The overview from your handout. Ask questions if you have any, either to Euler or to your instructor.
Change in Days (Δt)
The number of days between two columns of a table: the later day minus the earlier day. The tables skip some days, so Δt is not always 1.
Δt = later day − earlier day
Difference and ratio
Difference = New balance − Previous balance
Ratio = New balance ÷ Previous balance
Write ratios as decimals rounded to two decimal places, for example 1.67.
Linear vs. exponential
What sets an exponential function apart from a linear one is how it changes over equal steps.
LinearAdds a constant amount each step.constant difference
constant slope
ExponentialMultiplies by a constant factor each step.constant ratio
constant percent change
Function notation
f(x) is the output of the function f when the input is x. It is read "f of x," and the parentheses do not mean multiplication.
In this activity the input is the day t, and each client has a function for their balance:
k(t) Kanye South's balance on day tm(t) Michelle Omaha's balance on day t
So m(3) is Michelle's balance on day 3, and m(5) ÷ m(3) means her day 5 balance divided by her day 3 balance.
Linear function
This part of the overview opens after you try Question 3(c), so you can write Kanye's function yourself first.
A linear function has the form
f(x) = mx + b
Here, b = f(0) is the initial value, and m is the slope or rate of change: the amount added each step. Over any single step, f(x + 1) − f(x) = m.
Exponential function
This part of the overview opens after you try Question 6(c), so you can find the pattern yourself first.
An exponential function has the form
f(x) = a · bxa ≠ 0, b > 0, b ≠ 1
Here, a = f(0) is the initial value, and b is the base or growth factor. Over any single step, f(x + 1) ÷ f(x) = b.
This overview is keyed to the free, openly licensed text OpenStax Algebra and Trigonometry 2e, Chapters 4 and 6 (Linear Functions; Exponential and Logarithmic Functions).