct 1108:19


(ii) 2a - b = 11 a - b = 5

Oct 1108:19


(iii) 2d + 6v = 36 3d - 2v = -1

Oct 1108:19


(iv) 12q + 8p = -34 2q - 2p = 6

Oct 1108:19


Solving by substitution: To solve simultaneous equations by substitution, get one variable in terms of the other. e.g. y in terms of x and then substitute into the other equation.

Examples:

(i) 3x + y = 18

x = y ‐ 2

Oct 1108:22


Solving by substitution

(ii) 2x ‐ 5y = ‐12

x ‐ y = ‐3

Oct 2215:26

Questions:

Solve by substitution -

Q1. y - 2x = 3 and 3y - 2x = 17

Q2. y - 4x - 3 = 5 and y + 7x - 30 = 0

Q3. 2x - y = -1 and 3x - y = 2

Solve by elimination -

Q1. x + 4y = 9

2x - 2y = 3

Q2. 4m - 6n = 7

5m - 8n = 8

Q3. a = 2b + 1

a + b = 10

23.10.2013


4

December 12, 2013

Oct 2307:59

Questions:

Solve by substitution -

Q1. y - 2x = 3 and 3y - 2x = 17

Oct 2307:59

Q2. y - 4x - 3 = 5 and y + 7x - 30 = 0

Oct 2307:59

Q3. 2x - y = -1 and 3x - y = 2

Oct 2307:59

Solve by elimination -

Q1. x + 4y = 9

2x - 2y = 3

Q2. 4m - 6n = 7

5m - 8n = 8

Q3. a = 2b + 1

a + b = 10

Oct 2307:57

Examples:

1. A rectangular park is x metres long and y metres broad. The difference between the length and the breadth is 50m, and the perimeter of the park is 200m. Calculate its length and breadth.

Worded Simultaneous Equations 23.10.2013

Oct 2307:57

2. Robyn sold 30 tickets for a concert.She sold x tickets for £3 each, and y tickets for £4.50 each. She collected £123 in total.

a. Write down two equations connecting x and y.

b. Solve these simultaneous equations to find the numbers of the two different types of tickets sold.


5

December 12, 2013

Oct 2307:54

3. Sara bought three identical blouses and four skirts costing £60.May paid £33 for three blouses and one skirt.(a) Find the cost of each blouse(b) Find the cost of 5 skirts

Oct 2307:55

Daily practice 24.10.2013

Q1.

Q2.

Q3.

Q4.(3m - 1)(2m - 1)

Find the value of a flat bought for £122 000 that appreciates in value by 3.8% a year for 3 years?

What is the equation of this line?

What is the volume of this shape?

(0, 3)

(5, 0)

Oct 2411:00 Oct 2410:41

Oct 2410:43 Oct 2308:05

Example: Two engines are being filled using different methods. The first has 10 litres of fuel in it and is being filled at a rate of 10 litres per minute and the other has 20 litres of fuel in it and is being filled at a rate of 5 litres per minute

(a) Draw graphs to represent the tanks being filled over a period of 6 minutes

(b) After how many minutes do the tanks hold the same volume?


6

December 12, 2013

Grid medium

(a) Draw graphs to represent the tanks being filled over a

period of 6 minutes

(b) After how many minutes do the tanks hold the same volume?

Grid large

Oct 2814:01

Daily Practice 29.10.2013Q1. Simplify 7k x 3k3 x 2k2

Q2. Calculate the volume of a hemisphere with diameter 19cm

Q3. Solve 1/4(5x + 5) < 10

Q4. State the equation of the line with gradient 3 that passes through (1, 3)

Q5. Write with a rational denominator 6√2

Oct 2416:00

Today we will be learning about quadratic functions

Oct 2416:07

Quadratic Functions 29.10.2013

Quadratic functions are functions that contain an x2

If you sketch a quadratic function, it will make a curve known as a parabola.

The simplest form is y = kx2

Oct 2416:12

Sketching quadratic functions 29.10.2013

The best way to sketch a quadratic function is to make a table of values for ‐3 x 3.

This means that you are finding the corresponding y ‐ value for each x ‐value between 3 and negative 3.

