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: โก
2x โก 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. ๐ฅ2 and ๐ฅ are not like terms.
2๐ฅ + 3๐ฆ + 4๐ฅ โ 5๐ฆ + 3 = 6๐ฅ โ 2๐ฆ + 3 3๐ฅ + 4 โ ๐ฅ2 + 2๐ฅ โ 1 = 5๐ฅ โ ๐ฅ2 + 3
๐ฅ times ๐ฅ The answer is ๐ฅ2 not 2๐ฅ. Squaring is multiplying by itself, not by 2.
๐ ร ๐ ร ๐ The answer is ๐3 not 3๐ If p=2, then ๐3=2x2x2=8, not 2x3=6
๐ + ๐ + ๐ The answer is 3p not ๐3 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 + 7) = 3๐ฅ + 21
Factorise The reverse of expanding. Factorising is writing an expression as a product of terms by โtaking outโ a common factor.
6๐ฅ โ 15 = 3(2๐ฅ โ 5), where 3 is the common factor.
Essential Algebra
Topic/Skill Definition/Tips Example
Linear Sequence A number pattern with a common difference. 2, 5, 8, 11โฆ is a linear sequence
Term Each value in a sequence is called a term. In the sequence 2, 5, 8, 11โฆ, 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 โadd 3โ Sequence is: 2, 5, 8, 11โฆ
nth term A rule which allows you to calculate the term that is in the nth position of the sequence. Also known as the โposition-to-termโ rule. n refers to the position of a term in a sequence.
nth term is 3๐ โ 1 The 100th term is 3 ร 100 โ 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, 15โฆ 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 โ 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 โฆ
An example of a Fibonacci-type sequence is: 4, 7, 11, 18, 29 โฆ
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 โฆ
The common ratio is 5 Another example of a geometric sequence is: 81, โ27, 9, โ3, 1 โฆ
The common ratio is โ1
3
Quadratic Sequence
A sequence of numbers where the second difference is constant. A quadratic sequence will have a ๐2 term.
nth term of a geometric sequence
๐๐๐โ๐ where ๐ is the first term and ๐ is the common ratio
The nth term of 2, 10, 50, 250 โฆ. Is
2 ร 5๐โ1
nth term of a quadratic sequence
1. Find the first and second differences. 2. Halve the second difference and multiply this by ๐2. 3. Substitute ๐ = 1,2,3,4 โฆ 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 = 2๐2 Sequence: 4, 7, 14, 25, 40 2๐2 2, 8, 18, 32, 50 Difference: 2, -1, -4, -7, -10 Nth term of this set is โ3๐ + 5 Overall nth term: 2๐2 โ 3๐ + 5
Triangular numbers
The sequence which comes from a pattern of dots that form a triangle.
1, 3, 6, 10, 15, 21 โฆ
Sequences
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 2๐ฅ โ 3 = 7 Add 3 on both sides
2๐ฅ = 10 Divide by 2 on both sides
๐ฅ = 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 ๐ฆ =2๐ฅโ1
๐ง
Multiply both sides by z
๐ฆ๐ง = 2๐ฅ โ 1 Add 1 to both sides
๐ฆ๐ง + 1 = 2๐ฅ Divide by 2 on both sides
๐ฆ๐ง + 1
2= ๐ฅ
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.
๐ถ = 3๐ + 5 Where N=number of windows and C=cost
Substitution Replace letters with numbers. Be careful of 5๐ฅ2. You need to square first, then multiply by 5.
๐ = 3, ๐ = 2 ๐๐๐ ๐ = 5. Find: 1. 2๐ = 2 ร 3 = 6 2. 3๐ โ 2๐ = 3 ร 3 โ 2 ร 2 = 5 3. 7๐2 โ 5 = 7 ร 22 โ 5 = 23
Equations and Formulae
Topic/Skill Definition/Tips Example
Quadratic A quadratic expression is of the form
๐๐๐ + ๐๐ + ๐ where ๐, ๐ and ๐ are numbers, ๐ โ ๐
Examples of quadratic expressions: ๐ฅ2
8๐ฅ2 โ 3๐ฅ + 7 Examples of non-quadratic expressions:
2๐ฅ3 โ 5๐ฅ2 9๐ฅ โ 1
Factorising Quadratics
When a quadratic expression is in the form ๐ฅ2 + ๐๐ฅ + ๐ find the two numbers that add to give b and multiply to give c.
