Anatomy of an Equation
This term refers to the constant multiplier by which an output is multiplied every time the input increases by one
What is the growth factor?
For an exponential function to represent growth, the growth factor "b" must meet this numerical condition.
What is greater than 1 (b>1)?
Unlike exponential functions which have a constant multiplier, these functions are characterized by a constant difference over equal intervals.
What are linear functions?
This term refers to the stated or published annual interest rate, often used to determine monthly or weekly rates.
What is the nominal interest rate?
On a graph of f(x)=a⋅b^x , this is the specific coordinate of the vertical intercept.
What is (0, a)?
In the general form f(x)=a⋅b^x , this letter represents the starting value or the y-intercept of the function.
What is 'a'?
This term is used to describe situations where a quantity decreases by the same factor at regular intervals.
What is exponential decay?
If 500 bacteria double every hour, this equation models the population (p) after t hours.
What is p=500⋅2^t ?
This rate reflects the actual percentage of interest earned in one year after compounding is applied.
What is the effective interest rate?
For a linear function, this value is constant over any interval, but for an exponential function, it changes.
What is the average rate of change?
If a population increases by a growth rate of 20%, this is the resulting growth factor used in the equation.
What is 1.2?
Between two functions with growth factors of b = 2 and b = 1.5, this one will grow more quickly over time.
What is the function with b = 2?
In a model of population over time, a negative value for the domain (such as t=−5) represents this.
What is time before the initial measurement?
If a credit card has a nominal APR of 24% and interest is compounded monthly, this is the interest rate applied each month.
What is 2%?
In an exponential growth function, the average rate of change does this as the input value increases.
What is increase (or get larger)?
If a car loses one-third of its value every year, this number represents the decay factor.
What is 2/3?
In the equation f(x)=100⋅(0.85)^x , the function is decaying at this percentage rate per interval.
What is 15%?
In real-world contexts like medicine or population, the range of an exponential function usually excludes these types of numbers.
What are negative numbers?
This is the general result of calculating interest more frequently (e.g., monthly vs. annually) on the same initial balance.
What is a greater account balance/more money?
In an exponential decay function, the average rate of change gets closer to this value as time passes.
What is zero?
In an exponential expression like 5⋅3^0 , the value of the exponent part (30) is always equal to this number.
What is 1?
Because of repeated multiplication, exponential growth functions will eventually surpass this other type of function, even if the other starts with a much larger value.
What is a linear function?
When modeling data like ball bounces, you can find the height from which the ball was dropped by dividing the first bounce height by this.
What is the growth factor?
This is the expression for a $1,000 balance at 6% annual interest compounded quarterly for one year.
What is 1,000⋅(1+0.015)^4 ?
This must be carefully chosen to ensure an exponential graph shows the behavior of the function in an "informative or meaningful" way.
What is the graphing window?