≤ ≤


7

December 12, 2013

Oct 2416:20

y = x2 + 1 X Y

3 102 51 20 11 22 53 10

4 3 2 1 0 1 2 3 4

2

4

6

8

10

x

y

Oct 2915:19

X Y

3 42 11 00 11 42 93 164 2 0 2 4

5

10

15

20

x

y

Grid medium Oct 2416:19

(b) y = (x + 2)2 (c) y = x2 + 2x ‐ 3

(d) y = (x ‐ 1)(x ‐ 2)

(e) y = (x ‐ 1)2 + 3

Questions: Sketch the graph of the following

(a) y = ‐x2

(f) y = x2 + 1

(g) y = x2 ‐ 1

X Y

3 02 31 40 31 02 53 12

Oct 2915:30

X Y

3 02 31 60 31 02 53 12

4 3 2 1 0 1 2 3 4

5

5

10

x

y

Oct 2416:19

Daily Practice 30.10.2013Q1.�Calculate�the�volume�of�this�square�based�pyramid

Q2.�State�the�gradient�of�the�line�3x�-�4y�+�8�=�0

Q3.�Simplify�

Q4.�Write�as�a�single�fraction��

Q5.�Calculate�the�gradient�of�the�line�joining�(-1,�-7)�and�(2,�5)

5cm

7cm


8

December 12, 2013

Oct 3007:59

Quadratic Functions 30.10.2013An n‐shaped curve will always have a minus sign in front of the x2

The nature of a turning point means stating whether it is maximum or minimum.

An n‐shaped curve has a maximum turning point.

A u‐shaped curve (positive x2) has a minimum turning point.

Oct 3007:59


Roots (x ‐ intercepts)

Turning point (minimum in this case)y ‐ intercept (when x = 0)

axis of symmetry (always goes through the TP and lies halfway between the roots)

y

x

Oct 3008:00

Quadratic Functions 30.10.2013Functions of the form y = kx2 :‐ Have a turning point of (0, 0)‐ Axis of symmetry is the y‐axis (x = 0)

To state the equation given the graph, substitute any point from the graph (except (0,0)) into the equation

Oct 3008:00


(3, 9)

Example: State the equation of the function below if it is in the form y = kx2

Oct 3009:28

Quadratic Functions 30.10.2013Questions: State the equations for each of the functions below if they are in the form y = kx2

(3,18 ) (3 ,45)

(‐2 ,28)

(2 ,28)(‐1 , 6)

(‐2 , 12)

Oct 2416:19

Daily Practice 31.10.2013Q1. Round 6177 to 1 significant figure

Q2. A group of witches are ages 25, 37, 48, 43, 64, 77 and 82. Calculate their mean age

Q3. Solve ½(7x - 1) < x + 17

Q4. Ghostbusters blasted the ghost population (12000) by 30% every minute for 3 minutes. How many ghosts were there after 3 mins?

Q5. A witch's hat is made from a special material. The material is made into a hat by using a template in the sector of a circle. Calculate the area of material a witch needs to make her hat

71015cm


9

December 12, 2013

Oct 2814:06


y = x2 y = kx2 + a y = kx2 ‐ a y = k(x ‐ a)2

y = k(x + a)2 y = k(x ‐ a)2 + by = k(x + a)2 + b

y = ‐x2

Oct 3009:35

Quadratic Functions 31.10.2013Quadratic functions such as y = ax2 + bx + c are written in completed square from to identify the turning point and axis of symmetry easier.

For Example: Below is the graph of y = (x + 1)2

Below is the graph of y = (x + 1)2 + 2

Below is the graph of y = ‐(x + 8)2 + 2

What is the turning point?



What is the axis of symmetry?



