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Essential Algebra

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Topic/Skill Definition/Tips Example Expression A mathematical statement written using symbols, numbers or letters, 3x + 2 or 5y 2 Equation A statement showing that two expressions are equal 2y โ€“ 17 = 15 Identity An equation that is true for all values of the variables An identity uses the symbol: โ‰ก 2x โ‰ก x+x Formula Shows the relationship between two or more variables Area of a rectangle = length x width or A= LxW Simplifying Expressions Collect like terms. Be careful with negatives. 2 and are not like terms. 2 + 3 + 4 โˆ’ 5 + 3 = 6 โˆ’ 2 + 3 3 + 4 โˆ’ 2 + 2 โˆ’ 1 = 5 โˆ’ 2 +3 times The answer is 2 not 2. Squaring is multiplying by itself, not by 2. ร—ร— The answer is 3 not 3 If p=2, then 3 =2x2x2=8, not 2x3=6 ++ The answer is 3p not 3 If p=2, then 2+2+2=6, not 2 3 =8 Expand To expand a bracket, multiply each term in the bracket by the expression outside the bracket. 3( + 7) = 3 + 21 Factorise The reverse of expanding. Factorising is writing an expression as a product of terms by โ€˜taking outโ€™ a common factor. 6 โˆ’ 15 = 3(2 โˆ’ 5), where 3 is the common factor. Essential Algebra
Transcript
Page 1: Essential Algebra

Topic/Skill Definition/Tips Example

Expression A mathematical statement written using symbols, numbers or letters,

3x + 2 or 5y2

Equation A statement showing that two expressions are equal

2y โ€“ 17 = 15

Identity An equation that is true for all values of the variables An identity uses the symbol: โ‰ก

2x โ‰ก x+x

Formula Shows the relationship between two or more variables

Area of a rectangle = length x width or A= LxW

Simplifying Expressions

Collect like terms. Be careful with negatives. ๐‘ฅ2 and ๐‘ฅ are not like terms.

2๐‘ฅ + 3๐‘ฆ + 4๐‘ฅ โˆ’ 5๐‘ฆ + 3 = 6๐‘ฅ โˆ’ 2๐‘ฆ + 3 3๐‘ฅ + 4 โˆ’ ๐‘ฅ2 + 2๐‘ฅ โˆ’ 1 = 5๐‘ฅ โˆ’ ๐‘ฅ2 + 3

๐‘ฅ times ๐‘ฅ The answer is ๐‘ฅ2 not 2๐‘ฅ. Squaring is multiplying by itself, not by 2.

๐‘ ร— ๐‘ ร— ๐‘ The answer is ๐‘3 not 3๐‘ If p=2, then ๐‘3=2x2x2=8, not 2x3=6

๐‘ + ๐‘ + ๐‘ The answer is 3p not ๐‘3 If p=2, then 2+2+2=6, not 23 = 8

Expand To expand a bracket, multiply each term in the bracket by the expression outside the bracket.

3(๐‘š + 7) = 3๐‘ฅ + 21

Factorise The reverse of expanding. Factorising is writing an expression as a product of terms by โ€˜taking outโ€™ a common factor.

6๐‘ฅ โˆ’ 15 = 3(2๐‘ฅ โˆ’ 5), where 3 is the common factor.

Essential Algebra

Page 2: Essential Algebra

Topic/Skill Definition/Tips Example

Linear Sequence A number pattern with a common difference. 2, 5, 8, 11โ€ฆ is a linear sequence

Term Each value in a sequence is called a term. In the sequence 2, 5, 8, 11โ€ฆ, 8 is the third term of the sequence.

Term-to-term rule A rule which allows you to find the next term in a sequence if you know the previous term.

First term is 2. Term-to-term rule is โ€˜add 3โ€™ Sequence is: 2, 5, 8, 11โ€ฆ

nth term A rule which allows you to calculate the term that is in the nth position of the sequence. Also known as the โ€˜position-to-termโ€™ rule. n refers to the position of a term in a sequence.

nth term is 3๐‘› โˆ’ 1 The 100th term is 3 ร— 100 โˆ’ 1 = 299

Finding the nth term of a linear sequence

1. Find the difference. 2. Multiply that by ๐’. 3. Substitute ๐‘› = 1 to find out what number you need to add or subtract to get the first number in the sequence.

Find the nth term of: 3, 7, 11, 15โ€ฆ 1. Difference is +4 2. Start with 4๐‘› 3. 4 ร— 1 = 4, so we need to subtract 1 to get 3. nth term = 4๐‘› โˆ’ 1

Fibonacci type sequences

A sequence where the next number is found by adding up the previous two terms

The Fibonacci sequence is: 1,1,2,3,5,8,13,21,34 โ€ฆ

An example of a Fibonacci-type sequence is: 4, 7, 11, 18, 29 โ€ฆ

Geometric Sequence

A sequence of numbers where each term is found by multiplying the previous one by a number called the common ratio, r.

An example of a geometric sequence is: 2, 10, 50, 250 โ€ฆ

The common ratio is 5 Another example of a geometric sequence is: 81, โˆ’27, 9, โˆ’3, 1 โ€ฆ

The common ratio is โˆ’1

3

Quadratic Sequence

A sequence of numbers where the second difference is constant. A quadratic sequence will have a ๐‘›2 term.

nth term of a geometric sequence

๐’‚๐’“๐’โˆ’๐Ÿ where ๐‘Ž is the first term and ๐‘Ÿ is the common ratio

The nth term of 2, 10, 50, 250 โ€ฆ. Is

2 ร— 5๐‘›โˆ’1

nth term of a quadratic sequence

1. Find the first and second differences. 2. Halve the second difference and multiply this by ๐‘›2. 3. Substitute ๐‘› = 1,2,3,4 โ€ฆ into your expression so far. 4. Subtract this set of numbers from the corresponding terms in the sequence from the question. 5. Find the nth term of this set of numbers. 6. Combine the nth terms to find the overall nth term of the quadratic sequence. Substitute values in to check your nth term works for the sequence.

Find the nth term of: 4, 7, 14, 25, 40.. Answer: Second difference = +4 nth term = 2๐‘›2 Sequence: 4, 7, 14, 25, 40 2๐‘›2 2, 8, 18, 32, 50 Difference: 2, -1, -4, -7, -10 Nth term of this set is โˆ’3๐‘› + 5 Overall nth term: 2๐‘›2 โˆ’ 3๐‘› + 5

Triangular numbers

The sequence which comes from a pattern of dots that form a triangle.

