PreCalc Chapter 1 Test Part 2 Flashcards
a constant function can be horizontal and vertical
true or false
false–ONLY HORIZONTAL
f(1)=4 really is
an ordered pair (1,4)
f(x)=x^2
parent quadratic function
f(x)=square root x
parent radical function
f(x)=1/x
parent rational function (reciprocal function)
what is the difference between 1/x and x/1 in terms of functions
1/x is the parent rational function and x/1 is a linear function with slope 1/1
f(x)= [[x]]
greatest integer function (or, step function)
what is a piecewise function
a function with pieces (normally 2 or 3)
the greatest integer function takes the next integer ______
so -3.1 would go to __ and 2.9 would go to ___
-4, 2
it is also called the round-down function by some
f(x)=x
identity function
the constant function (f(x)=c) and the identity function (f(x)=x) are two special types of _____ functions
linear
a piecewise defined function will always have at least one x-intercept or at least one y-intercept
true or false
true- defined means that the domain is all real numbers so it will have a y intercept at least
what is a defined function
one that has a domain of all real numbers
when performing a piecewise function, always ______ your solutions and make sure that your functions work
test
a linear equation will always have an x intercept and a y intercept
true or false
false-constant functions will not have an x intercept
what are the three types of transformations
what are the four types of translations
translation, reflection, dilation
up, down, left, right
describe the translation f(x)+c f(x)-c f(x+c) f(x-c)
c units up
c units down
c units left
c units right
describe the graph
h(x)=(x+2)^3 +1
cubic function shifted two units to the left and shifted up one
when is there a reflection in the y-axis for a function?
x-axis?
y-axis: h(x)=f(-x)
x-axis: h(x)=-f(x)
describe the graph
j(x)=-(x+3)^2 +1
quadratic function reflected over the x-axis, shifted three units to the left and up one
when the shapes are congruent or unchanged after a transformation, the transformation is a _________
what kinds of transformations are included
rigid transformation
translations, reflections
transformations that cause shapes to change (horizontal or vertical stretches) are _____________
example?
nonrigid transformations
dilation
when will a function vertically stretch?
Shrink?
n*f(x) when n<-1(-infinty,-1) or n>1(1, infinity)
n*f(x) when -1<n<1 (-1,1)
g(x)=(x-1)^3 +2
what is the parent function?
use function notation to write g in terms of f
f(x)=x^3
g(x)=f(x-1) +2
***Remember not to include the ^3, as that is implied in f(x)
horizontal shifts, vertical shifts, and reflections are called _____ transformations
rigid
A reflection in the x-axis of y=f(x) is represented by h(x)= ________, while a reflection in the y-axis of y=f(x) is represented by h(x)=__________
-f(x)
f(-x)
A nonrigid transformation of y=f(x) represented by g(x)=cf(x) is a _________ when c<-1 or c>1 and a ______ _______ when -1<c<1
vertical stretch, vertical shrink
in (f of g)(x), what is the domain?
the domain of f of g is the set of all x in the domain of g such that g(x) is in the domain of f
what is the composition of functions
taking one function and plugging it into another function (not commutative)
is composition of functions commutative? what does this mean?
no- this means that you will get the same answer in reverse (think addition, multiplication [4+2=2+4])
is (f of g)(x) muliplication
No
how do you decompose a composite function? decompose h(x)=1/(x-2)^2
first find the simplified function (this is your f(x)), then think, what do i plug in (g(x)), to get what I have now?
f(x)=1/x^2 g(x)=x-2
***Note that the ^2 is outside of the parentheses. This is why you cannot have f(x)=1/x and g(x)=x-2^2–> you could have g(x)=(x-2)^2
what are transcendental functions
mix of two types of functions (1/x^2) quadratic and rational
what are the 4 types of functions
polynomial, rational, radical, trigonometric (i dont think we need to know this just in case though)
two functions f and g can be combined by the arithmetic operations of ________,_________,_________, and ____________to create new functions
addition, subtraction, multiplication, division
The _____ of the function f with g is (f of g)(x)=f(g(x))
composition
f(x)=x^2 +6 g(x) square root (1-x)
divide these
x^2 +6/square root of (1-x) —cannot have square roots on the bottom (multiply top and bottom by square root (1-x))—-
x^2 +6 square root (1-x)/1-x
what is (f-g)(0)
f(0)-g(0)
a set ordered pair (mapping, x/y chart, etc.)
