Properties of Exponents
The product rule for exponents
When do you add the exponents when multiplying like bases?
f(x) = a*b^x, where "a" is the initial base, and "b" is the base
What is an exponential function?
It starts at the y-intercept and then rises steeply as x increases
What does the graph of an exponential growth function look like?
Population growth, compound interest
What is an example of exponential growth in real life?
Add 1 to the growth rate expressed as a decimal
How do you calculate the multiplier for exponential growth?
The quotient rule for exponents
When do you subtract the exponents when dividing like bases?
Y=a*(1+r)^x, where "a" is the initial amount, "r" is the growth rate, and "x" is the time
What is the equation for exponential decay?
It starts at the y-intercept and then decreases gradually as x increases
What does the graph of an exponential decay function look like?
Radioactive decay, cooling of a hot object
What's an example of exponential decay in real life?
Subtract the decay rate expressed as a decimal from 1
How do you calculate the multiplier for exponential growth?
Power rule for exponents
When do you raise exponents to another exponent when you multiply exponents?
Y=a*(1-r)^x, where "a" is the initial amount, "r" is the decay rate, and "x" is the time
What is the equation for exponential decay?
It starts at the y-intercept and then decreases gradually as x increases
What is the asymptote of an exponential function?
A=Pert, where A is the amount of time "t", P is the principal amount, r is the rate, and t is the time
What is the formula for continuous exponential growth?
The multiplier is calculated as 1 + r for growth and 1 - r for decay, where r is the rate expressed as a decimal
What is the relationship between the multiplier and the growth/decay rate?
Any non-zero number raised to the power of zero equals 1
What is the zero exponent rule?
The initial value is the value of "a" in the function f(x)=a*bx
What is the initial value in an exponential function?
It is an increasing curve that is steep at first and then levels off as x increases
What is the shape of the graph of an exponential function with a base greater than 1?
The time it takes for the quantity to double
What is the doubling time in exponential growth?
compounding frequency affects the growth of an investment
How does frequent compounding leads to faster growth of compounding interest?
Any non-zero number raised to a negative exponent is equal to its reciprocal raised to the positive exponent
What is the negative exponent rule?
Changing the base and it affecting the graph of an exponential function
How does changing the base stretch or compress the graph horizontally?
It is a decreasing curve that gets closer to the x-axis but never touches it
What is the shape of the graph of an exponential function with a base between 0 and 1?
The time it takes for half a substance to decay
What's radioactive decay?
The growth rate in a savings account with 5% interest compounded annually
0.05, because it's 5% expressed as a decimal