1, 3, 6, 10, 15, 21 โ€ฆ

Sequences

Page 3: Essential Algebra

Topic/Skill Definition/Tips Example

Solve To find the answer/value of something Use inverse operations on both sides of the equation (balancing method) until you find the value for the letter.

Solve 2๐‘ฅ โˆ’ 3 = 7 Add 3 on both sides

2๐‘ฅ = 10 Divide by 2 on both sides

๐‘ฅ = 5

Inverse Opposite The inverse of addition is subtraction. The inverse of multiplication is division.

Rearranging Formulae

Use inverse operations on both sides of the formula (balancing method) until you find the expression for the letter.

Make x the subject of ๐‘ฆ =2๐‘ฅโˆ’1

๐‘ง

Multiply both sides by z

๐‘ฆ๐‘ง = 2๐‘ฅ โˆ’ 1 Add 1 to both sides

๐‘ฆ๐‘ง + 1 = 2๐‘ฅ Divide by 2 on both sides

๐‘ฆ๐‘ง + 1

2= ๐‘ฅ

We now have x as the subject.

Writing Formulae

Substitute letters for words in the question. Bob charges ยฃ3 per window and a ยฃ5 call out charge.

๐ถ = 3๐‘ + 5 Where N=number of windows and C=cost

Substitution Replace letters with numbers. Be careful of 5๐‘ฅ2. You need to square first, then multiply by 5.

๐‘Ž = 3, ๐‘ = 2 ๐‘Ž๐‘›๐‘‘ ๐‘ = 5. Find: 1. 2๐‘Ž = 2 ร— 3 = 6 2. 3๐‘Ž โˆ’ 2๐‘ = 3 ร— 3 โˆ’ 2 ร— 2 = 5 3. 7๐‘2 โˆ’ 5 = 7 ร— 22 โˆ’ 5 = 23

Equations and Formulae

Page 4: Essential Algebra

Topic/Skill Definition/Tips Example

Quadratic A quadratic expression is of the form

๐’‚๐’™๐Ÿ + ๐’ƒ๐’™ + ๐’„ where ๐‘Ž, ๐‘ and ๐‘ are numbers, ๐’‚ โ‰  ๐ŸŽ

Examples of quadratic expressions: ๐‘ฅ2

8๐‘ฅ2 โˆ’ 3๐‘ฅ + 7 Examples of non-quadratic expressions:

2๐‘ฅ3 โˆ’ 5๐‘ฅ2 9๐‘ฅ โˆ’ 1

Factorising Quadratics

When a quadratic expression is in the form ๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘ find the two numbers that add to give b and multiply to give c.

๐‘ฅ2 + 7๐‘ฅ + 10 = (๐‘ฅ + 5)(๐‘ฅ + 2) (because 5 and 2 add to give 7 and multiply to give 10)

๐‘ฅ2 + 2๐‘ฅ โˆ’ 8 = (๐‘ฅ + 4)(๐‘ฅ โˆ’ 2) (because +4 and -2 add to give +2 and multiply to give -8)

Difference of Two Squares

An expression of the form ๐’‚๐Ÿ โˆ’ ๐’ƒ๐Ÿ can be factorised to give (๐’‚ + ๐’ƒ)(๐’‚ โˆ’ ๐’ƒ)

๐‘ฅ2 โˆ’ 25 = (๐‘ฅ + 5)(๐‘ฅ โˆ’ 5) 16๐‘ฅ2 โˆ’ 81 = (4๐‘ฅ + 9)(4๐‘ฅ โˆ’ 9)

Solving Quadratics (๐‘Ž๐‘ฅ2 = ๐‘)

Isolate the ๐‘ฅ2 term and square root both sides. Remember there will be a positive and a negative solution.

2๐‘ฅ2 = 98 ๐‘ฅ2 = 49 ๐‘ฅ = ยฑ7

Solving Quadratics (๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ = 0)

Factorise and then solve = 0. ๐‘ฅ2 โˆ’ 3๐‘ฅ = 0 ๐‘ฅ(๐‘ฅ โˆ’ 3) = 0

๐‘ฅ = 0 ๐‘œ๐‘Ÿ ๐‘ฅ = 3

Solving Quadratics by Factorising (๐‘Ž = 1)

Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.

Solve ๐‘ฅ2 + 3๐‘ฅ โˆ’ 10 = 0 Factorise: (๐‘ฅ + 5)(๐‘ฅ โˆ’ 2) = 0

๐‘ฅ = โˆ’5 ๐‘œ๐‘Ÿ ๐‘ฅ = 2

Factorising Quadratics when ๐‘Ž โ‰  1

When a quadratic is in the form ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘

1. Multiply a by c = ac 2. Find two numbers that add to give b and multiply to give ac. 3. Re-write the quadratic, replacing ๐‘๐‘ฅ with the two numbers you found. 4. Factorise in pairs โ€“ you should get the same bracket twice 5. Write your two brackets โ€“ one will be the repeated bracket, the other will be made of the factors outside each of the two brackets.

Factorise 6๐‘ฅ2 + 5๐‘ฅ โˆ’ 4

1. 6 ร— โˆ’4 = โˆ’24 2. Two numbers that add to give +5 and multiply to give -24 are +8 and -3 3. 6๐‘ฅ2 + 8๐‘ฅ โˆ’ 3๐‘ฅ โˆ’ 4 4. Factorise in pairs:

2๐‘ฅ(3๐‘ฅ + 4) โˆ’ 1(3๐‘ฅ + 4) 5. Answer = (3๐‘ฅ + 4)(2๐‘ฅ โˆ’ 1)

8. Solving Quadratics by Factorising (๐‘Ž โ‰  1)

Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.

Solve 2๐‘ฅ2 + 7๐‘ฅ โˆ’ 4 = 0 Factorise: (2๐‘ฅ โˆ’ 1)(๐‘ฅ + 4) = 0

๐‘ฅ =1

2 ๐‘œ๐‘Ÿ ๐‘ฅ = โˆ’4

Solving Quadratics by Factorising

Page 5: Essential Algebra

Topic/Skill Definition/Tips Example

Quadratic A quadratic expression is of the form

๐’‚๐’™๐Ÿ + ๐’ƒ๐’™ + ๐’„ where ๐‘Ž, ๐‘ and ๐‘ are numbers, ๐’‚ โ‰  ๐ŸŽ

Examples of quadratic expressions: ๐‘ฅ2

8๐‘ฅ2 โˆ’ 3๐‘ฅ + 7 Examples of non-quadratic expressions:

2๐‘ฅ3 โˆ’ 5๐‘ฅ2 9๐‘ฅ โˆ’ 1

Factorising Quadratics

When a quadratic expression is in the form ๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘ find the two numbers that add to give b and multiply to give c.

