**2.Algebra and Functions**

**-Law of indices for rational exponents**

x ^{p} × x^{q} = x^{p+q}

\(\frac{x^p}{x^q}\) = \(x^{p-q}\)

\((x^{p)q}\) = \(x^{pq}\)

\(^p\sqrt{x^q}\) = \(x^{\frac{q}{p}}\)

**– Surds and rationalization**

\((\sqrt{p})^2\) = p

\(\sqrt{pqr}\) = \(\sqrt{p}\) × \(\sqrt{q}\) × \((\sqrt{r}\)

\(p-q\)= \((\sqrt{p}\) – \(\sqrt{q})\) \((\sqrt{p}\) + \(\sqrt{q})\) )

**– Quadratic functions and Graphs**

ax^{2 }+ bx + c = 0

The discriminant is

∆ = b^{2} – 4ac

If ∆ >0 => 2 real roots

∆=0 => 1 real root

∆ < 0 => No real roots

f(x) = ax^{2 }+ bx + c = 0

By completing the square, f(x) = \(a(x +\frac{b}{2a})^2 + (c -\frac{b^2}{4a})\)

Solve quadratic equations by factorisation, using the discriminant or by completing the square.

If α and β are the roots of the equation

ax^{2}+bx+c=0

Then α + β = –\(\frac{b}{a}\) and α β = \(\frac{c}{a}\)

**– Simultaneous equations**

Solving by elimination, substitution, or graphically including one linear equation and one quadratic equation

Example : y=2x+1 and y = x^{2}– 4x – 1

**– Inequalities**

if x > y for x > 0 and y > 0 ; then -x < -y and \(\frac{1}{x}\) < \(\frac{1}{y}\)

Solve quadratic and linear inequalities in one variable and their graphical interpretation.

px+q < rx+t

ax^{2}+bx+c ≤ 0

ax^{2}+bx+c > px+q

Express the answers in the form x ∈ (a,b) ∪ (d,e)

**– Polynomials**

Manipulation of polynomials

If f(x)=0 when x=a , then (x-a) is a factor of f(x)

Factorization of cubic expressions such as

f(x)=x^{3}+3x^{2}-5

f(x)=2x^{3}+9x^{2}+13x+6

Simplification of expressions such as

\(\frac{a^3 - b^3}{a^2 - b^2}\)**– Graphs of functions**

Sketch the graph of y = 2x^{2} +(3x – 1 )^{2}

y= |px+q|

Sketch the graph of y=|3x-2| and use it to solve

|3x-2|=x

Or

|3x-2| < x

The graphs of y = \(\frac{b}{x}\) and y = \(\frac{b}{x^2}\)

have asymptotes parallel to the axes.

The asymptotes to the curve y = \(\frac{3}{x +p}+q\) are

y=q and x=-p

Use the proportion symbol ∝ to express the relation between 2 variables.

Area of a circle and the radius A∝ r^{2}

The graph of A v/s r^{2} is a straight line passing through the Origin.

**– Composition of functions**

A one to one function mapped as ℝ to ℝ is often written as

f∶ x ↦

will be used

A composite function of f(x) and g(x) is

f g(x) and this is different from gf(x)

fg(x)≠ gf(x)

f^{-2} f(x)=ff^{-2} (x)=x

The graph of y=f^{-1} (x) is the graph of y=f(x) reflected in the line y=x

**– Transformation of Graphs**

The following are the transformations of the graph y=f(x)

y=af(x) this is a vertical stretch by a factor a

y=f(x)+k this is vertical shift of k

y=f(x+h) this is a horizontal shift

y=f(bx) this is a horizontal stretch by a factor b

Graph of y=f(x) is given , calculate the graph of y=|f(x)| and y=|f(-x)|

Examples : y=2+ sin3x ; y=cos (x- \(\frac{3π}{4}\))

**– Partial fractions**

The following decomposition must be known

\(\frac{1}{(ax +b)^2(cx^2 + d)}\) = \(\frac{A}{ax +b}\) + \(\frac{B}{(ax +b)^2}\) + \(\frac{Cx +D}{(cx^2 +d)}\)

**– Functions in modelling**

Exponential functions for radioactive decay or population growth

Trigonometric functions for waves

Projectile’s trajectory