Exponential functions.

ƒ(x) = bx

b is the base, any/every positive real number: b>0. (0 as a base would have 00 and e.g. 0-1=1/0 problems) (negative bases would have e.g. -1(1/2)=√-1 complex numbers).
There are an infinite number of different exponential functions, one for every positive real number.
The variable x is the exponent.
Domain is ℝ.
NB. These are not polynomial functions.

Each exponential function's graph in the XY Cartesian plane is an exponential curve, continuous and smooth.
It crosses the y axis at 1 (f(0)=b0=1 for all b); i.e. the y-intercept is 1.
It does not cross the X axis (y is never 0 or negative); i.e. X axis is a one-sided horizontal asymptote (the other end goes to +∞).
The range is (0,∞), i.e. all real numbers >0.
No X-intercepts, no turning points, no extrema, no "wiggle", no domain issues.
f(-1)=1/b     f(0)=1     f(1)=b

If b>1, the curve is increasing (goes up to the right):

The larger the base b, the steeper/faster the increase.
Usually in applications, the negative x values don't have meaning and so aren't used.

2x is the doubling function. For every one more unit of X, the y doubles (i.e. x increases by 1, y doubles (f(x+1)=f(x)*2)). It increases by 100%. (NB. the linear function f(x)=2x is "increase by 2")
Powers of 2 worksheet

3x is the tripling function. For every one more unit of X, the y triples (i.e. x increases by 1, y triples (f(x+1)=f(x)*3)). It increases by 200%.

1.5x is the "increase by 50%" function.

The 1.1x above might look flat, but eventually every exponential function of base b>1 goes "exponential", this is the essence of exponential growth. 1.1x represents 10% (compounded) growth.

b is the growth factor.
Exponential: the bigger it is, the more it grows. It increases faster as x increases.
Every increasing exponential function eventually surpasses every increasing polynomial function.

Base e ≈2.718281828
ex is the exponential function.    Sometimes denoted: exp(x)

e-1=1/e=0.3678

More about e

bx = ex ln b

If b=1, 1x=1 for all x, so it's the horizontal line y=1, which isn't considered to be exponential.

If b<1, the curve is decreasing (goes down to the right):

.5x is the halving function. For every one more unit of X, the y halves (i.e. x increases by 1, y halves (f(x+1)=f(x)/2)). It decreases by 50%.
It's the reflection of f(x)=2x across the y-axis.

Negative exponent:    b-x    = 1/bx = (1/b)x
Exs. 2-x = (1/2)x    decaying
e-x = (1/e)x    decaying!

bx and b-x are reflections of each other across the Y-axis.

Negative exponent with coefficient:

The larger the coefficient, the faster the decay.

General exponential functions: ƒ(x)= a·bkx-d + c
a If >1, stretchs the graph up; if <1, compresses down. Will be y-intercept / initial value (unless c). If negative, flips graph over X-axis.
k If >1, steepens the curve; if <1, flattens/broadens it. If negative, switches between increasing and decreasing exponential (see below), flipping the graph over the Y-axis.
d Horizontal shift.
c raises or lowers the resulting graph.
Has a one-sided horizontal asymptote.
 
Derivative: akbkx ln b
Integral: abkx / (k ln b) + cx

Doubling time of increasing exponential functions:
ƒ(t)= a·bkt has a doubling time td = (ln 2) / (k ln b)
ƒ(t)= a·ekt has a doubling time td = (ln 2) / k ≈ .6931/k
The function's value doubles in (every/any) td time units.
If at time tnow the function value is y, then at time tnow+td the function value is 2y.
Exponential functions are the only kind of function that have this property.
Halving time of decreasing exponential functions is (ln (1/2)) / k = -(ln 2) / k ≈ -.6931/k

Exponential exercises
answers

Compounded Interest    worksheet
Radioactive decay
Newton's Law of cooling/warming calculations
Logistic growth    Logistic exercises


ex inverse is the natural logarithm function: ln

ln(exp(x))=x and exp(ln(x))=x
bx=ex ln b     2x=ex ln 2=e.6931x     3x=eln(3)x=e1.098x
eax=(ea)x

Two points (x1,y1) and (x2,y2) determine an exponential function y=abx.
a=y1 / (y2/y1)x1/(x2-x1)
b=(y2/y1)1/(x2-x1)