๐‘ฅ2 + 7๐‘ฅ + 10 = (๐‘ฅ + 5)(๐‘ฅ + 2) (because 5 and 2 add to give 7 and multiply to give 10)

๐‘ฅ2 + 2๐‘ฅ โˆ’ 8 = (๐‘ฅ + 4)(๐‘ฅ โˆ’ 2) (because +4 and -2 add to give +2 and multiply to give -8)

Difference of Two Squares

An expression of the form ๐’‚๐Ÿ โˆ’ ๐’ƒ๐Ÿ can be factorised to give (๐’‚ + ๐’ƒ)(๐’‚ โˆ’ ๐’ƒ)

๐‘ฅ2 โˆ’ 25 = (๐‘ฅ + 5)(๐‘ฅ โˆ’ 5) 16๐‘ฅ2 โˆ’ 81 = (4๐‘ฅ + 9)(4๐‘ฅ โˆ’ 9)

Solving Quadratics (๐‘Ž๐‘ฅ2 = ๐‘)

Isolate the ๐‘ฅ2 term and square root both sides. Remember there will be a positive and a negative solution.

2๐‘ฅ2 = 98 ๐‘ฅ2 = 49 ๐‘ฅ = ยฑ7

Solving Quadratics (๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ = 0)

Factorise and then solve = 0. ๐‘ฅ2 โˆ’ 3๐‘ฅ = 0 ๐‘ฅ(๐‘ฅ โˆ’ 3) = 0

๐‘ฅ = 0 ๐‘œ๐‘Ÿ ๐‘ฅ = 3

Solving Quadratics by Factorising (๐‘Ž = 1)

Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.

Solve ๐‘ฅ2 + 3๐‘ฅ โˆ’ 10 = 0 Factorise: (๐‘ฅ + 5)(๐‘ฅ โˆ’ 2) = 0

๐‘ฅ = โˆ’5 ๐‘œ๐‘Ÿ ๐‘ฅ = 2

Quadratic Graph

A โ€˜U-shapedโ€™ curve called a parabola. The equation is of the form

๐‘ฆ = ๐’‚๐’™๐Ÿ + ๐’ƒ๐’™ + ๐’„, where ๐‘Ž, ๐‘ and ๐‘ are numbers, ๐’‚ โ‰  ๐ŸŽ. If ๐’‚ < ๐ŸŽ, the parabola is upside down.

Roots of a Quadratic

A root is a solution. The roots of a quadratic are the ๐’™-intercepts of the quadratic graph.

Turning Point of a Quadratic

A turning point is the point where a quadratic turns. On a positive parabola, the turning point is called a minimum. On a negative parabola, the turning point is called a maximum.

Factorising Quadratics when ๐‘Ž โ‰  1

When a quadratic is in the form ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘

1. Multiply a by c = ac

Factorise 6๐‘ฅ2 + 5๐‘ฅ โˆ’ 4

1. 6 ร— โˆ’4 = โˆ’24

Further Quadratics

Page 6: Essential Algebra

2. Find two numbers that add to give b and multiply to give ac. 3. Re-write the quadratic, replacing ๐‘๐‘ฅ with the two numbers you found. 4. Factorise in pairs โ€“ you should get the same bracket twice 5. Write your two brackets โ€“ one will be the repeated bracket, the other will be made of the factors outside each of the two brackets.

2. Two numbers that add to give +5 and multiply to give -24 are +8 and -3 3. 6๐‘ฅ2 + 8๐‘ฅ โˆ’ 3๐‘ฅ โˆ’ 4 4. Factorise in pairs:

2๐‘ฅ(3๐‘ฅ + 4) โˆ’ 1(3๐‘ฅ + 4) 5. Answer = (3๐‘ฅ + 4)(2๐‘ฅ โˆ’ 1)

Solving Quadratics by Factorising (๐‘Ž โ‰  1)

Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.

Solve 2๐‘ฅ2 + 7๐‘ฅ โˆ’ 4 = 0 Factorise: (2๐‘ฅ โˆ’ 1)(๐‘ฅ + 4) = 0

๐‘ฅ =1

2 ๐‘œ๐‘Ÿ ๐‘ฅ = โˆ’4

Completing the Square (when ๐‘Ž = 1)

A quadratic in the form ๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘ can be

written in the form (๐’™ + ๐’‘)๐Ÿ + ๐’’ 1. Write a set of brackets with ๐‘ฅ in and half the value of ๐‘. 2. Square the bracket.

3. Subtract (๐‘

2)

2and add ๐‘.

4. Simplify the expression. You can use the completing the square form to help find the maximum or minimum of quadratic graph.

Complete the square of ๐‘ฆ = ๐‘ฅ2 โˆ’ 6๐‘ฅ + 2

Answer: (๐‘ฅ โˆ’ 3)2 โˆ’ 32 + 2

= (๐‘ฅ โˆ’ 3)2 โˆ’ 7 The minimum value of this expression occurs when (๐‘ฅ โˆ’ 3)2 = 0, which occurs when ๐‘ฅ = 3 When ๐‘ฅ = 3, ๐‘ฆ = 0 โˆ’ 7 = โˆ’7

Minimum point = (3, โˆ’7)

Completing the Square (when ๐‘Ž โ‰  1)

A quadratic in the form ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘ can be

written in the form p(๐’™ + ๐’’)๐Ÿ + ๐’“ Use the same method as above, but factorise out ๐‘Ž at the start.

Complete the square of 4๐‘ฅ2 + 8๐‘ฅ โˆ’ 3

Answer: 4[๐‘ฅ2 + 2๐‘ฅ] โˆ’ 3

= 4[(๐‘ฅ + 1)2 โˆ’ 12] โˆ’ 3 = 4(๐‘ฅ + 1)2 โˆ’ 4 โˆ’ 3

= 4(๐‘ฅ + 1)2 โˆ’ 7 Solving Quadratics by Completing the Square

Complete the square in the usual way and use inverse operations to solve.

