AS Maths

Ms Parr:C1 - Algebraic expressions- Inequalities- Differentiation- Applications of

differentiation- Transforming graphs

Mr Corbridge:C1- Surds and indices- Equations and quadratic

functions- Simultaneous equations- Coordinate geometry and

the straight line- Coordinate geometry of the

circle

S1 C2

Organising your filesC1 File- Topics

- Algebraic expressions- ....- ....- Coordinate geometry of

the circle

- Assessment sheets and progress tests

- Completed past papers- Course overview/spec?

For each topic....- Lesson notes, worked

examples, handouts etc.- Exercises (classwork and

homework)*- Further independent

study/practice

* on separate sheets of paper

What you need to bring to my lessons

• Textbook• Formula book?• A4 paper, pen, pencil, rubber, ruler, calculator • C1 folder or as a minimum

– current topic(s) section(s)– current tracker/assessment sheet

• Complete C1 folder when requested• Homework when required

C1 ALGEBRAIC EXPRESSIONS

Wed 10/9/14LO: (i) review test(ii) Recap components of algebraic expressions(iii) understand function notation

Starting A level test

• Mark my paper

• Review your paper

Algebraic expressions

Write down an example of:(i) An expression(ii) An equation(iii) An identity(iv) A formula(v) An inequality

Some important terminology

• Variable• Constant• Term• Expression• Polynomial• Coefficient

C1 ALGEBRAIC EXPRESSIONS

Fri 12/9/14LO: (i)Understand function notation(ii)Add, subtract, multiply polynomials

Mappings

Sketch the following curves:1)y = 2x + 12)y = x3

3)y = 1/x4)x2 + y2 = 9

These all represent relationships between x and y.

The first 3 are formulae for calculating y given x.

Function notation

The formulae can also be written using function notation.1)y = 2x + 12)y = x3

3)y = 1/x

f(x) = 2x + 1g(x) = x3

h(x) = 1/x

or f:x →2x + 1or g:x → x3

or h:x → 1/x

Examples

f(x) = 3x2 + 4

f(y) =

f(2) =

f(-3) =

f(x) - 2 =

f(a – 2) =

f(x) = x(x+3)

f(y) =

f(2) =

f(-3) =

f(x) - 2 =

f(a – 2) =

PolynomialsA polynomial in x is an expression of the form

where a, b, c, … are constant coefficients and n is a positive integer.

1 2 2+ + +...+ + +n n nax bx cx px qx r

Examples of polynomials include:

Polynomials are usually written in descending powers of x.

3x7 + 4x3 – x + 8 x11 – 2x8 + 9x 5 + 3x2 – 2x3.and

The value of a is called the leading coefficient.

They can also be written in ascending powers of x, especially when the leading coefficient is negative, as in the last example.

Polynomials

A polynomial of degree 1 is called linear and has the general form ax + b.

A polynomial of degree 2 is called quadratic and has the general form ax2 + bx + c.

A polynomial of degree 3 is called cubic and has the general form ax3 + bx2 + cx + d.

A polynomial of degree 4 is called quartic and has the general form ax4 + bx3 + cx2 + dx + e.

The degree, or order, of a polynomial is given by the highest power of the variable.

Using function notationPolynomials are often expressed using function notation.

For example,

f(x) = 2x2 – 7

However, for polynomials, we often use the letter p instead of f, hence

p(x) = 2x2 – 7

Adding and subtracting polynomialsWhen two or more polynomials are added, subtracted or multiplied, the result is another polynomial.

Find a) f(x) + g(x) b) f(x) – g(x)

a) f(x) + g(x)

= 2x3 – 5x + 4 + 2x – 4

Polynomials are added and subtracted by collecting like terms.

= 2x3 – 3x

For example: f(x) = 2x3 – 5x + 4 and g(x) = 2x – 4

b) f(x) – g(x)

= 2x3 – 5x + 4 – (2x – 4)

= 2x3 – 5x + 4 – 2x + 4

= 2x3 – 7x + 8

Suppose p(x) has degree 4 and q(x) has degree 5. What is the degree of f(x) + g(x)?What about f(x) – g(x)?

Suppose f(x) and g(x) both have order 5. What is the order of f(x) + g(x)?