Grid large

Examples: State the i) equation ii) turning point iii)nature of the turning point iv) axis of symmetry of the following graphs


Grid medium

Oct 3010:51

Quadratic Functions 31.10.2013From our knowledge of the graph, we can now state the turning point of a function from it's equation alone.

y = k(x ‐ a)2 + b the turning point = (a, b)or if y = k(x + a)2 + b then the turning point = (‐a, b)

k determines whether a graph is n‐shaped or u‐shaped, if k > 0, then the graph is u ‐ shaped, if k < 0 then it is n‐shaped

Oct 3108:02


Example: State the turning point, the nature of the turning point and the axis of symmetry for the graph

(i) y = ( x ‐ 1)2 + 2 (ii) y = ‐(x + 7)2


10

December 12, 2013

Oct 3108:07


Questions: State the turning point, the nature of the turning point and the axis of symmetry for each of the equations below

(a) y = ( x ‐ 2)2 + 7 (b) y = (x + 1)2 (c) y = (x ‐ 8)2

(d) y = (x + 3)2 ‐ 5 (e) y = ‐(x ‐ 2)2 + 1 (f) y = 5x2

(g) Write x2 + 12x + 15 in completed square from, then state the turning point, axis of symmetry and nature of the turning point

Oct 3110:57

Daily Practice 5.11.2013Q1. Round 74.77 to 1 significant figure

Q2. Calculate the volume of a cylinder with

diameter 4cm and height 7cm

Q3. State the gradient of the line joining (-3, 4) and (2, -6)

Q4. State the turning point of y = -(x - 1)2 + 3

Q5. Write x2 + 12x + 15 in completed square form,

then state the turning point, axis of symmetry

and nature of the turning point

Nov 414:05

Today we are going to

learn about intercepts

Nov 414:08

Quadratic Functions 5.11.2013y ‐ intercept of quadratic function => x = 0

Examples: State the y ‐ intercept of the following

(i) y = (x ‐ 1)2 + 3 (ii) y = 2x2 ‐ 3x + 4 (iii) y = (x + 7)2

Nov 414:12

Quadratic Functions 5.11.2013x ‐ intercepts (Roots) of quadratic function => y = 0

Examples: State the roots of the following

(i) y = x2 ‐ 4

(ii) y = 4x2 ‐ 16

(iii) y = x2 + 7x + 12

(iv) y = (x + 3)2

Nov 414:24


Example:(i) State the y ‐ intercept of the function y = x2 ‐ 9x + 20(ii) State the roots of the function y = x2 ‐ 9x + 20(iii) Find the turning point of the equation x2 ‐ 9x + 20

Using the roots to find the turning point.


11

December 12, 2013

Nov 414:32


Q1. Draw and label the vector =

Q2. The length and breadth of a rectangle are (x + 2)cm and 5cm respectively. If the area of this rectangle is 60cm2, state the value of x

Q3. Solve 8x + 3y = 2 and 5x = 1 2y for x and y

Q4. Write (x + 5) + (x 3) as a single fraction

Q5. Calculate the area of triangle PQR

5 10x

1210

15.2cm

12.3cm

P

Q R

Grid large

Nov 614:00

Q2. The length and breadth of a rectangle are (x + 2)cm and 5cm respectively. If the area of this rectangle is 60cm2, state the value of x

Q3. Solve 8x + 3y = 2 and 5x = 1 2y for x and y

Nov 613:45

Q4. Write (x + 5) + (x 3) as a single fraction5 10x

Nov 520:38

Today we are going to continue to practise questions that involve finding the roots of quadratic functions.

Nov 520:41

Quadratic Functions 6.11.2013Using the roots to state the equation of a quadratic function

Rewrite the quadratic

in factorised form

(4, 0)(2, 0)


12

December 12, 2013

Nov 608:06

Quadratic Functions 6.11.2013Finding the roots of a quadratic that is in completed square form

Example: State the roots of the function y = (x ‐ 2)2 + 3

* Multiply the function out so that it is in the form y = ax2 + bx + c

then factorise

Nov 608:13

Nov 608:14 Nov 620:43

Daily Practice 7.11.2013Q1. State the turning point and axis of symmetry of y = ‐(x + 2)2 ‐ 3

Q2. Calculate the value of a house that was originally worth £148 000 and appreciated in value by 2.7% per annum over 3 years

Q3. Calculate the gradient of the line joining (‐1, 5) and (3, ‐4)

Q4. Rearrange 6x + 4 + 2y = 0 so that y is the subject

Q5. Simplify √75

Nov 708:11

Today we are going to be practising more questions on quadratics.