Solve ๐‘ฅ2 + 8๐‘ฅ + 1 = 0 Answer:

(๐‘ฅ + 4)2 โˆ’ 42 + 1 = 0 (๐‘ฅ + 4)2 โˆ’ 15 = 0

(๐‘ฅ + 4)2 = 15

(๐‘ฅ + 4) = ยฑโˆš15

๐‘ฅ = โˆ’4 ยฑ โˆš15 Solving Quadratics using the Quadratic Formula

A quadratic in the form ๐‘Ž๐‘ฅ2 + ๐‘๐‘ฅ + ๐‘ = 0 can be solved using the formula:

๐’™ =โˆ’๐’ƒ ยฑ โˆš๐’ƒ๐Ÿ โˆ’ ๐Ÿ’๐’‚๐’„

๐Ÿ๐’‚

Use the formula if the quadratic does not factorise easily.

Solve 3๐‘ฅ2 + ๐‘ฅ โˆ’ 5 = 0 Answer: ๐‘Ž = 3, ๐‘ = 1, ๐‘ = โˆ’5

๐‘ฅ =โˆ’1 ยฑ โˆš12 โˆ’ 4 ร— 3 ร— โˆ’5

2 ร— 3

๐‘ฅ =โˆ’1 ยฑ โˆš61

6

๐‘ฅ = 1.14 ๐‘œ๐‘Ÿ โˆ’ 1.47 (2 ๐‘‘. ๐‘. )

Page 7: Essential Algebra

Topic/Skill Definition/Tips Example

Simultaneous Equations

A set of two or more equations, each involving two or more variables (letters). The solutions to simultaneous equations satisfy both/all of the equations.

2๐‘ฅ + ๐‘ฆ = 7 3๐‘ฅ โˆ’ ๐‘ฆ = 8

๐‘ฅ = 3 ๐‘ฆ = 1

Variable A symbol, usually a letter, which represents a number which is usually unknown.

In the equation ๐‘ฅ + 2 = 5, ๐‘ฅ is the variable.

Coefficient A number used to multiply a variable. It is the number that comes before/in front of a letter.

6z 6 is the coefficient

z is the variable

Solving Simultaneous Equations (by Elimination)

1. Balance the coefficients of one of the variables. 2. Eliminate this variable by adding or subtracting the equations (Same Sign Subtract, Different Sign Add) 3. Solve the linear equation you get using the other variable. 4. Substitute the value you found back into one of the previous equations. 5. Solve the equation you get. 6. Check that the two values you get satisfy both of the original equations.

5๐‘ฅ + 2๐‘ฆ = 9 10๐‘ฅ + 3๐‘ฆ = 16

Multiply the first equation by 2.

10๐‘ฅ + 4๐‘ฆ = 18 10๐‘ฅ + 3๐‘ฆ = 16

Same Sign Subtract (+10x on both) ๐‘ฆ = 2

Substitute ๐‘ฆ = 2 in to equation.

5๐‘ฅ + 2 ร— 2 = 9 5๐‘ฅ + 4 = 9

5๐‘ฅ = 5 ๐‘ฅ = 1

Solution: ๐‘ฅ = 1, ๐‘ฆ = 2

Solving Simultaneous Equations (by Substitution)

1. Rearrange one of the equations into the form ๐‘ฆ =. .. or ๐‘ฅ =. .. 2. Substitute the right-hand side of the rearranged equation into the other equation. 3. Expand and solve this equation. 4. Substitute the value into the ๐‘ฆ =. .. or ๐‘ฅ =. .. equation. 5. Check that the two values you get satisfy both of the original equations.

๐‘ฆ โˆ’ 2๐‘ฅ = 3 3๐‘ฅ + 4๐‘ฆ = 1

Rearrange: ๐‘ฆ โˆ’ 2๐‘ฅ = 3 โ†’ ๐‘ฆ = 2๐‘ฅ + 3 Substitute: 3๐‘ฅ + 4(2๐‘ฅ + 3) = 1 Solve: 3๐‘ฅ + 8๐‘ฅ + 12 = 1

11๐‘ฅ = โˆ’11 ๐‘ฅ = โˆ’1

Substitute: ๐‘ฆ = 2 ร— โˆ’1 + 3

๐‘ฆ = 1 Solution: ๐‘ฅ = โˆ’1, ๐‘ฆ = 1

Simultaneous Equations

Page 8: Essential Algebra

Solving Simultaneous Equations (Graphically)

Draw the graphs of the two equations. The solutions will be where the lines meet. The solution can be written as a coordinate.

๐‘ฆ = 5 โˆ’ ๐‘ฅ and ๐‘ฆ = 2๐‘ฅ โˆ’ 1. They meet at the point with coordinates (2,3) so the answer is ๐‘ฅ = 2 and ๐‘ฆ = 3

Solving Linear and Quadratic Simultaneous Equations

Method 1: If both equations are in the same form (eg. Both ๐‘ฆ =โ€ฆ): 1. Set the equations equal to each other. 2. Rearrange to make the equation equal to zero. 3. Solve the quadratic equation. 4. Substitute the values back in to one of the equations. Method 2: If the equations are not in the same form: 1. Rearrange the linear equation into the form ๐‘ฆ =. .. or ๐‘ฅ =. .. 2. Substitute in to the quadratic equation. 3. Rearrange to make the equation equal to zero. 4. Solve the quadratic equation. 5. Substitute the values back in to one of the equations. You should get two pairs of solutions (two values for ๐‘ฅ, two values for ๐‘ฆ.) Graphically, you should have two points of intersection.

