Studied by 3 people
Area of a circle
A=πr²
Area of a rectangle
A=lw
Area of a triangle
A=1/2bh
Volume of a cylinder
V=πr²h
Volume of a rectangular solid
V=lwh
Volume of a triangular prism
V=(1/2bh)h
Circumference of a circle
C=2πr
Pythagorean theorem
a^2+b^2=c^2
Even symmetry: if _, then _
if (a,b), then (-a, b)
Odd symmetry: If _, then _
if (a,b), then (-a,-b)
Symmetric about the line y=x: If _, then _
If (a,b), then (b,a)
Describe the symmetry between f(x) and the inverse of f(x)
symmetric about the line y=x
When evaluating a problem and you do not find an answer then it is
undefined
When solving a problem and you do not find an answer then it is
no solution
Is it okay for multiple inputs to give the same output?
yes
Is it okay for one input to give multiple outputs?
no
What are the domain restrictions for a log function?
x > 0
What is the criteria for functions to be identical/equivalent?
they must have the same domain and ALL the same points
List the asymptotes for y=2^x
horizontal
List the asymptotes for y=1/x
horizontal and vertical
List the asymptotes for a log function
vertical
What is a correspondence that maps EACH element in x to a UNIQUE element in y?
function
Name the other terms for input (there's 4)
x, domain, independent variable, argument
Name the other terms for output (there's four)
y, range, dependent variable, value
What is evaluating?
finding an output for the given input (at most one answer)
What is solving?
finding all inputs for a given output (infinite answers)
What is finding?
either evaluating, solving, or finding an expression
What is an increasing function?
What is a decreasing function?
What is a concave up function?
What is a concave down function?
If a function has even symmetry then f(-x) =
f(x)
If a function has odd symmetry then f(-x) =
-f(x)
What are three domain restrictions?
can't divide by zero, can't take the square root of a negative number, can't take the log of a negative number or zero
When do you need a sign chart?
if multiple linear factors are in the same even root or the same log
What is an equation for the thing you want to maximize or minimize called?
objective function
If the objective function has more than one input variable, you must find any equations that relate to your input variables. What are these called?
constraint equations
General equation for a vertical shift
f(x) + c
How do you demonstrate a vertical shift on a set of points?
Either add or subtract c from the y-values
General equation for a vertical stretch/compression
cf(x)
When is cf(x) a vertical stretch?
when c is greater than one
When is cf(x) a vertical compression?
when c is greater than zero, but less than one
How do you demonstrate a vertical stretch on a set of points?
multiply y values by c
How do you demonstrate a vertical compression on a set of points?
divide y values by c
General equation for horizontal shift
f(x+c)
How do you demonstrate a horizontal shift on a set of points?
Either add or subtract c from the x-values
If c is positive during a horizontal shift, do you add or subtract c from the x-values?
subtract
If c is negative during a horizontal shift, do you add or subtract c from the x-values?
add
General equation for a horizontal compression/stretch
f(cx)
When is f(cx) a compression?
when c is greater than one
When is f(cx) a stretch?
when c is greater than one but less than zero
How do you demonstrate a horizontal stretch on a set of points?
multiply x-values by the reciprocal of c
How do you demonstrate a horizontal compression on a set of points?
divide x-values by c
What is -f(x)?
vertical flip, reflection about the x-axis
How do the points of a function change when there is a vertical flip/reflection about the x-axis?
x remains unchanged, y switches signs
What is f(-x)?
horizontal flip, reflection about the y-axis
How do the points of a function change when there is horizontal flip/reflection about the y-axis?
x switches signs, y remains unchanged
What must you do if you have a horizontal shift and another horizontal transformation?
factor the argument to see the true shift
Steps for a sequence of transformation
horizontal shift -> stretches/compressions -> reflections -> vertical shifts
When should you plug points into the equation to check you did the transformation correctly?
always
___________ is the set of numbers in the domain in the domain of a resulting composite function AND the domain of the inner function.
domain of a composite function
In order to ensure a composite function remains a function, you need _____
domain restrictions
How do you find inverses algebraically?
switch x and y and solve for y
The input of f is the _____ of f inverse.
output
The input of f inverse is the _____ of f.
output
If the point (a,b) is on f, then the point _____ is on f inverse.
(b,a)
Are the domain and range of f and f inverse the same?
no, they switch
f^1(x) is the _____. (f(x))^1 is the ______.
inverse, reciprocal
What do you do if a function is not one-to-one?
restrict it's domain
Draw y = |x|
Draw y=x^2
Draw y=x^3
Draw y=√x
Draw y=2^x
Draw y=1/x
Draw y=sin(x)
Note
Flashcard