HW Online Due 13.11.2013

Nov 608:04


13

December 12, 2013

Nov 608:05 Nov 608:05

Nov 608:08 Nov 608:10

Nov 520:50

Quadratic Functions 7.11.2013Finding the roots by factorising

More examples:

(i) x2 +15x + 56 = 0 (ii) x2 ‐ 100 = 0 (iii) 3x2 ‐ 17x ‐ 28 = 0

(iv) x2 ‐ 7x = 0

Nov 520:53

Quadratic Functions 7.11.2013Questions: Finding the roots by factorising

(i) x2 + 17x + 72 = 0 (ii) 5x2 ‐ 10x = 0 (iii) 6x2 + 3x ‐ 63 = 0

(iv) 4x ‐ x2 = 0 (v) x2 + 3x = 0 (vi) 8x2 = 3 ‐ 2x


14

December 12, 2013

Nov 608:03

Quadratic Equations 7.11.2013

Nov 613:44

Daily Practice 12.11.2013Q1. Simplify 27 + 5 3 ‐ 63

Q2. Calculate the capacity of this cylinder

Q3. Calculate the gradient of the line joining (0, 2) and (4, ‐3)

Q4. Calculate the value of a house bought for £95000 that increases in value by 2.4% per annum for 3 years

Q5. State the roots of the function y = x2 + 7x ‐ 44

16 cm

22 cm

Nov 1114:20

Today we are going to learn how to find the roots of a function that cannot be factorised.

HW due tomorrow

Nov 1114:21

The Quadratic Formula 12.11.2013If you cannot find the roots of a function by factorising, it usually means that the roots aren't whole numbers.

We then use the quadratic formula:

Given ax2 + bx + c = 0

then x = ‐b b2 ‐ 4ac2a

+

Nov 1114:31

The Quadratic Formula 12.11.2013Examples: Find the roots of the following, give your answers to 1 d.p

(a) x2 + 7x ‐ 1 = 0

(b) 2x2 ‐ 5x + 4 = 0(c) 7x2 ‐ 10x ‐ 3 = 0

Nov 1114:31

The Quadratic Formula 12.11.2013Questions: Solve and give your answers to 1 d.p

(a) 2x2 + 4x + 1 = 0 (b) x2 + 3x ‐ 3 = 0 (c) 2x2 ‐ 7x + 4 = 0

(d) x2 ‐ 8x ‐ 1 = 0 (e) x2 + 3x ‐ 2 = 0 (f) 3x2 + 5x ‐ 7 = 0

(g) x2 ‐ 2x ‐ 2 = 0 (h) 4x2 ‐ 5x ‐ 3 = 0 (i) 5x2 ‐ 9x + 3 = 0

(j) Find the roots of the equation 2x2 ‐ 3x ‐ 4 = 0 to 2 d.p.

Hence solve the equation 2(x ‐ 1)2 ‐ 3(x ‐ 1) ‐ 4 = 0


15

December 12, 2013

Nov 1114:31

The Quadratic Formula 12.11.2013Questions: Solve and give your answers to 1 d.p

(a) 2x2 + 4x + 1 = 0 (b) x2 + 3x ‐ 3 = 0 (c) 2x2 ‐ 7x + 4 = 0

(d) x2 ‐ 8x ‐ 1 = 0 (e) x2 + 3x ‐ 2 = 0 (f) 3x2 + 5x ‐ 7 = 0

(j) Find the roots of the equation 2x2 ‐ 3x ‐ 4 = 0 to 2 d.p.


Nov 1215:48

Daily Practice 13.11.2013Q1.

Draw the resultant vector where

Q2. State the turning point of y = (x + 2)2 ‐ 3 and then state its nature

Q3. State the equation of the line parallel to 2x + y ‐ 3 = 0 and passes through (3, ‐4)

Q4. Find the roots of the equation 2x2 ‐ 3x ‐ 4 = 0 to 2 d.p.


Grid large Nov 1114:36

The Discriminant

The discriminant ‐ Gives us information about the roots of a function.