๐ฅ2 + 7๐ฅ + 10 = (๐ฅ + 5)(๐ฅ + 2) (because 5 and 2 add to give 7 and multiply to give 10)
๐ฅ2 + 2๐ฅ โ 8 = (๐ฅ + 4)(๐ฅ โ 2) (because +4 and -2 add to give +2 and multiply to give -8)
Difference of Two Squares
An expression of the form ๐๐ โ ๐๐ can be factorised to give (๐ + ๐)(๐ โ ๐)
๐ฅ2 โ 25 = (๐ฅ + 5)(๐ฅ โ 5) 16๐ฅ2 โ 81 = (4๐ฅ + 9)(4๐ฅ โ 9)
Solving Quadratics (๐๐ฅ2 = ๐)
Isolate the ๐ฅ2 term and square root both sides. Remember there will be a positive and a negative solution.
2๐ฅ2 = 98 ๐ฅ2 = 49 ๐ฅ = ยฑ7
Solving Quadratics (๐๐ฅ2 + ๐๐ฅ = 0)
Factorise and then solve = 0. ๐ฅ2 โ 3๐ฅ = 0 ๐ฅ(๐ฅ โ 3) = 0
๐ฅ = 0 ๐๐ ๐ฅ = 3
Solving Quadratics by Factorising (๐ = 1)
Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.
Solve ๐ฅ2 + 3๐ฅ โ 10 = 0 Factorise: (๐ฅ + 5)(๐ฅ โ 2) = 0
๐ฅ = โ5 ๐๐ ๐ฅ = 2
Factorising Quadratics when ๐ โ 1
When a quadratic is in the form ๐๐ฅ2 + ๐๐ฅ + 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 ๐๐ฅ 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 6๐ฅ2 + 5๐ฅ โ 4
1. 6 ร โ4 = โ24 2. Two numbers that add to give +5 and multiply to give -24 are +8 and -3 3. 6๐ฅ2 + 8๐ฅ โ 3๐ฅ โ 4 4. Factorise in pairs:
2๐ฅ(3๐ฅ + 4) โ 1(3๐ฅ + 4) 5. Answer = (3๐ฅ + 4)(2๐ฅ โ 1)
8. Solving Quadratics by Factorising (๐ โ 1)
Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.
Solve 2๐ฅ2 + 7๐ฅ โ 4 = 0 Factorise: (2๐ฅ โ 1)(๐ฅ + 4) = 0
๐ฅ =1
2 ๐๐ ๐ฅ = โ4
Solving Quadratics by Factorising
Topic/Skill Definition/Tips Example
Quadratic A quadratic expression is of the form
๐๐๐ + ๐๐ + ๐ where ๐, ๐ and ๐ are numbers, ๐ โ ๐
Examples of quadratic expressions: ๐ฅ2
8๐ฅ2 โ 3๐ฅ + 7 Examples of non-quadratic expressions:
2๐ฅ3 โ 5๐ฅ2 9๐ฅ โ 1
Factorising Quadratics
When a quadratic expression is in the form ๐ฅ2 + ๐๐ฅ + ๐ find the two numbers that add to give b and multiply to give c.
๐ฅ2 + 7๐ฅ + 10 = (๐ฅ + 5)(๐ฅ + 2) (because 5 and 2 add to give 7 and multiply to give 10)
๐ฅ2 + 2๐ฅ โ 8 = (๐ฅ + 4)(๐ฅ โ 2) (because +4 and -2 add to give +2 and multiply to give -8)
Difference of Two Squares
An expression of the form ๐๐ โ ๐๐ can be factorised to give (๐ + ๐)(๐ โ ๐)
๐ฅ2 โ 25 = (๐ฅ + 5)(๐ฅ โ 5) 16๐ฅ2 โ 81 = (4๐ฅ + 9)(4๐ฅ โ 9)
Solving Quadratics (๐๐ฅ2 = ๐)
Isolate the ๐ฅ2 term and square root both sides. Remember there will be a positive and a negative solution.
