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):
This is exponential growth.
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);
i.e. x→x+1, y→2y).
It increases by 100%. (NB. the linear function f(x)=2x is "increase by 2";
i.e. x→x+1, y→y+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); i.e. x→x+1, y→3y)). It increases by 200%.
1.5x is the "increase by 50%" function. For every one more unit of X, the y increases by half its current value (i.e. x increases by 1, y (f(x+1)=f(x)*1.5); i.e. x→x+1, y→y+½y)).
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.
(It's actually "exponential" everywhere; it just doesn't look like it is at samll x values.)
1.1x represents 10% (compounded) growth.
If the horizontal axis is time, you have to give it some time before it "rockets" up.
The world has been on such a curve since the start of the Industrial Revolution: economy,
population, science/technology.
b is the growth factor.
Exponential: the bigger it is, the more it grows. It increases faster/more 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
Convert any base b expression to base 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);
i.e. x→x+1, y→y/2).
It decreases by 50%.
It's the reflection of f(x)=2x across the y-axis.
Exponenetial decay curves:
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.
Always has a one-sided horizontal asymptote. And restricted range.
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.
The time between every doubling (or halving) is the same.
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 functions worksheet
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)