Nov 1208:00

The Discriminant

b2 ‐ 4ac > 0

b2 ‐ 4ac = 0

b2 ‐ 4ac < 0

2 real roots

If b2 ‐ 4ac is a

perfect square, roots are rational

1 real root

No real roots

(Imaginary roots)

Nov 1208:10

The DiscriminantExamples:

1. Find the discriminant for the following quadratics and state the nature of the roots.

(a) 3x2 ‐ 7x + 2 = 0 (b) x2 ‐ 3x + 4 = 0


16

December 12, 2013

Nov 1208:10

The DiscriminantExamples:

2. Find p given that 2x2 + 4x + p = 0 has real roots

3. Find the value of a such that the roots of x2 ‐ 8x + a = 0 are equal

Nov 1313:41

��

Q1. Rearrange the formula c2 = a2 + b2 such that b is the subject

Q2. Calculate the volume of a sphere with diameter 18cm and give your answer to 3 s.f.

Q3. Simplify 2m1/2(5m3/2 ‐ m5/2)

Q4. Factorise x2 + 8x ‐ 9

Q5. Calculate the area of the minor sector AOB

A

OB

7cm1180

Nov 1411:03

New HW online due 20.11.2013

Nov 1313:51

The Discriminant

Questions:

Find the discriminant for each of the following and state the nature of the roots

(a) x2 + 5x + 3 = 0

(b) 2x2 ‐ 5x ‐ 3 = 0

(c) 6x2 + 10 = 19x

(d) 2x ‐ 1 = x2

(e) The equation x2 + qx + 25 = 0 has one real root, state the value of q

(f) The roots of the equation 2x2 + ax + 2a = 0 are real and equal, find the value of a

Nov 1321:55

Maximum/Minimum problems in quadratic functions

Nov 1321:55

Maximum/Minimum problems in quadratic functions


17

December 12, 2013

Nov 1313:50

Today we are going to revise over Pythagoras' Theoremand apply our knowledge to some 3D problems.

Nov 1320:06

Nov 1410:54

Pythagoras' Theorem

c2 = a2 + b2 where c is the longest side in a right‐angled triangle. The formula can then be rearranged to find a shorter side.

a

b

c

The converse of this is that if the square of longest side is equal to the sum of the square of the two shorter sides, then the triangle must be right ‐ angled.

Nov 1320:32

Pythagoras' Theorem

Examples:

1. Show that the triangle PQR is right ‐ angled

2. Calculate the distance between C(1, ‐1) and D(4, 4)

13 cm

12 cm

5 cm

P

Q

R

Nov 1814:00


Q1. State the equation of the line that passes through (‐3, 2) and

(2, ‐4)

Q2. State the turning point of the function y = ‐(x + 2)2 ‐ 1

Q3. Calculate the volume of a cone with diameter 16cm and height 12cm to 3 s.f.

Q4. State the point of intersection of the lines 3x ‐ 4y ‐ 1 = 0 and y = 5

Q5. State the roots of the function y = (x ‐ 3)2 ‐ 49

Nov 2008:51


18

December 12, 2013

Nov 2008:53 Nov 2008:55

Nov 2008:56 Nov 1320:16

Pythagoras' Theorem Q1. Calculate the length of the missing side for each (a) (b) (c)

Q2. Prove that the triangle ABC is right‐angled

Q3. Calculate the distance between the following pairs of points(a) A(2, 0) and B(6, 3)(b) E(2, 2) and F(‐3, ‐10)(c) I(3, ‐1) and J(‐3, ‐5)

Q4. Find the length of the sloping side of this shape

7cm

6.5cm

k cmk cm 10 cm

15 cm2.5 m

g m

1.95 m

4cm

2cm

1cmA

B

CD

1m

20m

3.6m

k cm

6.5cm

7cm

Nov 2009:05


Q1. A function of the form y = kx2 and passes throught he point (3, 18), state the value of k

Q2. State the equation of the line that passes through (‐3, 4) and (2, ‐5)

Q3. Calculate the interior angle of this polygon

Q4. State the roots of the function x2 + 7x ‐ 30 = 0

Q5. State the value of each of the missing angles 1350

Nov 1320:06

Pythagoras' Theorem Pythagoras 3‐Dimensional Problems

Examples:

1. Calculate the length of AG

2. Calculate the perpendicular height of this cone Slant height = 25cm

hcm

20cm


19

December 12, 2013

Nov 1321:02

Pythagoras' Theorem Pythagoras 3‐Dimensional Problems Questions

Q1. Calculate the length of PQ Q2. Calculate height of this rectangular based pyramid

Q3. Calculate the length of AC

A

BC

D

FE

13cm15cm

6cm

Sep 1816:49

Today we are going to practise sketching straight lines and familiarising ourselves with the various terms.