2๐ฅ2 = 98 ๐ฅ2 = 49 ๐ฅ = ยฑ7
Solving Quadratics (๐๐ฅ2 + ๐๐ฅ = 0)
Factorise and then solve = 0. ๐ฅ2 โ 3๐ฅ = 0 ๐ฅ(๐ฅ โ 3) = 0
๐ฅ = 0 ๐๐ ๐ฅ = 3
Solving Quadratics by Factorising (๐ = 1)
Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.
Solve ๐ฅ2 + 3๐ฅ โ 10 = 0 Factorise: (๐ฅ + 5)(๐ฅ โ 2) = 0
๐ฅ = โ5 ๐๐ ๐ฅ = 2
Quadratic Graph
A โU-shapedโ curve called a parabola. The equation is of the form
๐ฆ = ๐๐๐ + ๐๐ + ๐, where ๐, ๐ and ๐ are numbers, ๐ โ ๐. If ๐ < ๐, 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 โ 1
When a quadratic is in the form ๐๐ฅ2 + ๐๐ฅ + ๐
1. Multiply a by c = ac
Factorise 6๐ฅ2 + 5๐ฅ โ 4
1. 6 ร โ4 = โ24
Further Quadratics
2. Find two numbers that add to give b and multiply to give ac. 3. Re-write the quadratic, replacing ๐๐ฅ 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. 6๐ฅ2 + 8๐ฅ โ 3๐ฅ โ 4 4. Factorise in pairs:
2๐ฅ(3๐ฅ + 4) โ 1(3๐ฅ + 4) 5. Answer = (3๐ฅ + 4)(2๐ฅ โ 1)
Solving Quadratics by Factorising (๐ โ 1)
Factorise the quadratic in the usual way. Solve = 0 Make sure the equation = 0 before factorising.
Solve 2๐ฅ2 + 7๐ฅ โ 4 = 0 Factorise: (2๐ฅ โ 1)(๐ฅ + 4) = 0
๐ฅ =1
2 ๐๐ ๐ฅ = โ4
Completing the Square (when ๐ = 1)
A quadratic in the form ๐ฅ2 + ๐๐ฅ + ๐ can be
written in the form (๐ + ๐)๐ + ๐ 1. Write a set of brackets with ๐ฅ 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 ๐ฆ = ๐ฅ2 โ 6๐ฅ + 2
Answer: (๐ฅ โ 3)2 โ 32 + 2
= (๐ฅ โ 3)2 โ 7 The minimum value of this expression occurs when (๐ฅ โ 3)2 = 0, which occurs when ๐ฅ = 3 When ๐ฅ = 3, ๐ฆ = 0 โ 7 = โ7
Minimum point = (3, โ7)
Completing the Square (when ๐ โ 1)
A quadratic in the form ๐๐ฅ2 + ๐๐ฅ + ๐ can be
written in the form p(๐ + ๐)๐ + ๐ Use the same method as above, but factorise out ๐ at the start.
Complete the square of 4๐ฅ2 + 8๐ฅ โ 3
Answer: 4[๐ฅ2 + 2๐ฅ] โ 3
= 4[(๐ฅ + 1)2 โ 12] โ 3 = 4(๐ฅ + 1)2 โ 4 โ 3
= 4(๐ฅ + 1)2 โ 7 Solving Quadratics by Completing the Square
Complete the square in the usual way and use inverse operations to solve.
Solve ๐ฅ2 + 8๐ฅ + 1 = 0 Answer:
(๐ฅ + 4)2 โ 42 + 1 = 0 (๐ฅ + 4)2 โ 15 = 0
(๐ฅ + 4)2 = 15
(๐ฅ + 4) = ยฑโ15
๐ฅ = โ4 ยฑ โ15 Solving Quadratics using the Quadratic Formula
A quadratic in the form ๐๐ฅ2 + ๐๐ฅ + ๐ = 0 can be solved using the formula:
๐ =โ๐ ยฑ โ๐๐ โ ๐๐๐
๐๐
Use the formula if the quadratic does not factorise easily.
Solve 3๐ฅ2 + ๐ฅ โ 5 = 0 Answer: ๐ = 3, ๐ = 1, ๐ = โ5
๐ฅ =โ1 ยฑ โ12 โ 4 ร 3 ร โ5
2 ร 3
๐ฅ =โ1 ยฑ โ61
6
๐ฅ = 1.14 ๐๐ โ 1.47 (2 ๐. ๐. )
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.