Adding and subtracting polynomials

Adding and subtracting polynomials

PRACTICEEx 2B Q 1-3

Multiplying polynomials

Don’t forget DOTS!

Remember how to multiply two linear expressions together to form a quadratic - for example,

(3x – 2)(2x – 1) =

Multiplying polynomials

When two polynomials are multiplied together every term in the first polynomial must by multiplied by every term in the second polynomial.

(3x3 – 2)(x3 + 5x – 1) =

(Check: do you have the right number of terms?)

Try this:

Multiplying polynomials

Suppose p(x) = (3x3 – 2). Find -2xp(x)

Let p(x) = 3x3 – 2 and q(x) = x3 + 5x – 1. Find p(x)q(x).

Example 1

Example 2

Find (3x + 2)(x - 5)(x2 – 1)

Example 3

(What is the degree of p(x)q(x)?)

Multiplying polynomials

Find the coefficient of x2 in the expansion of (2x + 3)(3x2 – 2x + 4).

Example 4

Multiplying polynomials

The product (Ax + B)(2x – 9) = 6x2 – 19x – 36, where A and B are constants.Find A and B.

Example 5

Function Notation and Multiplying Polynomials

Homework (from sheet)- Ex 3A(excl Q 5 & 7)- Miscellaneous Exercise 9

HomeworkI will...• set on Friday• collect on Wednesday• mark Wed/Thu• return on Friday

You will...• complete it• mark it (usually)• re-attempt if necessary• present it properly• hand it in on time

Individual study• Go over material from

lessons.• Follow up problem

areas.

• Practise by completing further questions.

C1 ALGEBRAIC EXPRESSIONS

Wed 17/9/14LO: Factorise expressions (including quadratics)

Warm-up: Mini-test

Marking/Collecting...Function Notation and Multiplying Polynomials- Ex 3A- Misc. Ex. 9

Factorising expressions (mini-test)

Factorising quadraticsax2 + bx + c

c = 0

Example 1: 10x2 – 2x

Factorising quadraticsax2 + bx + c

b = 0

Example 2: x2 – 9

Factorising quadraticsax2 + bx + c

b = 0

Example 3: 25x2 – 81

Factorising quadraticsax2 + bx + c

b = 0

Example 4: 5x2 – 80

Factorising quadraticsax2 + bx + c

b = 0

Example 5: x2 + 9

Factorising quadraticsax2 + bx + c

b = 0

Example 6: x2 + 9

Factorising quadraticsax2 + bx + c

a = 1

Example 7a: x2 + 5x + 6

Example 7b: x2 - 5x + 6

Example 7c: x2 - x - 6

a = -1

Example 7d: 12 + 4x - x2

Factorising quadraticsax2 + bx + c

a ≠ ± 1 but a or c is a prime number

Example 8a: 5x2 – 8x - 4

Example 8b: 6x2 + 13x - 5

Factorising quadraticsax2 + bx + c

Worst case scenario: neither a nor c is prime

Example 9: 4x2 - 5x - 6

Either try all combinations:

Or.....

Example: 4x2 - 5x - 6

i) Find 2 numbers that multiply to give ac and add to give b.

Factorising ax2 + bx + c – Guaranteed method

ii) Split the “x” term.iii) Factorise in pairs.iv) Complete the factorisation.

Factorising

Practice/Homework Ex 2C Q 2-5Ex 2D all QsFirst and last part of each Q only

C1 ALGEBRAIC EXPRESSIONS

Fri 19/9/14LO: Multiply and divide algebraic fractions

Reviewing...Factorising- Ex 2C Q 2-5- Ex 2D all QsAny issues?

Returning... Function Notation and Multiplying Polynomials- Ex 3A- Misc. Ex. 9

Homework review• Missing and mystery homework• C3/X see me at end of lesson• Questions not attempted• Careless errors

– “-” signs– Expanding brackets– Collecting like terms– Simple arithmetic

• Incorrect marking

-4 ≠ 4 4.1346... ≠ 42x2 – 4x + 4 ≠ x2 – 2z + 2

Presentation

• Your name• Title/Exercise number• Margin and question numbers (and a, b etc.)• Questions in the right order• Sequence within a question (↓)• Legible writing• Not cramped• Presentable. Rewrite if necessary!

