Principles (Sem 1)

Graphs

• Quadratic equations are in the general form ax + by + c = 0,  where a, b and c are constants, and a is not zero.

• In a graph, the gradient and y-intercept form the equation y = mx + c,  where the gradient of the line is m and the line intersects the y-axis at c

• Graph of x = c -> vertical line intersects x-axis at c

• Graph of y = c  -> horizontal line intersects y-axis at c

• Graph of "Double intercept form" -> (x / a) + (y / b) = 1 -> Line intersects x-axis at a and y-axis at b

Simultaneous Linear Equations

• A pair of simultaneous linear equations can have 1 solution, no solution, or an infinite number of solutions.

• 1 solution: "The 2 lines intersect at (3 , 4), so the solution is x = 3 and y = 4."

• No solution: "The 2 lines are parallel and they never meet, so the simultaneous equations have no solution."

• Infinite no, of solutions: "The two lines are identical / coincident, so they have an infinite number of solutions."

Gradient

• To measure the gradient of a curve at point P, draw a tangent to P. Gradient of line = gradient of curve at P

• To measure the gradient of a line, choose two points on the line, use (y2 - y1) / (x2 - x1) or (y1 - y2) / (x1 - x2)

Quadratic Function

• Equation of a quadratic function -> y = ax^2 + bx + c

• Since this is the equation, for any given value of y, the statement will be true.

Algebra

• The greatest exponent of the variable in a polynomial is the degree of the polynomial. (e.g. 3x^2 has a degree of 2)

• The degree of a polynomial represents its maximum number of solutions (e.g. 3x^2 has a maximum of 2 solutions)

• For polynomials with more than 1 variable, to calculate its degree, add all powers of all variables in a term and see which one is largest (e.g. 3x^2y^4 + 7x^3y has a degree of 6)

Laws of Indices

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Factorisation (methods)

1. Identify HCF

2. Grouping in pairs

3. Using algebraic identities: 

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

(a + b)(a - b) = a^2 - b^2

4. Inspection