2.Algebra and Functions
-Law of indices for rational exponents
x p × xq = xp+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
ax2 + bx + c = 0
The discriminant is
∆ = b2 – 4ac
If ∆ >0 => 2 real roots
∆=0 => 1 real root
∆ < 0 => No real roots
f(x) = ax2 + 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
ax2+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 = x2– 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
ax2+bx+c ≤ 0
ax2+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)=x3+3x2-5
f(x)=2x3+9x2+13x+6
Simplification of expressions such as
\(\frac{a^3 - b^3}{a^2 - b^2}\)– Graphs of functions
Sketch the graph of y = 2x2 +(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∝ r2
The graph of A v/s r2 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