Nov 1410:42 Nov 1410:43

Daily Practice 26.11.2013Q1. (x + 5)(2x2 + 3x + 1)

Q2. State the turning point and axis of symmetry of the function

y = (x ‐ 5)2 + 2

Q3. State the y intercept of the function y = (x + 1)(x ‐ 3)

Q4. Solve 5k(2k ‐ 1) = 0

Q5. Write as a single fraction x + 2 x ‐ 1 2 x

+

Nov 1814:04

Today we are going to learn how to sketch the graphs of Trigonometric Functions.

Nov 2009:16

Graphs of Trigonometric Functions

Trig. Graphs show the values of Sin, Cos and Tan of an angle usually between 00 and 3600. When drawn, you can see that Sin and Cos are wave functions.

The amplitude of a function is the distance between the x ‐ axis and the max/min. turning point

The period of a function is how far the graph reaches on the x ‐ axis before it repeats itself.


20

December 12, 2013

Grid large

xsin(x)

30 60 90 120 150 180 210 240 270 300 330 3600

30 60 90 120 150 180 210 240 270 300 330 3600

1

1

The Sine Function: The graph of y = Sinx where 00 ≤ x ≤ 3600

Grid large

xcos(x)

30 60 90 120 150 180 210 240 270 300 330 3600

30 60 90 120 150 180 210 240 270 300 330 3600

1

1

The Cosine Function: The graph of y = Cosx where 00 ≤ x ≤ 3600

Nov 2709:30


Q1. Multiply out and simplify 3m + (2m ‐ 1)(m + 4)

Q2. Calculate the gradient of the line joining (3, ‐1) and (7, 6)

Q3. Factorise 81 ‐ 100b2

Q4. Calculate the volume of a sphere with diameter 10cm to 2 s.f.

Q5. Rearrange 2t2 + b = c2 such that t is the subject

Grid large

xtan(x)

30 60 90 120 150 180 210 240 270 300 330 3600

30 60 90 120 150 180 210 240 270 300 330 3600

1

2

3

1

2

3

The Tangent Function: The graph of y = Tanx where 00 ≤ x ≤ 3600

Nov 2709:32

Today we will be practising sketching trig. graphs

Homework Online Due 4.12.2013

Pythagoras

Nov 2009:28


Trig. functions are written in the form

atanbx0 acosbx0 asinbx0

where a is the amplitude and b is how many times the graph repeats itself in 3600 (3600 ÷ period)

* When sketching graphs, draw the axis first, then the wave and then label your axes.


21

December 12, 2013

Nov 2009:31

00

Example: The graph of y = -sinx where 00 ≤ x ≤ 3600


Nov 2009:05

00


Example 2: Sketch the graph of y = cos3x0

Nov 2009:36


Example 3: Sketch the graph of y = 2sin2x0

Nov 2709:39


Questions:

Sketch the graphs of the following‐

(a) y = 4sin5x0

(b) y = 2cos2x0

(c) y = ‐cosx0

(d) y = 3sin6x0

(e) y = 2sinx0

(f) y = 3sin2x0

(g) y = cos5x0

(h) y = 2sin3x0

Nov 2009:32

00 900 1800 2700 3600

Nov 2009:32


22

December 12, 2013

Nov 2009:33

00 1800 3600

Nov 2009:33

00 1800 3600

1

2

3

1

2

3

Nov 2009:33

00 3600

1

1

24001200

Nov 2009:33

00 1800

1

2

1

2

3600

3

4

3

4

Nov 2009:34

00

1

1

2400 3600600 1200 1800 3000

Nov 2009:35

00 3600

1

1

2700


23

December 12, 2013

Nov 2009:36

00 3600

2

2

Nov 2715:50

�� Q1. Calculate the height of a cylinder with radius 7cm and volume 14000cm3

Q2. Calculate the value of a house that is worth £160 000 and appreciates in value by 12% a year over 3 years