Example 1 Solve ๐‘ฆ = ๐‘ฅ2 โˆ’ 2๐‘ฅ โˆ’ 5 and ๐‘ฆ = ๐‘ฅ โˆ’ 1

๐‘ฅ2 โˆ’ 2๐‘ฅ โˆ’ 5 = ๐‘ฅ โˆ’ 1 ๐‘ฅ2 โˆ’ 3๐‘ฅ โˆ’ 4 = 0

(๐‘ฅ โˆ’ 4)(๐‘ฅ + 1) = 0 ๐‘ฅ = 4 and ๐‘ฅ = โˆ’1 ๐‘ฆ = 4 โˆ’ 1 = 3 and ๐‘ฆ = โˆ’1 โˆ’ 1 = โˆ’2 Answers: (4,3) and (-1,-2) Example 2 Solve ๐‘ฅ2 + ๐‘ฆ2 = 5 and ๐‘ฅ + ๐‘ฆ = 3

๐‘ฅ = 3 โˆ’ ๐‘ฆ (3 โˆ’ ๐‘ฆ)2 + ๐‘ฆ2 = 5

9 โˆ’ 6๐‘ฆ + ๐‘ฆ2 + ๐‘ฆ2 = 5 2๐‘ฆ2 โˆ’ 6๐‘ฆ + 4 = 0 ๐‘ฆ2 โˆ’ 3๐‘ฆ + 2 = 0

(๐‘ฆ โˆ’ 1)(๐‘ฆ โˆ’ 2) = 0 ๐‘ฆ = 1 and ๐‘ฆ = 2 ๐‘ฅ = 3 โˆ’ 1 = 2 and ๐‘ฅ = 3 โˆ’ 2 = 1

Answers: (2,1) and (1,2)

Page 9: Essential Algebra

Topic/Skill Definition/Tips Example

Inequality An inequality says that two values are not equal. ๐‘Ž โ‰  ๐‘ means that a is not equal to b.

7 โ‰  3 ๐‘ฅ โ‰  0

Inequality symbols

๐’™ > ๐Ÿ means x is greater than 2 ๐’™ < ๐Ÿ‘ means x is less than 3 ๐’™ โ‰ฅ ๐Ÿ means x is greater than or equal to 1 ๐’™ โ‰ค ๐Ÿ” means x is less than or equal to 6

State the integers that satisfy โˆ’2 < ๐‘ฅ โ‰ค 4.

-1, 0, 1, 2, 3, 4

Inequalities on a Number Line

Inequalities can be shown on a number line. Open circles are used for numbers that are less than or greater than (< ๐‘œ๐‘Ÿ >) Closed circles are used for numbers that are less than or equal or greater than or equal (โ‰ค ๐‘œ๐‘Ÿ โ‰ฅ)

๐‘ฅ โ‰ฅ 0

๐‘ฅ < 2

โˆ’5 โ‰ค ๐‘ฅ < 4 Graphical Inequalities

Inequalities can be represented on a coordinate grid. If the inequality is strict (๐‘ฅ > 2) then use a dotted line. If the inequality is not strict (๐‘ฅ โ‰ค 6) then use a solid line. Shade the region which satisfies all the inequalities.

Shade the region that satisfies: ๐‘ฆ > 2๐‘ฅ, ๐‘ฅ > 1 ๐‘Ž๐‘›๐‘‘ ๐‘ฆ โ‰ค 3

Quadratic Inequalities

Sketch the quadratic graph of the inequality. If the expression is > ๐’๐’“ โ‰ฅ then the answer will be above the x-axis. If the expression is < ๐’๐’“ โ‰ค then the answer will be below the x-axis. Look carefully at the inequality symbol in the question. Look carefully if the quadratic is a positive or negative parabola.

Solve the inequality ๐‘ฅ2 โˆ’ ๐‘ฅ โˆ’ 12 < 0 Sketch the quadratic:

The required region is below the x-axis, so the final answer is:

โˆ’3 < ๐‘ฅ < 4 If the question had been > 0, the answer would have been:

๐‘ฅ < โˆ’3 ๐‘œ๐‘Ÿ ๐‘ฅ > 4 Set Notation

A set is a collection of things, usually numbers, denoted with brackets { } {๐‘ฅ | ๐‘ฅ โ‰ฅ 7} means โ€˜the set of all xโ€™s, such that x is greater than or equal to 7โ€™ The โ€˜xโ€™ can be replaced by any letter. Some people use โ€˜:โ€™ instead of โ€˜|โ€™

{3, 6, 9} is a set.

{๐‘ฅ โˆถ โˆ’2 โ‰ค ๐‘ฅ < 5}

Inequalities

Page 10: Essential Algebra

Topic/Skill Definition/Tips Example

Function Machine

Takes an input value, performs some operations and produces an output value.

Function A relationship between two sets of values. ๐‘“(๐‘ฅ) = 3๐‘ฅ2 โˆ’ 5 โ€˜For any input value, square the term, then multiply by 3, then subtract 5โ€™.

Function notation

๐‘“(๐‘ฅ) ๐’™ is the input value ๐’‡(๐’™) is the output value.

๐‘“(๐‘ฅ) = 3๐‘ฅ + 11 Suppose the input value is ๐‘ฅ = 5 The output value is ๐‘“(5) = 3 ร— 5 + 11 =26

Inverse function ๐‘“โˆ’1(๐‘ฅ) A function that performs the opposite process of the original function. 1. Write the function as ๐‘ฆ = ๐‘“(๐‘ฅ) 2. Rearrange to make ๐‘ฅ the subject.

3. Replace the ๐’š with ๐’™ and the ๐’™ with ๐’‡โˆ’๐Ÿ(๐’™)

๐‘“(๐‘ฅ) = (1 โˆ’ 2๐‘ฅ)5. Find the inverse.

๐‘ฆ = (1 โˆ’ 2๐‘ฅ)5

โˆš๐‘ฆ5 = 1 โˆ’ 2๐‘ฅ

1 โˆ’ โˆš๐‘ฆ5 = 2๐‘ฅ

1โˆ’ โˆš๐‘ฆ5

2= ๐‘ฅ

๐‘“โˆ’1(๐‘ฅ) =1 โˆ’ โˆš๐‘ฅ

5

2

Composite function

A combination of two or more functions to create a new function. ๐’‡๐’ˆ(๐’™) is the composite function that substitutes the function ๐’ˆ(๐’™) into the function ๐’‡(๐’™). ๐’‡๐’ˆ(๐’™) means โ€˜do g first, then fโ€™ ๐’ˆ๐’‡(๐’™) means โ€˜do f first, then gโ€™

๐‘“(๐‘ฅ) = 5๐‘ฅ โˆ’ 3, ๐‘”(๐‘ฅ) =1

2๐‘ฅ + 1

What is ๐‘“๐‘”(4)?

๐‘”(4) =1

2ร— 4 + 1 = 3

๐‘“(3) = 5 ร— 3 โˆ’ 3 = 12 = ๐‘“๐‘”(4) What is ๐‘“๐‘”(๐‘ฅ)?

๐‘“๐‘”(๐‘ฅ) = 5 (1

2๐‘ฅ + 1) โˆ’ 3 =

5

2๐‘ฅ + 2

Functions

Page 11: Essential Algebra

Topic/Skill Definition/Tips Example

1. Algebraic Fraction

A fraction whose numerator and denominator are algebraic expressions.

