Algebraic Fractions When adding or subtracting algebraic fractions, the first thing to do is to put them onto a common denominator (by cross multiplying). e.g. 1 + 4 (x + 1) (x + 6) = 1(x + 6) + 4(x + 1) (x + 1)(x + 6) = x + 6 + 4x + 4 (x + 1)(x + 6) = 5x + 10 (x + 1)(x + 6) Solving equations When solving equations containing algebraic fractions, first multiply both sides by a number/expression which removes the fractions. Example: Solve 10 - 2 = 1 (x + 3) x multiply both sides by x(x + 3):\ 10x(x + 3) - 2x(x + 3) = x(x + 3) (x + 3) x \ 10x - 2(x + 3) = x² + 3x [after cancelling]\ 10x - 2x - 6 = x² + 3x\ x² - 5x + 6 = 0\ (x - 3)(x - 2) = 0\ either x = 3 or x = 2algebra
Functions A function is a rule which indicates an operation to perform. e.g. if f(x) = x² + 3 f(2) = 2² + 3 = 7 (i.e. replace x with 2) Functions can be graphed. For example, the graph of f(x) = 1/x is as follows: This is the same graph as y = 1/x, although the y axis is f(x) instead of y. Types of graphs The graph of y = k/x (f(x) = k/x) is known as a hyperbola. Asymptotes are lines on a graph which the graph gets very close to, but never touches. Therefore in the case of y = 1/x, the x and y axes are asymptotes. Parabolas are graphs of the form y = ax² + bx + c (where a, b and c are numbers). They can be 'U' shaped, when a is positive, or 'n' shaped, when a is negative. Graph ShiftingIf you add 1 to f(x), this will shift the graph up 1 unit. i.e. f(x) + n shifts the graph upwards by n units. f(x - 1) will shift the graph 1 unit to the right. i.e. f(x - n) shifts the graph n units to the right. f(x + n) will shift the graph n units to the left. Inverse Functions The inverse function of y = 2x is y = ½x . The inverse of a function does the opposite of the function. To find the inverse of a function, follow the following procedures: let y = f(x). Swap all y's and x's . Rearrange to give y = . This is the inverse function. Example: Find the inverse of f(x), where f(x) = 3x - 7 f(x) = 3x - 7 y = 3x - 7 (let f(x) = y) x = 3y - 7 (swap x's and y's)\ y = x + 7 3 Combining Functions If f(x) = 3x + 1 and g(x) = x² + 2\ f(x) + g(x) = 7\ 3x + 1 + x² + 2 = 7\ (x - 1)(x + 4) = 0\ x = 1 or -4
Expanding Brackets Brackets should be expanded in the following ways: For an expression of the form a(b + c), the expanded version is ab + ac, i.e., multiply the term outside the bracket by everything inside the bracket (e.g. 2x(x + 3) = 2x² + 6x [remember x × x is x²]) For an expression of the form (a + b)(c + d), the expanded version is ac + ad + bc + bd, in other words everything in the first bracket should be multiplied by everything in the second. Example: Expand (2x + 3)(x - 1):(2x + 3)(x - 1) = 2x² - 2x + 3x - 3 = 2x² + x - 3 Factorising Factorising is the reverse of expanding brackets, so it is putting 2x² + x - 3 into the form (2x + 3)(x - 1). This is an important way of solving quadratic equations. The first step of factorising an expression is to 'take out' any common factors which the terms have. So if you were asked to factorise x² + x, since x goes into both terms, you would write x(x + 1) . Factorising Quadratics There is no simple method of factorising a quadratic expression. One way, however, is as follows: Example: Factorise 12y² - 20y + 3 12y² - 18y - 2y + 3 [here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers]. The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y. 6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y] Now, make the last two expressions look like the expression in the bracket: 6y(2y - 3) -1(2y - 3)The answer is (2y - 3)(6y - 1) Example: Factorise x² + 2x - 8 We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and -2. x² + 4x - 2x - 8 x(x + 4) - 2x - 8 x(x + 4)- 2(x + 4) (x + 4)(x - 2) Once you work out what is going on, this method makes factorising any expression easy. It is worth studying these examples further if you do not understand what is happening. Unfortunately, the only other method of factorising is by trial and error. The Difference of Two Squares If you are asked to factorise an expression which is one square number minus another, you can factorise it immediately. This is because a² - b² = (a + b)(a - b) . Example: Factorise 25 - x² = (5 + x)(5 - x) [imagine that a = 5 and b = x]
Flow charts are a diagrammatic representation of a set of instructions which must be followed. Flow charts are made up of different boxes, which each have different functions. The flow chart above says think of a number, add 5 and multiply by 2. If the number is negative, make it positive. Flow charts are usually equivalent to some function. In the example above, the flow chart is equivalent to the function f(x) = | 2(x + 5) | . (the vertical lines around 2(x + 5) means take the magnitude of the answer).
Indices/ Powers 3³ ('3 to the power of 3') and 5² (5 'to the power' of 2) are example of numbers in index form. 3³ = 3×3×3 2¹ = 2 2² = 2×2 2³ = 2×2×2 etc. The ² and ³ are known as indices. Indices are useful (for example they allow us to represent numbers in standard form) and have a number of properties. These properties only hold, however, when the same number is being raised to a certain power. For example, we cannot easily work out what 2³×5² is, whereas we can simplify 3²×3³ .