transformations test

sine function: even or odd

odd?

sine function shape and point it passes through

The sine function has a wave-like shape that oscillates between -1 and 1. It passes through the origin at the point (0, 0).

sine function domain

The domain of the sine function is all real numbers, represented as $(-\infty,\infty)$ .

sine function range

The range of the sine function is all real numbers between -1 and 1, represented as $[-1, 1]$.

meaning of a periodic function

a function where outputs repeat in regular intervals/over a constant interval

example of a periodic function

sine and cosine functions

cosine function shape and y intercept

W, (0,1)

is a cosine function periodic

yes

domain of cosine function

infinity and neg infinity

range cosine function

[-1, 1]

cosine function odd or even

even

even function meaning

reflected/symmetrical over y axis

even function algebraic rule

f(-x) = f(x)

odd function meaning

symmetrical over the origin, 180 degree rotation

odd function algebraic rule

f(-x) = -f(x)

process to test if even or odd function algebraically

1. substitute -x for x everywhere in f(x)

2. simplify the expression

3. see what it matches

a function is even when after simplifying… (algebraically)

it matches f(x), the original

a function is odd when after simplifying… (algebraically)

it matches -f(x)

greatest integer/step function

f(x)= [x] or int(x), find the greatest integer less than or equal to x

domain of greatest integer/step function

infinity

range of greatest integer/step function

{y: y=EZZ} y such that y is equal to all integers

greatest integer/step function, odd even or neither?

neither

is greatest integer/step function a continuous graph

no, range is only integers

is greatest integer/step function a one to one function

no

what is the opposite of a greatest integer/step function

least integer/ceiling function

constant function

f(x) = any real number

can a constant function be vertical?

no, y needs to be constant

identity function

f(x) = x

quadratic function

x²

square root function

sq root x

cubic function

x³

cube root func

cubic root x

rational func

1/x

rational square func

1/x²

absolute value func

lxl

exponential func

2^x

^{(any pos base)}

expression for transformations of f(x)

af(b(x-h))+k

expression for transformations of f(x): f(x-h)

shift right h

expression for transformations of f(x): f(x)+k

shift up k

expression for transformations of f(x): f(bx)

dilate horizontally b

when b or a is negative in expression for transformations of f(x):

reflect over axes

expression for transformations of f(x): af(x)

dilate vertically by a

why are horizontal transformations the opposite of what you think?

difference between f(b(x-h)) and f(bx-bh)