6๐‘ฅ

3๐‘ฅ โˆ’ 1

2. Adding/ Subtracting Algebraic Fractions

For ๐‘Ž

๐‘ยฑ

๐‘

๐‘‘ , the common denominator is ๐‘๐‘‘

๐‘Ž

๐‘ยฑ

๐‘

๐‘‘=

๐‘Ž๐‘‘

๐‘๐‘‘ยฑ

๐‘๐‘

๐‘๐‘‘=

๐‘Ž๐‘‘ ยฑ ๐‘๐‘

๐‘๐‘‘

1

๐‘ฅ+

๐‘ฅ

2๐‘ฆ

=1(2๐‘ฆ)

2๐‘ฅ๐‘ฆ+

๐‘ฅ(๐‘ฅ)

2๐‘ฅ๐‘ฆ

=2๐‘ฆ + ๐‘ฅ2

2๐‘ฅ๐‘ฆ

3. Multiplying Algebraic Fractions

Multiply the numerators together and the denominators together.

๐‘Ž

๐‘ร—

๐‘

๐‘‘=

๐‘Ž๐‘

๐‘๐‘‘

๐‘ฅ

3ร—

๐‘ฅ + 2

๐‘ฅ โˆ’ 2

=๐‘ฅ(๐‘ฅ + 2)

3(๐‘ฅ โˆ’ 2)

=๐‘ฅ2 + 2๐‘ฅ

3๐‘ฅ โˆ’ 6

4. Dividing Algebraic Fractions

Multiply the first fraction by the reciprocal of the second fraction.

๐‘Ž

๐‘รท

๐‘

๐‘‘=

๐‘Ž

๐‘ร—

๐‘‘

๐‘=

๐‘Ž๐‘‘

๐‘๐‘

๐‘ฅ

3รท

2๐‘ฅ

7

=๐‘ฅ

3ร—

7

2๐‘ฅ

=7๐‘ฅ

6๐‘ฅ=

7

6

5. Simplifying Algebraic Fractions

Factorise the numerator and denominator and cancel common factors.

๐‘ฅ2 + ๐‘ฅ โˆ’ 6

2๐‘ฅ โˆ’ 4=

(๐‘ฅ + 3)(๐‘ฅ โˆ’ 2)

2(๐‘ฅ โˆ’ 2)=

๐‘ฅ + 3

2

Algebraic Fractions

Page 12: Essential Algebra

Topic/Skill Definition/Tips Example

Iteration The act of repeating a process over and over again, often with the aim of approximating a desired result more closely.

Recursive Notation: ๐‘ฅ๐‘›+1 = โˆš3๐‘ฅ๐‘› + 6

๐‘ฅ1 = 4

๐‘ฅ2 = โˆš3 ร— 4 + 6 = 4.242640 โ€ฆ

๐‘ฅ3 = โˆš3 ร— 4.242640 โ€ฆ + 6= 4.357576 โ€ฆ

Iterative Method

To create an iterative formula, rearrange an equation with more than one x term to make one of the x terms the subject. You will be given the first value to substitute in, often called ๐’™๐Ÿ. Keep substituting in your previous answer until your answers are the same to a certain degree of accuracy. This is called converging to a limit. Use the โ€˜ANSโ€™ button on your calculator to keep substituting in the previous answer.

Use an iterative formula to find the positive root of ๐‘ฅ2 โˆ’ 3๐‘ฅ โˆ’ 6 = 0 to 3 decimal places. ๐‘ฅ1 = 4 Answer:

๐‘ฅ2 = 3๐‘ฅ + 6

๐‘ฅ = โˆš3๐‘ฅ + 6

So ๐‘ฅ๐‘›+1 = โˆš3๐‘ฅ๐‘› + 6

๐‘ฅ1 = 4

๐‘ฅ2 = โˆš3 ร— 4 + 6 = 4.242640 โ€ฆ

๐‘ฅ3 = โˆš3 ร— 4.242640 โ€ฆ + 6= 4.357576 โ€ฆ

Keep repeatingโ€ฆ

๐‘ฅ7 = 4.372068. . = 4.372 (3๐‘‘๐‘) ๐‘ฅ8 = 4.372208 โ€ฆ = 4.372 (3๐‘‘๐‘)

So answer is ๐‘ฅ = 4.372 (3๐‘‘๐‘)

Topic: Iteration

Page 13: Essential Algebra

Topic/Skill Definition/Tips Example

Expression A mathematical statement written using symbols, numbers or letters,

3x + 2 or 5y2

Equation A statement showing that two expressions are equal

2y โ€“ 17 = 15

Identity An equation that is true for all values of the variables An identity uses the symbol: โ‰ก

2x โ‰ก x+x

Formula Shows the relationship between two or more variables

Area of a rectangle = length x width or A= LxW

Coefficient A number used to multiply a variable. It is the number that comes before/in front of a letter.

6z 6 is the coefficient z is the variable

Odds and Evens An even number is a multiple of 2 An odd number is an integer which is not a multiple of 2.

If n is an integer (whole number): An even number can be represented by 2n or 2m etc. An odd number can be represented by 2n-1 or 2n+1 or 2m+1 etc.

Consecutive Integers

Whole numbers that follow each other in order.

If n is an integer: n, n+1, n+2 etc. are consecutive integers.

Square Terms A term that is produced by multiply another term by itself.

If n is an integer: ๐‘›2, ๐‘š2 etc. are square integers

Sum The sum of two or more numbers is the value you get when you add them together.

The sum of 4 and 6 is 10

Product The product of two or more numbers is the value you get when you multiply them together.

The product of 4 and 6 is 24

Multiple To show that an expression is a multiple of a number, you need to show that you can factor out the number.

4๐‘›2 + 8๐‘› โˆ’ 12 is a multiple of 4 because it can be written as:

4(๐‘›2 + 2๐‘› โˆ’ 3)

Algebraic Proofs

Page 14: Essential Algebra

Topic/Skill Definition/Tips Example

Coordinates Written in pairs. The first term is the x-coordinate (movement across). The second term is the y-coordinate (movement up or down)

A: (4,7) B: (-6,-3)

Midpoint of a Line

Method 1: add the x coordinates and divide by 2, add the y coordinates and divide by 2 Method 2: Sketch the line and find the values half way between the two x and two y values.