2๐ฅ + ๐ฆ = 7 3๐ฅ โ ๐ฆ = 8
๐ฅ = 3 ๐ฆ = 1
Variable A symbol, usually a letter, which represents a number which is usually unknown.
In the equation ๐ฅ + 2 = 5, ๐ฅ 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.
5๐ฅ + 2๐ฆ = 9 10๐ฅ + 3๐ฆ = 16
Multiply the first equation by 2.
10๐ฅ + 4๐ฆ = 18 10๐ฅ + 3๐ฆ = 16
Same Sign Subtract (+10x on both) ๐ฆ = 2
Substitute ๐ฆ = 2 in to equation.
5๐ฅ + 2 ร 2 = 9 5๐ฅ + 4 = 9
5๐ฅ = 5 ๐ฅ = 1
Solution: ๐ฅ = 1, ๐ฆ = 2
Solving Simultaneous Equations (by Substitution)
1. Rearrange one of the equations into the form ๐ฆ =. .. or ๐ฅ =. .. 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 ๐ฆ =. .. or ๐ฅ =. .. equation. 5. Check that the two values you get satisfy both of the original equations.
๐ฆ โ 2๐ฅ = 3 3๐ฅ + 4๐ฆ = 1
Rearrange: ๐ฆ โ 2๐ฅ = 3 โ ๐ฆ = 2๐ฅ + 3 Substitute: 3๐ฅ + 4(2๐ฅ + 3) = 1 Solve: 3๐ฅ + 8๐ฅ + 12 = 1
11๐ฅ = โ11 ๐ฅ = โ1
Substitute: ๐ฆ = 2 ร โ1 + 3
๐ฆ = 1 Solution: ๐ฅ = โ1, ๐ฆ = 1
Simultaneous Equations
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.
๐ฆ = 5 โ ๐ฅ and ๐ฆ = 2๐ฅ โ 1. They meet at the point with coordinates (2,3) so the answer is ๐ฅ = 2 and ๐ฆ = 3
Solving Linear and Quadratic Simultaneous Equations
Method 1: If both equations are in the same form (eg. Both ๐ฆ =โฆ): 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 ๐ฆ =. .. or ๐ฅ =. .. 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 ๐ฅ, two values for ๐ฆ.) Graphically, you should have two points of intersection.
Example 1 Solve ๐ฆ = ๐ฅ2 โ 2๐ฅ โ 5 and ๐ฆ = ๐ฅ โ 1
๐ฅ2 โ 2๐ฅ โ 5 = ๐ฅ โ 1 ๐ฅ2 โ 3๐ฅ โ 4 = 0
(๐ฅ โ 4)(๐ฅ + 1) = 0 ๐ฅ = 4 and ๐ฅ = โ1 ๐ฆ = 4 โ 1 = 3 and ๐ฆ = โ1 โ 1 = โ2 Answers: (4,3) and (-1,-2) Example 2 Solve ๐ฅ2 + ๐ฆ2 = 5 and ๐ฅ + ๐ฆ = 3
๐ฅ = 3 โ ๐ฆ (3 โ ๐ฆ)2 + ๐ฆ2 = 5
9 โ 6๐ฆ + ๐ฆ2 + ๐ฆ2 = 5 2๐ฆ2 โ 6๐ฆ + 4 = 0 ๐ฆ2 โ 3๐ฆ + 2 = 0
(๐ฆ โ 1)(๐ฆ โ 2) = 0 ๐ฆ = 1 and ๐ฆ = 2 ๐ฅ = 3 โ 1 = 2 and ๐ฅ = 3 โ 2 = 1
Answers: (2,1) and (1,2)
Topic/Skill Definition/Tips Example
Inequality An inequality says that two values are not equal. ๐ โ ๐ means that a is not equal to b.