Q3. Multiply out and simplify (m ‐ 7)(2m2 + 5m ‐ 1)

Q4. State the point where the line 2x + y = 12 crosses the x ‐ axis (remember y = 0, when a line crosses the x ‐ axis)

Q5. Factorise 2x2 + 9x + 4

Nov 2808:25

Today we will be learning how to identify the equation of tan graphs and the transformation of a trig. graph.

Nov 2808:24

The graph of Tan x0 has a period of 1800

Nov 2808:25 Nov 2513:12

Transformations of graphsA graph being up, down, left or right is known as its transformation.

A graph that moves up or down will be of the form y = asinbx0 + k.

Be careful as the amplitude will still be the same (think about the amplitude this time as being half the distance between the max. and min. turning points). Below is the graph of y = sinx0 ‐ 0.5

1

1

00 3600

0.5

0.5


24

December 12, 2013

Nov 2808:22 Nov 2716:02

Page 79 Q4.

Pg. 80 Q9 12

Dec 209:09


Q1. State the size of angle BAC

Q2. State the gradient of the

line joining (‐2, 7) and (3, ‐5)

Q3. State the axis of symmetry, turning point and the y ‐ intercept of the function y = (x ‐ 3)(x + 1)

Q4. State the equation of the function

B

A C410

7

7

x

y

1800 3600

Dec 209:13

Today we will be learning about more transformations of graphs.

Dec 308:10

So far, we have learned about graphs that shift up or down

2

6

00 1800

Dec 308:13

10

4

00 600


25

December 12, 2013

Dec 308:14

18

4

00 900

Dec 308:15

15

00 12003

Nov 2808:17

When the graph is moved left or right, the difference in x0 that it moves is known as the phase angle.

i.e. Given the graph y = Sin(x0 + 450), the phase angle is 450 and the graph of sin x0 is moved 450 to the left.

Similar to quadratic functions, if the phase angle is negative, the graph moves to the right, if it is positive, the graph moves to the left.

Transformations of graphs

Nov 2513:30

Transformations of graphs: Examples

Nov 2715:57 Dec 408:24


Q1. Write as a single fraction

Q2. Calculate the height of a cylinder with volume 3600cm3 and a radius of 5.3cm

Q3. Solve 3x2 + 2x ‐ 12 = 0 and give your answer to 1 d.p.

Q4. Two garden ponds are similar. The dimensions of the larger pond are 3 times as big as the smaller pond. The smaller pond holds 200litres of water. How many litres does the larger one hold?


26

December 12, 2013

Dec 409:29

Today we are going to consolidate our knowledge of Trig. Graphs

Homework Due

Dec 409:02

The relationship between the sin and cos graphs

The graph of Cosx0 is actually the same as the graph of Sin(x0 + 900)

You will see more of the relationships between sin, cos and tan when we are doing trigonometric identities.

Dec 413:23

Daily Practice 5.12.2013Q1. Solve the set of equations y = 2x ‐ 3 and 4x ‐ y = 15

Q2. (i) Calculate the mean, median, mode and range of

17, 3, 12, 4, 7, 5, 6, 8 and 10

(ii) State Q1 and Q3 and calculate the SIQR

Q3. AB is a chord 28cm long in a circle, centre O. The radius of the circle is 15cm. Find the length of OC

O

CA B

Dec 511:18

Dec 912:35

��

Q1. Simplify

Q2. Multiply out and simplify (3k ‐ 1)(k2 + 2k ‐ 7)

Q3. Rearrange the formula vmk2 = y such that k is the subject

Q4. Factorise 3m2 ‐ 19m ‐ 14

Q5. State the equation of the graph

4

4

x

y

1800 3600-500

Dec 409:33

Today we are going to learn how to solve trigonometric equations.

Homework Online Due 11.12.2013


27

December 12, 2013

Dec 409:15

Solving Trigonometric Equations

Similar to solving regular equations.