Find the midpoint between (2,1) and (6,9) 2+6

2= 4 and

1+9

2= 5

So, the midpoint is (4,5)

Linear Graph Straight line graph. The general equation of a linear graph is

๐’š = ๐’Ž๐’™ + ๐’„ where ๐’Ž is the gradient and ๐‘ is the y-intercept. The equation of a linear graph can contain an x-term, a y-term and a number.

Example: Other examples: ๐‘ฅ = ๐‘ฆ ๐‘ฆ = 4 ๐‘ฅ = โˆ’2 ๐‘ฆ = 2๐‘ฅ โˆ’ 7 ๐‘ฆ + ๐‘ฅ = 10 2๐‘ฆ โˆ’ 4๐‘ฅ = 12

Plotting Linear Graphs

Method 1: Table of Values Construct a table of values to calculate coordinates. Method 2: Gradient-Intercept Method (use when the equation is in the form ๐‘ฆ = ๐‘š๐‘ฅ + ๐‘) 1. Plots the y-intercept 2. Using the gradient, plot a second point. 3. Draw a line through the two points plotted. Method 3: Cover-Up Method (use when the equation is in the form ๐‘Ž๐‘ฅ + ๐‘๐‘ฆ = ๐‘) 1. Cover the ๐‘ฅ term and solve the resulting equation. Plot this on the ๐‘ฅ โˆ’ ๐‘Ž๐‘ฅ๐‘–๐‘ . 2. Cover the ๐‘ฆ term and solve the resulting equation. Plot this on the ๐‘ฆ โˆ’ ๐‘Ž๐‘ฅ๐‘–๐‘ . 3. Draw a line through the two points plotted.

Coordinates and Linear Graphs

Page 15: Essential Algebra

Gradient The gradient of a line is how steep it is. Gradient =

๐‘ช๐’‰๐’‚๐’๐’ˆ๐’† ๐’Š๐’ ๐’š

๐‘ช๐’‰๐’‚๐’๐’ˆ๐’† ๐’Š๐’ ๐’™=

๐‘น๐’Š๐’”๐’†

๐‘น๐’–๐’

The gradient can be positive (sloping upwards) or negative (sloping downwards)

Finding the Equation of a Line given a point and a gradient

Substitute in the gradient (m) and point (x,y) in to the equation ๐’š = ๐’Ž๐’™ + ๐’„ and solve for c.

Find the equation of the line with gradient 4 passing through (2,7).

๐‘ฆ = ๐‘š๐‘ฅ + ๐‘ 7 = 4 ร— 2 + ๐‘

๐‘ = โˆ’1

๐‘ฆ = 4๐‘ฅ โˆ’ 1 Finding the Equation of a Line given two points

Use the two points to calculate the gradient. Then repeat the method above using the gradient and either of the points.

Find the equation of the line passing through (6,11) and (2,3)

๐‘š =11 โˆ’ 3

6 โˆ’ 2= 2

๐‘ฆ = ๐‘š๐‘ฅ + ๐‘

11 = 2 ร— 6 + ๐‘ ๐‘ = โˆ’1

๐‘ฆ = 2๐‘ฅ โˆ’ 1

Parallel Lines If two lines are parallel, they will have the same gradient. The value of m will be the same for both lines.

Are the lines ๐‘ฆ = 3๐‘ฅ โˆ’ 1 and 2๐‘ฆ โˆ’ 6๐‘ฅ +10 = 0 parallel? Answer: Rearrange the second equation in to the form ๐‘ฆ = ๐‘š๐‘ฅ + ๐‘

2๐‘ฆ โˆ’ 6๐‘ฅ + 10 = 0 โ†’ ๐‘ฆ = 3๐‘ฅ โˆ’ 5 Since the two gradients are equal (3), the lines are parallel.

Perpendicular Lines

If two lines are perpendicular, the product of their gradients will always equal -1. The gradient of one line will be the negative reciprocal of the gradient of the other line. You may need to rearrange equations of lines to compare gradients (they need to be in the form ๐‘ฆ = ๐‘š๐‘ฅ + ๐‘)

Find the equation of the line perpendicular to ๐‘ฆ = 3๐‘ฅ + 2 which passes through (6,5) Answer: As they are perpendicular, the gradient of

the new line will be โˆ’1

3 as this is the

negative reciprocal of 3.

๐‘ฆ = ๐‘š๐‘ฅ + ๐‘

5 = โˆ’1

3ร— 6 + ๐‘

๐‘ = 7

๐‘ฆ = โˆ’1

3๐‘ฅ + 7

Or 3๐‘ฅ + ๐‘ฅ โˆ’ 7 = 0

Page 16: Essential Algebra

Topic/Skill Definition/Tips Example

Real Life Graphs Graphs that are supposed to model some real-life situation. The actual meaning of the values depends on the labels and units on each axis. The gradient might have a contextual meaning. The y-intercept might have a contextual meaning. The area under the graph might have a contextual meaning.

A graph showing the cost of hiring a ladder for various numbers of days. The gradient shows the cost per day. It costs ยฃ3/day to hire the ladder. The y-intercept shows the additional cost/deposit/fixed charge (something not linked to how long the ladder is hired for). The additional cost is ยฃ7.

Conversion Graph

A line graph to convert one unit to another. Can be used to convert units (eg. miles and kilometres) or currencies ($ and ยฃ) Find the value you know on one axis, read up/across to the conversion line and read the equivalent value from the other axis.

8 ๐‘˜๐‘š = 5 ๐‘š๐‘–๐‘™๐‘’๐‘  Depth of Water in Containers

Graphs can be used to show how the depth of water changes as different shaped containers are filled with water at a constant rate.

Real Life Graphs

Page 17: Essential Algebra

Topic/Skill Definition/Tips Example

Coordinates Written in pairs. The first term is the x-coordinate (movement across). The second term is the y-coordinate (movement up or down)

A: (4,7) B: (-6,-3)

Linear Graph Straight line graph. The equation of a linear graph can contain an x-term, a y-term and a number.

Example: Other examples: ๐‘ฅ = ๐‘ฆ ๐‘ฆ = 4 ๐‘ฅ = โˆ’2 ๐‘ฆ = 2๐‘ฅ โˆ’ 7 ๐‘ฆ + ๐‘ฅ = 10 2๐‘ฆ โˆ’ 4๐‘ฅ = 12

Quadratic Graph

A โ€˜U-shapedโ€™ curve called a parabola. The equation is of the form

๐‘ฆ = ๐’‚๐’™๐Ÿ + ๐’ƒ๐’™ + ๐’„, where ๐‘Ž, ๐‘ and ๐‘ are numbers, ๐’‚ โ‰  ๐ŸŽ. If ๐’‚ < ๐ŸŽ, the parabola is upside down.