7 โ 3 ๐ฅ โ 0
Inequality symbols
๐ > ๐ means x is greater than 2 ๐ < ๐ means x is less than 3 ๐ โฅ ๐ means x is greater than or equal to 1 ๐ โค ๐ means x is less than or equal to 6
State the integers that satisfy โ2 < ๐ฅ โค 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 (< ๐๐ >) Closed circles are used for numbers that are less than or equal or greater than or equal (โค ๐๐ โฅ)
๐ฅ โฅ 0
๐ฅ < 2
โ5 โค ๐ฅ < 4 Graphical Inequalities
Inequalities can be represented on a coordinate grid. If the inequality is strict (๐ฅ > 2) then use a dotted line. If the inequality is not strict (๐ฅ โค 6) then use a solid line. Shade the region which satisfies all the inequalities.
Shade the region that satisfies: ๐ฆ > 2๐ฅ, ๐ฅ > 1 ๐๐๐ ๐ฆ โค 3
Quadratic Inequalities
Sketch the quadratic graph of the inequality. If the expression is > ๐๐ โฅ then the answer will be above the x-axis. If the expression is < ๐๐ โค 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 ๐ฅ2 โ ๐ฅ โ 12 < 0 Sketch the quadratic:
The required region is below the x-axis, so the final answer is:
โ3 < ๐ฅ < 4 If the question had been > 0, the answer would have been:
๐ฅ < โ3 ๐๐ ๐ฅ > 4 Set Notation
A set is a collection of things, usually numbers, denoted with brackets { } {๐ฅ | ๐ฅ โฅ 7} means โthe set of all xโs, such that x is greater than or equal to 7โ The โxโ can be replaced by any letter. Some people use โ:โ instead of โ|โ
{3, 6, 9} is a set.
{๐ฅ โถ โ2 โค ๐ฅ < 5}
Inequalities
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(๐ฅ) = 3๐ฅ2 โ 5 โFor any input value, square the term, then multiply by 3, then subtract 5โ.
Function notation
๐(๐ฅ) ๐ is the input value ๐(๐) is the output value.
๐(๐ฅ) = 3๐ฅ + 11 Suppose the input value is ๐ฅ = 5 The output value is ๐(5) = 3 ร 5 + 11 =26
Inverse function ๐โ1(๐ฅ) A function that performs the opposite process of the original function. 1. Write the function as ๐ฆ = ๐(๐ฅ) 2. Rearrange to make ๐ฅ the subject.
3. Replace the ๐ with ๐ and the ๐ with ๐โ๐(๐)
๐(๐ฅ) = (1 โ 2๐ฅ)5. Find the inverse.
๐ฆ = (1 โ 2๐ฅ)5
โ๐ฆ5 = 1 โ 2๐ฅ
1 โ โ๐ฆ5 = 2๐ฅ
1โ โ๐ฆ5
2= ๐ฅ
๐โ1(๐ฅ) =1 โ โ๐ฅ
5
2
Composite function
A combination of two or more functions to create a new function. ๐๐(๐) is the composite function that substitutes the function ๐(๐) into the function ๐(๐). ๐๐(๐) means โdo g first, then fโ ๐๐(๐) means โdo f first, then gโ
๐(๐ฅ) = 5๐ฅ โ 3, ๐(๐ฅ) =1
2๐ฅ + 1
What is ๐๐(4)?
๐(4) =1
2ร 4 + 1 = 3
๐(3) = 5 ร 3 โ 3 = 12 = ๐๐(4) What is ๐๐(๐ฅ)?
๐๐(๐ฅ) = 5 (1
2๐ฅ + 1) โ 3 =
5
2๐ฅ + 2
Functions
Topic/Skill Definition/Tips Example
1. Algebraic Fraction
A fraction whose numerator and denominator are algebraic expressions.
6๐ฅ
3๐ฅ โ 1
2. Adding/ Subtracting Algebraic Fractions
For ๐
๐ยฑ
๐
๐ , the common denominator is ๐๐
๐
๐ยฑ
๐
๐=
๐๐
๐๐ยฑ
๐๐
๐๐=
๐๐ ยฑ ๐๐
๐๐
1
๐ฅ+
๐ฅ
2๐ฆ
=1(2๐ฆ)
2๐ฅ๐ฆ+
๐ฅ(๐ฅ)
2๐ฅ๐ฆ
=2๐ฆ + ๐ฅ2
2๐ฅ๐ฆ
3. Multiplying Algebraic Fractions
Multiply the numerators together and the denominators together.