Given p(x) = x2 - 4x + 3, q(x) = x2 – x – 4,find p(x) – q(x)

p(x) – q(x) = x2 - 4x + 3 - x2 – x – 4 p(x) – q(x) = x2 - 4x + 3 – (x2 – x – 4)

Given p(x) = x2 - 4x + 3, q(x) = x2 – x – 4,find p(x)q(x)

x4 – x3 - 4x2 - 4x3 + 4x2 + 16x + 3x2 – 3x - 12

x4 – 5x3 + 3x2 + 13x - 12

p(x)q(x) = x4 – x3 - 4x2 - 4x3 + 4x2 + 16x + 3x2 – 3x - 12

= x4 – 5x3 + 3x2 + 13x - 12

3 x -1 = -32 = 9 + 2 = 11

Given f(x) = 3x2 + 2, find f(-1).

3 x -1 = -3

Should have squared the -1 first (BIDMAS)

2( )

(-3)2 not -32

= 9

-32 = -9 not 9

- + 2≠

(-3)2 ≠ 9 + 2

f(-1) =f(-1) = is missing

= 11≠

3 x -1 ≠ -32

f(-1) = 3 x (-1)2 + 2= 3 x 1 + 2= 5

Homework review

• Ex 3A Q 4c• Misc 9 q 3b, 4 (We did one just like it!!!)

Simplify fractions

• See teaching notes 19/9

Multiply fractions

• See teaching notes 19/9

Divide by a fraction

• See teaching notes 19/9

Algebraic fractions

ReadP 35-37 Examples 22-25

Practice Ex 2E Q 4 all, Q5 a-e

Factorising/Fractions

Homework Ex 2C Q 2-5Ex 2D all Qs(from Wed)Ex 2E Q 4 all, Q5 a-e

C1 ALGEBRAIC EXPRESSIONS

Wed 24/9/14LO: Add and subtract algebraic fractions

Returning... Function Notation and Multiplying Polynomials- Ex 3A- Misc. Ex. 9 (Late entries)

SEE ME:

Peravin, Zendell, John, Ehimen, Shaquille, Cameron

Collecting...FactorisingEx 2C Q 2-5Ex 2D all Qsx and ÷ fractions Ex 2E Q 4 all, Q5 a-e

Adding and subtracting fractions

Consider:

34

35 =+ 20

+=

2720

Write as a single fraction in its lowest terms.3

y x

x y

2 23=

3 3

y x y x

x y xy

The LCM of 3x and y is ___.3xy

15 12

2 2

2 3 2 + 3+ =

x

x x x

Adding and subtracting fractions

Write as a single fraction in its lowest terms.2

2 3+

x x

The LCM of x2 and x is ___.x2

Consider:

35

415 =+ 15

+=

1315

9 4

Write as a single fraction in its lowest terms.2 1

++ 3 2 + 6

x

x x

The LCM of x + 3 and 2(x + 3) is ______.2(x + 3)

Start by factorizing where possible:2 1

++ 3 2( + 3)

x

x x

2 1+ =

+ 3 2( + 3)

x

x x

4 1+

2( + 3) 2( + 3)

x

x x

4 +1=

2( + 3)

x

x

Adding and subtracting fractions

Write as a single fraction in its lowest terms.2

2 + 83

+ 5

x

x

2

2 + 83 =

+ 5

x

x

2

2 2

3( + 5) 2 + 8

+ 5 + 5

x x

x x

2

2

3 +15 2 8=

+ 5

x x

x

Notice that this becomes – 8.

2

2

3 2 7=

+ 5

x x

x

Adding and subtracting fractions

Adding and subtracting fractions

Read pp 34-35 Examples 20&21PracticeEx 2E Q2

Alegebraic expressions - review

Revision homework for weekend P 39 Ex 2F middle column

Returning... Function Notation and Multiplying Polynomials- Ex 3A- Misc. Ex. 9 (Late entries)

SEE ME:

Peravin, John, Ehimen, Cameron,Tim, Shanil

C1 ALGEBRAIC EXPRESSIONS

Wed 1/10/14

Write the following as single fractions in their lowest terms.

Adding and subtracting fractions – trickier examples

1

a +bc

a)

1

- zxy

b)

c) 1 -1a

x + 1

1 -1x3

d)

e)

1b

+1a

ab

-ba