NJC 2017 H2 Maths Exponential Logarithm and Modulus Functions and their Graphs Notes

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 1 of 9 National Junior College 2016 – 2017 H2 Mathematics Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Key Questions to Answer:  What is an exponential function?  What is the range of values for which an exponential function is well-defined  What laws do exponential functions follow?  What does the graph of an exponential function look like?  What are the key characteristics of the graph of an exponential function?  What is a logarithmic function?  What is the range of values for which a logarithmic function is well-defined?  What laws do logarithmic functions follow?  What does the graph of a logarithmic function look like?  What are the key characteristics of the graph of a logarithmic function?  What is the modulus function?  When do we use the modulus function?  How do I manipulate the modulus function?  How can I draw a graph of a modulus function? §1 Exponential Functions Definition 1.0.1 (Exponential Function) A function , 0 1,xy a a a  , is known as an exponential function . It is a function used to model a relationship for which a constant increase in the independent variable (here denoted by x) gives the same proportional change in the dependent variable (here denoted by y). The most common exponential function is exy . UNDERSTAND What is meant by ‘increasing exponentially’?

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 2 of 9 1.1 Laws of Indices If , , , , 0, 0a b m n a b   (i) mn a a = mna  (ii) mna a = mna  (iii)   nma  mna (iv)   mmma b ab (v) m m a b  m a b   (vi) 1 0n n a , aa  (vii) 0 10a , a (viii) 1/n naa  (ix)   m mn /n aa mna (x) , where 1,0,1xn a x na a    WONDER Why is that for (x), 1,0,1a ? Exercise 1 Without the use of a calculator, simplify the following expressions: (a) 11 369 9 (b) 12 24 2 n n n   (c) 1 0.528 2 (d) 3312 6 (a) 11 1 36 299 9 3   (b) (c) 1 3 1 0.5 22 2 28 2 42 2     (d)

National Junior College Mathematics Department 2016 Revision: Exponential, Logarithmic and Modulus Functions and their Graphs Page 3 of 9 1.2 Graph of the Exponential Function Key features of the graph of the exponential function: Axial Intercepts  0,1 Asymptotes 0y Exercise 2 Identify the key features of the graphs of the following exponential functions and sketch them. (Hint: use your graphic calculator) (a) 2, 1xya a  (b) 23 xy (c) e1xy   (d) 21e1xy   (a) (b) (c) (d) WONDER How should the graph of xya look like if 01