๐
๐ร
๐
๐=
๐๐
๐๐
๐ฅ
3ร
๐ฅ + 2
๐ฅ โ 2
=๐ฅ(๐ฅ + 2)
3(๐ฅ โ 2)
=๐ฅ2 + 2๐ฅ
3๐ฅ โ 6
4. Dividing Algebraic Fractions
Multiply the first fraction by the reciprocal of the second fraction.
๐
๐รท
๐
๐=
๐
๐ร
๐
๐=
๐๐
๐๐
๐ฅ
3รท
2๐ฅ
7
=๐ฅ
3ร
7
2๐ฅ
=7๐ฅ
6๐ฅ=
7
6
5. Simplifying Algebraic Fractions
Factorise the numerator and denominator and cancel common factors.
๐ฅ2 + ๐ฅ โ 6
2๐ฅ โ 4=
(๐ฅ + 3)(๐ฅ โ 2)
2(๐ฅ โ 2)=
๐ฅ + 3
2
Algebraic Fractions
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: ๐ฅ๐+1 = โ3๐ฅ๐ + 6
๐ฅ1 = 4
๐ฅ2 = โ3 ร 4 + 6 = 4.242640 โฆ
๐ฅ3 = โ3 ร 4.242640 โฆ + 6= 4.357576 โฆ
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 ๐๐. 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 โANSโ button on your calculator to keep substituting in the previous answer.
Use an iterative formula to find the positive root of ๐ฅ2 โ 3๐ฅ โ 6 = 0 to 3 decimal places. ๐ฅ1 = 4 Answer:
๐ฅ2 = 3๐ฅ + 6
๐ฅ = โ3๐ฅ + 6
So ๐ฅ๐+1 = โ3๐ฅ๐ + 6
๐ฅ1 = 4
๐ฅ2 = โ3 ร 4 + 6 = 4.242640 โฆ
๐ฅ3 = โ3 ร 4.242640 โฆ + 6= 4.357576 โฆ
Keep repeatingโฆ
๐ฅ7 = 4.372068. . = 4.372 (3๐๐) ๐ฅ8 = 4.372208 โฆ = 4.372 (3๐๐)
So answer is ๐ฅ = 4.372 (3๐๐)
Topic: Iteration
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: โก
2x โก 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: ๐2, ๐2 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.
4๐2 + 8๐ โ 12 is a multiple of 4 because it can be written as:
4(๐2 + 2๐ โ 3)
Algebraic Proofs
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 = ๐๐ + ๐ where ๐ 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: ๐ฅ = ๐ฆ ๐ฆ = 4 ๐ฅ = โ2 ๐ฆ = 2๐ฅ โ 7 ๐ฆ + ๐ฅ = 10 2๐ฆ โ 4๐ฅ = 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 ๐ฆ = ๐๐ฅ + 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 ๐๐ฅ + ๐๐ฆ = ๐) 1. Cover the ๐ฅ term and solve the resulting equation. Plot this on the ๐ฅ โ ๐๐ฅ๐๐ . 2. Cover the ๐ฆ term and solve the resulting equation. Plot this on the ๐ฆ โ ๐๐ฅ๐๐ . 3. Draw a line through the two points plotted.
Coordinates and Linear Graphs
Gradient The gradient of a line is how steep it is. Gradient =
๐ช๐๐๐๐๐ ๐๐ ๐
๐ช๐๐๐๐๐ ๐๐ ๐=
๐น๐๐๐
๐น๐๐
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 = ๐๐ + ๐ and solve for c.
Find the equation of the line with gradient 4 passing through (2,7).
๐ฆ = ๐๐ฅ + ๐ 7 = 4 ร 2 + ๐
๐ = โ1
๐ฆ = 4๐ฅ โ 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 =11 โ 3
6 โ 2= 2
๐ฆ = ๐๐ฅ + ๐
11 = 2 ร 6 + ๐ ๐ = โ1
๐ฆ = 2๐ฅ โ 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 ๐ฆ = 3๐ฅ โ 1 and 2๐ฆ โ 6๐ฅ +10 = 0 parallel? Answer: Rearrange the second equation in to the form ๐ฆ = ๐๐ฅ + ๐
2๐ฆ โ 6๐ฅ + 10 = 0 โ ๐ฆ = 3๐ฅ โ 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 ๐ฆ = ๐๐ฅ + ๐)
Find the equation of the line perpendicular to ๐ฆ = 3๐ฅ + 2 which passes through (6,5) Answer: As they are perpendicular, the gradient of
the new line will be โ1
3 as this is the
negative reciprocal of 3.
