APPC Unit 2a

sequence

a function from the whole numbers to the real numbers — we are only able to plug in whole numbers but we can get any real number

connect

when we graph a sequence, we cannot ___ the points

arithmetic and geometric

2 types of sequences

common difference

arithmetic sequence - successive terms have a ___ (constant rate of change)

arithmetic sequence formula

• a_n = a₀ + dn

• a_n = a_k + d(n-k)

linear functions

arithmetic sequences behave like ________, except they are not continuous

equally

increasing arithmetic sequences increase ____ each step. (slope always stays the same)

common ratio

geometric sequence - successive terms have a ____, or constant proportional change

geometric sequence formula

• g_n = g₀rⁿ

• g_n = g_kr^(n-k)

initial value

a₀ / g₀

kth term

a_k / g_k

exponential functions

geometric sequences behave like _____, except they are not continuous

larger

increasing geometric sequences increase by a ____ amount each step. (% increase always stays the same)

constant rate

linear functions - over equal-length input-value intervals, the output values change at a…

addition

linear functions - the change in y is based on…

proportionately

geometric sequence - over equal-length input-value intervals, the output values change…

multiplication

geometric sequence - the change in y is based on…

two points

if you have ___, you can write the equation of a linear function exponential function, geometric sequence, or arithmetic sequence

exponential growth

a > 0 and b > 1

exponential decay

a > 0 and 0 < b < 1

end behavior

for exponential functions in general form, as the input values increase/decrease without bound, the output values will increase/decrease without bound or approach 0

vertical translation

an exponential function might not display a proportional growth pattern due to a…

k

add or subtract __ from the output values ri reveal a proportional growth pattern, given the og was also exponential

product property

b^mbⁿ = b^m+n

power property

(b^m)ⁿ = b^mn

negative exponent property

b^-n = 1/bⁿ

horizontal translation; vertical dilation

every ___ of an exponential function is equivalent to a ____ of the exponential function

horizontal dilation; base

every ___ of an exponential function is equivalent to changing the ___ of the exponential function

linear

us a ___ model when the data reveals a relatively constant rate of change

quadratic

use a ___ model when the rates of change are increasing/decreasing at a relatively constant rate

exponential

use a ___ model when the output values are roughly proportional

without pattern

if a model for a given set of data is appropriate, the residual plot should appear…

communtative

compositions are NOT ____. So generally, f(g(x)) doesn’t equal g(f(x))

switching

every inverse relation can be found by ___ each x and y value

one

every input must have exactly ___ output

y=x

the graphs of inverses are reflections over the line ___

f^-1(x)

the inverse function of f(x) is written as…

x

to show that two functions are inverses, we must show that the f(f^-1(x)) and f^-1(f(x)) BOTH equal…

restricted

sometimes, we must use a ___ domain to ensure two functions are inverses

domain; range

the __ of f is the ____ of f^-1(x)