Cubic Graph The equation is of the form ๐’š = ๐’‚๐’™๐Ÿ‘ + ๐’Œ, where

๐’Œ is an number. If ๐’‚ > ๐ŸŽ, the curve is increasing. If ๐’‚ < ๐ŸŽ, the curve is decreasing.

Reciprocal Graph

The equation is of the form ๐’š =๐‘จ

๐’™, where ๐‘จ is a

number and ๐’™ โ‰  ๐ŸŽ. The graph has asymptotes on the x-axis and y-axis.

Asymptote A straight line that a graph approaches but never

touches.

Graphs and Graph Transformations

Page 18: Essential Algebra

Exponential Graph

The equation is of the form ๐’š = ๐’‚๐’™, where ๐‘Ž is a number called the base. If ๐’‚ > ๐Ÿ the graph increases. If ๐ŸŽ < ๐’‚ < ๐Ÿ, the graph decreases. The graph has an asymptote which is the x-axis.

๐‘ฆ = sin ๐‘ฅ Key Coordinates:

(๐ŸŽ, ๐ŸŽ), (๐Ÿ—๐ŸŽ, ๐Ÿ), (๐Ÿ๐Ÿ–๐ŸŽ, ๐ŸŽ), (๐Ÿ๐Ÿ•๐ŸŽ, โˆ’๐Ÿ), (๐Ÿ‘๐Ÿ”๐ŸŽ, ๐ŸŽ) ๐‘ฆ is never more than 1 or less than -1. Pattern repeats every 360ยฐ.

๐‘ฆ = cos ๐‘ฅ Key Coordinates:

(๐ŸŽ, ๐Ÿ), (๐Ÿ—๐ŸŽ, ๐ŸŽ), (๐Ÿ๐Ÿ–๐ŸŽ, โˆ’๐Ÿ), (๐Ÿ๐Ÿ•๐ŸŽ, ๐ŸŽ), (๐Ÿ‘๐Ÿ”๐ŸŽ, ๐Ÿ) ๐‘ฆ is never more than 1 or less than -1. Pattern repeats every 360ยฐ.

๐‘ฆ = tan ๐‘ฅ Key Coordinates:

(๐ŸŽ, ๐ŸŽ), (๐Ÿ’๐Ÿ“, ๐Ÿ), (๐Ÿ๐Ÿ‘๐Ÿ“, โˆ’๐Ÿ), (๐Ÿ๐Ÿ–๐ŸŽ, ๐ŸŽ), (๐Ÿ๐Ÿ๐Ÿ“, ๐Ÿ), (๐Ÿ‘๐Ÿ๐Ÿ“, โˆ’๐Ÿ), (๐Ÿ‘๐Ÿ”๐ŸŽ, ๐ŸŽ)

Asymptotes at ๐’™ = ๐Ÿ—๐ŸŽ and ๐’™ = ๐Ÿ๐Ÿ•๐ŸŽ Pattern repeats every 360ยฐ.

๐‘“(๐‘ฅ) + ๐‘Ž Vertical translation up a units. (

0๐‘Ž

)

๐‘“(๐‘ฅ + ๐‘Ž) Horizontal translation left a units. (

โˆ’๐‘Ž0

)

1 โˆ’๐‘“(๐‘ฅ) Reflection over the x-axis.

๐‘“(โˆ’๐‘ฅ) Reflection over the y-axis.

Page 19: Essential Algebra

Topic/Skill Definition/Tips Example

Area Under a Curve

To find the area under a curve, split it up into simpler shapes โ€“ such as rectangles, triangles and trapeziums โ€“ that approximate the area.

Tangent to a Curve

A straight line that touches a curve at exactly one point.

Gradient of a Curve

The gradient of a curve at a point is the same as the gradient of the tangent at that point. 1. Draw a tangent carefully at the point. 2. Make a right-angled triangle. 3. Use the measurements on the axes to calculate the rise and run (change in y and change in x) 4. Calculate the gradient.

๐บ๐‘Ÿ๐‘Ž๐‘‘๐‘–๐‘’๐‘›๐‘ก =๐ถโ„Ž๐‘Ž๐‘›๐‘”๐‘’ ๐‘–๐‘› ๐‘ฆ

๐ถโ„Ž๐‘Ž๐‘›๐‘”๐‘’ ๐‘–๐‘› ๐‘ฅ

=16

2= 8

Area Under Graph and Gradient of Curve

Page 20: Essential Algebra

Rate of Change The rate of change at a particular instant in time is represented by the gradient of the tangent to the curve at that point.

Distance-Time Graphs

You can find the speed from the gradient of the line (Distance รท Time) The steeper the line, the quicker the speed. A horizontal line means the object is not moving (stationary).

Velocity-Time Graphs

You can find the acceleration from the gradient of the line (Change in Velocity รท Time) The steeper the line, the quicker the acceleration. A horizontal line represents no acceleration, meaning a constant velocity. The area under the graph is the distance.

Page 21: Essential Algebra

Topic/Skill Definition/Tips Example

Equation of a Circle

The equation of a circle, centre (0,0), radius r, is:

๐‘ฅ2 + ๐‘ฆ2 = ๐‘Ÿ2

๐‘ฅ2 + ๐‘ฆ2 = 25 Tangent A straight line that touches a circle at exactly

one point, never entering the circleโ€™s interior. A radius is perpendicular to a tangent at the point of contact.

Gradient Gradient is another word for slope.

๐‘ฎ = ๐‘น๐’Š๐’”๐’†

๐‘น๐’–๐’=

๐‘ช๐’‰๐’‚๐’๐’ˆ๐’† ๐’Š๐’ ๐’š

๐‘ช๐’‰๐’‚๐’๐’ˆ๐’† ๐’Š๐’ ๐’™=

๐’š๐Ÿ โˆ’ ๐’š๐Ÿ

๐’™๐Ÿ โˆ’ ๐’™๐Ÿ

Circle Theorems A tangent is perpendicular to the radius at

the point of contact.

๐‘ฆ = 5๐‘๐‘š (Pythagorasโ€™ Theorem)

Equation of a Circle and Tangent


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