๐ฆ = ๐๐ฅ + ๐
5 = โ1
3ร 6 + ๐
๐ = 7
๐ฆ = โ1
3๐ฅ + 7
Or 3๐ฅ + ๐ฅ โ 7 = 0
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 ๐๐ = 5 ๐๐๐๐๐ 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
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: ๐ฅ = ๐ฆ ๐ฆ = 4 ๐ฅ = โ2 ๐ฆ = 2๐ฅ โ 7 ๐ฆ + ๐ฅ = 10 2๐ฆ โ 4๐ฅ = 12
Quadratic Graph
A โU-shapedโ curve called a parabola. The equation is of the form
๐ฆ = ๐๐๐ + ๐๐ + ๐, where ๐, ๐ and ๐ are numbers, ๐ โ ๐. If ๐ < ๐, the parabola is upside down.
Cubic Graph The equation is of the form ๐ = ๐๐๐ + ๐, where
๐ is an number. If ๐ > ๐, the curve is increasing. If ๐ < ๐, the curve is decreasing.
Reciprocal Graph
The equation is of the form ๐ =๐จ
๐, where ๐จ is a
number and ๐ โ ๐. 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
Exponential Graph
The equation is of the form ๐ = ๐๐, where ๐ is a number called the base. If ๐ > ๐ the graph increases. If ๐ < ๐ < ๐, the graph decreases. The graph has an asymptote which is the x-axis.
๐ฆ = sin ๐ฅ Key Coordinates:
(๐, ๐), (๐๐, ๐), (๐๐๐, ๐), (๐๐๐, โ๐), (๐๐๐, ๐) ๐ฆ is never more than 1 or less than -1. Pattern repeats every 360ยฐ.
๐ฆ = cos ๐ฅ Key Coordinates:
(๐, ๐), (๐๐, ๐), (๐๐๐, โ๐), (๐๐๐, ๐), (๐๐๐, ๐) ๐ฆ is never more than 1 or less than -1. Pattern repeats every 360ยฐ.
๐ฆ = tan ๐ฅ Key Coordinates:
(๐, ๐), (๐๐, ๐), (๐๐๐, โ๐), (๐๐๐, ๐), (๐๐๐, ๐), (๐๐๐, โ๐), (๐๐๐, ๐)
Asymptotes at ๐ = ๐๐ and ๐ = ๐๐๐ Pattern repeats every 360ยฐ.
๐(๐ฅ) + ๐ Vertical translation up a units. (
0๐
)
๐(๐ฅ + ๐) Horizontal translation left a units. (
โ๐0
)
1 โ๐(๐ฅ) Reflection over the x-axis.
๐(โ๐ฅ) Reflection over the y-axis.
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.
๐บ๐๐๐๐๐๐๐ก =๐ถโ๐๐๐๐ ๐๐ ๐ฆ
๐ถโ๐๐๐๐ ๐๐ ๐ฅ
=16
2= 8
Area Under Graph and Gradient of Curve
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.
Topic/Skill Definition/Tips Example
Equation of a Circle
The equation of a circle, centre (0,0), radius r, is:
๐ฅ2 + ๐ฆ2 = ๐2
๐ฅ2 + ๐ฆ2 = 25 Tangent A straight line that touches a circle at exactly
one point, never entering the circleโs interior. A radius is perpendicular to a tangent at the point of contact.
Gradient Gradient is another word for slope.
๐ฎ = ๐น๐๐๐
๐น๐๐=
๐ช๐๐๐๐๐ ๐๐ ๐
๐ช๐๐๐๐๐ ๐๐ ๐=
๐๐ โ ๐๐
๐๐ โ ๐๐
Circle Theorems A tangent is perpendicular to the radius at
the point of contact.
๐ฆ = 5๐๐ (Pythagorasโ Theorem)
Equation of a Circle and Tangent