2.x Algebra

Algebra and Functions

2.1 - Laws of indices

Understand and use all laws of indices for all rational exponents.

$a^m * a^n = a^{m+n}$ $(a^{m})^{n} = a^{mn}$ $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

2.2 - Using surds

Use and manipulate surds, including rationalising the denominator.

$(\sqrt{x})^2 = x$ $\sqrt{xy} = \sqrt{x}\sqrt{y}$ $(\sqrt{x} + \sqrt{y})(\sqrt{x}-\sqrt{y}) = x-y$

2.3 - Quadratics

Be comfortable working with quadratic functions and their graphs ($f(x)$)

Discriminant

An equation to find the number of roots of a function

$b^2-4ac > 0,$ The roots are real and unequal (curve crosses the x axis twice at different points) $b^2-4ac = 0,$ The roots are real and equal (curve crosses x axis once at the turning point) $b^2-4ac < 0,$ The roots are non real (curve never crosses x axis)

Completing the square

The process of converting a quadratic to a perfect square form with a remainder.

$ax^2+bx+c = a(x+\frac{b}{2a})^2 + (c-\frac{b^2}{4a})$

Methods of solving quadratic equations

Must be able to solve quadratics (Find roots, turning points etc) by the following methods:

  • Factorisation
  • Use of quadratic formula ($x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$)
  • Use of calculator (Classwiz)
  • Completing the square

2.4 - Solving simultaneous equations

Must be able to solve simultaneous equations in two variables by elimination and by substitution, including one linear and one qudratic equation.

Example:

Solve : $$ y=2x+3, y=x^2-4x+8 $$

$2x+3 = x^2-4x+8$ $x^2-6x+5 = 0$ $(x-5)(x-1) = 0$ $\text{Therefore}$ $x = 5, x = 1$

2.5 - Linear and quadratic inequalities

Must be able to solve and interpret inequalities, e.g. $ax+b > cx+d$ $px^2+qx+r \ge 0$ $px^2+qx+r < ax+b$

Rearranging

$\frac{a}{x} < b;$ becomes $;ax < bx^2$

Expressing solutions

Correct use of ‘and’ and ‘or’, “$x < a\text{ or }x > b$ is the same as ${x: x < a} \cup {x: x > b}$

2.8 - Understand and use composite/inverse functions

A function is a one to one mapping from $\mathbb{R}$ to

Example:

Find $f^-1(x)$ of the function $$ f(x) = x^2+2x-3,; x \ge 0 $$

Step 1: Complete the square Using the above formula: $f(x) = (x+1)^2-4$ Step 2: Rearrange for x $y = (x+1)^2-4$ $y+4 = (x+1)^2$ $\sqrt{y+4}-1=x,; y \ge -4$