At first, we are going to just find the reference angle

Examples: Solve

i) 5cosx = 4 ii) 1 + 2 sinx0 = 2 iii) 9sinx0 + 6 = 2

Dec 409:29


Solve each of these to find the reference angle for 00 ≤ x ≤ 3600

(i) tanx + 1 = 0

(ii) cosx ‐ 0.6 = 0

(iii) 2cosx ‐ 1 = 0

(iv) 2tanx ‐ 2 = 0

(v) 5cosx + 4 = 0

Dec 409:19


y = sinx0

You may have noticed that Sin, Cos and Tan graphs repeat themselves over and over. Therefore there is more than one solution to a trig. equation.

Dec 409:19

y =cosx0

Dec 409:20

y = tanx0

Dec 914:15


Q1. Find the volume of a hemisphere with radius 18cm

Q2. Factorise x2 - 15x + 50

Q3. Sketch the graph of y = 3sin2x0

Q4. Simplify √300

Q5. Solve ¾x - 5 = 2x - 30


28

December 12, 2013

Dec 1113:51

Q1. Find the volume of a hemisphere with radius 18cm

Q2. Factorise x2 - 15x + 50

Q3. Sketch the graph of y = 3sin2x0

Q4. Simplify √300

Q5. Solve ¾x - 5 = 2x - 30

Dec 1113:55

Dec 1015:39 Dec 409:28


00, 3600

900

1800

2700

Dec 409:18


Finding more than one solution using graph symmetry.

Examples:

i) 6sinx0 + 1 = 0

Dec 409:22


ii) 18 + 4cosx0 = 15


29

December 12, 2013

Dec 409:22

iii) 2tanx0 ‐ 7 = 0


Dec 409:28

Solutions:

(a)300, 1500

(b) 41.40, 318.60

(c) 67.30, 247.30

(d) 19.50, 160.50

(e) 72.90, 252.90

(f) 33.60 , 326.40

(g) 131.80, 228.20

(h) 156.80, 336.80

(i) 236.40, 303.60

(j) 233.10, 126.90

Dec 914:15

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

Dec 1016:34


Q1. State the equation of the line joining (0, ‐3) and (4, 5)

Q2. Sketch the graph y = ‐(x ‐ 2)2 + 4

Q3. State the size of the external angle of anoctagon

Q4. Calculate the radius of a sphere with volume123800cm3

Q5. Solve the equation 3 + 5sinx = ‐1

Dec 1115:57

Today we will be learning

about trigonometric identities.

Dec 1115:57

Trigonometric Identities

Trig. Identities are relationships between Sin, Cos and Tan.

x

y

r

p0


30

December 12, 2013

Dec 1207:43


Using your knowledge from the last proof... sinp0 = y cosp0 = x tanp0 = y

x

y

r

p0

r r x

Dec 1207:47


Examples:

1. Show that cosxtanx = sinx

2. Show that (sinx ‐ cosx)2 + 2sinxcosx = 1

Dec 1211:50

2cos2A ‐ 1 = 1 ‐2sin2A

Dec 1212:01


Dec 1211:57


cos2x + sin2x = 1which meanscos2x = 1 ‐ sin2x sin2x = 1 ‐ cos2x

sinx = tanxcosx

Prove:

2cos2A ‐ 1 = 1 ‐2sin2A

Dec 1208:00

Scale Factor

*2 Shapes are congruent if they are the same shape and the same size.

*2 Shapes are similar if they are the same shape but one is an enlargement or reduction of the other.

Similar shapes have equal corresponding angles and their corresponding sides are in the same ratio

Enlargement Scale Factor = A dimension of the larger shape ÷ the same dimension of the smaller shape.

Reduction Scale Factor = A dimension of the smaller shape ÷ the same dimension of the larger shape.


31

December 12, 2013

Dec 1208:07

Scale Factor

Examples:

1. The trees below are similar. Calculate the size of the miniture tree.

1.8m

0.5m0.05m

? m

2. These triangles are similar. Calculate the lengths of x and y

x 18

20

30

15 y

s4 unit 2 relationships.notebook - weebly

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