Unit 9 Cambridge IGCSE Maths 0607 Functions with Asymptotes and Logarithms.

Unit 9 notes booklet.
E2.4 Indices — Add indices when multiplying, subtract when dividing, multiply when raising a power to a power. a0=1a^0 = 1
a0=1, a−n=1/ana^{-n} = 1/a^n
a−n=1/an, am/n=amna^{m/n} = \sqrt[n]{a^m}
am/n=nam​. Solve ax=ba^x = b
ax=b by writing both sides as the same base.
E2.5 Equations & Subjects — Clear fractions by multiplying by the LCD. To change subject, isolate the term and undo in reverse order; if the new subject appears twice, collect, factorise, divide.
E2.8 Proportion — Direct: y=kxy = kx
y=kx. Inverse: y=k/xy = k/x
y=k/x. Find kk
k from a given pair, then use the equation. For inverse proportion, xy=kxy = k
xy=k is constant.
E3.1 Reciprocal & Exponential — y=a/x+by = a/x + b
y=a/x+b has asymptote y=by = b
y=b; y=a⋅bx+cy = a \cdot b^x + c
y=a⋅bx+c has asymptote y=cy = c
y=c. Two points pin down the parameters.
E3.3 Functions — Domain = allowed inputs; range = actual outputs. Find f−1f^{-1}
f−1 by swapping xx
x and yy
y then solving. f(g(x))f(g(x))
f(g(x)) applies gg
g first, then ff
f.
E3.5 Asymptotes — Vertical: where the function blows up (denominator zero, log argument zero). Horizontal: the constant the curve approaches as x→±∞x \to \pm\infty
x→±∞.
E3.6 Transformations — f(x)+kf(x) + k
f(x)+k shifts up by kk
k; f(x−k)f(x - k)
f(x−k) shifts right by kk
k (sign flips!). Asymptotes move with the graph.
E3.10 Logarithms — y=ax  ⟺  x=log⁡ayy = a^x \iff x = \log_a y
y=ax⟺x=loga​y. Laws: log⁡(xy)=log⁡x+log⁡y\log(xy) = \log x + \log y
log(xy)=logx+logy, log⁡(x/y)=log⁡x−log⁡y\log(x/y) = \log x - \log y
log(x/y)=logx−logy, log⁡(xn)=nlog⁡x\log(x^n) = n \log x
log(xn)=nlogx. Solve ax=ba^x = b
ax=b by taking logs of both sides.
Test calculator and Non-calculator.