Exponential Functions

We are interested in the graphs of  f(x) = a^x , where a >1.

Example 1: If f(x) = 2^x . Then

(1) plot f;

(2) what is f(0);

(3) what happens to f when x gets large (infinity ) ?

(4) what happens to f when x gets small (-infinity )?

> f:=x->2^x;

f := proc (x) options operator, arrow; 2^x end proc

> plot(f,-5..5,-1..10);

[Plot]

> f(100);

1267650600228229401496703205376

> evalf(f(-100));

0.7888609052e-30

Exercise 1: Answer the questions listed in "Example 1" by using the function f(x) = 10^x .

Example 2: Use the shifting/reflection techniques to graph the each of the following functions together with f(x) = 2^x .[Do them by hands first and verify your answers with Maple].

(1) f1(x) = 2^x+2

(2) f2(x) = 2^x-2

(3) f3(x) = 2^(x-2)

(4) f4(x) = 2^(x+2)

(5) f5(x) = 2^(-x)

(6) f6(x) = -2^x

> f1:=x->2^x+2;f2:=x->2^x-2;f3:=x->2^(x-2);f4:=x->2^(x+2);f5:=x->2^(-x);f6:=x->-2^x;

f1 := proc (x) options operator, arrow; 2^x+2 end proc

f2 := proc (x) options operator, arrow; 2^x-2 end proc

f3 := proc (x) options operator, arrow; 2^(x-2) end proc

f4 := proc (x) options operator, arrow; 2^(x+2) end proc

f5 := proc (x) options operator, arrow; 2^(-x) end proc

f6 := proc (x) options operator, arrow; -2^x end proc

Note. The following Maple syntax is to plot two functions together. [1] Do you see that f1 is a vertical shifting of f? [2] What about f2, f3, f4, f5, f6 in relation to f?

> plot({f,f1},-3..6,-1..10,thickness=2);

[Plot]

Exercise 3. Given that f(x) = 3^x+2 .

(a) Find  g  that is being horizontally shifted to the right 3 units from f.

(b) Find  h  that  is being vertically shifted down 5 units from g.

Note: You should be able to do this by hand first and verify your answer with Maple or ClassPad.

Exercise 4:  Given that f(x) = 3^(-x)+4 .

(a) Find  g  that is being horizontally shifted to the right 3 units from f.

(b) Find  h that  is being vertically shifted down 5 units from g.

Note: You should be able to do this by hand first and verify your answer with Maple or ClassPad.