relation
Two techniques for fitting models to data are called direct and iverse _______ and least squares ________
variation, regression
Statisticians use a measure called the ______ of _____ _______to find a model that approximates a set of data most accurately
sum, square differences
The linear model with the least sum of square differences is called the ______ ______ _______ line
line of regression
An r value of a set of data, also called a ________ _________, gives a measure of how well a model fits a set of data.
what is the worst of these? best?
correlation coefficient
0, 1
direct variation models can be described as “y varies directly as x,” or “y is _______ ________ to x”
directly proportional
In direct variation models of the form y=kx, k is called the ____ of ____
constant, variation
The direction variation model y=kx^n can be described as “y varies directly as the nth power of x,” or “y is ____ _____ to the nth power of x”
directly proportional
The mathematical model y=k/x is an example of _____ variation
inverse
The joint variation model z=kxy can be described as “z varies jointly as x and y,” or “z is ________ ________ to x and y.”
directly proportional
what is mathematic modeling
coming up with the equation
what is a model?
an equation
y varies directly as x
y is directly proportional to x
y=kx for some nonzero constant k
direct variation
what is k in y=kx?
constant of variation, also the rate
what is the equation for state income tax
what kind of variation does it have
state income tax=k(gross income) (T=k*g where T is the dependent variable and g is independent)
direct
what must you have in order to find k-the constant of variation
initial condition
inverse variation says as one gets bigger, _________________________
the other gets smaller
y=k/x is ______ variation. It is the opposite of _____
inverse, direct
z=kxy
joint variation (z varies jointly as x and y)
what is the equation for Interest
I=Prt
since k is also rate, you could say I=k(P)(t
how do you find a lines equation and graph a scatter plot on your calculator?
lines equation given a lot of points- hit stat, edit, enter x values in L1 and y values in L2, make sure you have the same number of data entries, hit stat again, calc, LinReg, make sure Xlist says L1 and Ylist says L2, calculate
this will give you your equation!!!
to plot a scatter graph- go to y=, graph the line you got above^ (may have to adjust windows) then hit Stat Plot (second y=), plot 1 on, choose type, graph (should get a line with plots)
what kind of variation will these ordered pairs have?
5, -3.5)(10, -7)(15, -10.5)(20, -14)(25, -17.5
Direct—although technically the numbers are getting smaller, the positive values are increasing
z varies directly with the square of x and inversely with y with a constant variation of 2/3
how would you write this?
z=2x^2/3y (separate your fraction)
S=4pi r^2
how would you use variation terminology to say this aloud
The surface area of a sphere varies directly as the square of the radius r (your constant of variation is 4pi [you will never say a number])
how would you use variation terminology to say A=1/2bh
the area of a triangle is jointly proportional to its base and height
Inverse operations _________ each other
undo
what is f^-1(x)
inverse of f (-1 has new mathematical value!)
what does a graph’s inverse do
what kind of symmetry do they have
switches x and y
reflectional symmetry over the line y=x
for a function to have an inverse, it must pass (Horizontal/Vertical) Line Test
BOTH- it must pass vertical to be a legitimate function. It must pass horizontal to have an inverse
an inverse cannot fail (Horizontal/Vertical) line test
I think just Vertical???
The domain of a function = the _____ of its inverse.
the range of a function = the _____ of its inverse
range, domain
how do you prove f(x) and g(x) are inverses of eachother
use composite functions (analytically) plug g(x) into f(x) [f(g(x))] and you will get x
a function is __________ if it passes Horizontal and Vertical line test
one to one
only __________ have an inverse function
one to one
how would you make f(x)=x^2 a function with an inverse
restrict the domain (x>_ 0)
in a piecewise function, the ranges of the starting functions are the _________ of the inverse functions
domains
If the composite functions f(g(x)) and g(f(x)) both equal x, then the function g is the ______ function of f
inverse
The inverse function of f is denoted by
f^-1
The domain of f is the ____ of f^-1 and the, and the ______ of f^-1 is the range of f
range, domain
The graphs of f and f^-1 are reflections of each other in the line ___
y=x
A function is _____ when each value of the dependent variable corresponds to exactly one value of the independent variable
one-to-one
A graphical test for the existence of an inverse function of f is called the _____ Line Test
Horizontal
To reflect over x axis, make _ values negative
Vice versa
y
Be careful with distributing negatives in reflection cases
Square root of (x+6) reflected in both x and y axes is…
-square root of -x-6
Remember when a graph is up 3 and a point is (1,7), it’s technically
(1,4)
If f(x)=x^2 and you plug in 4t, what do you get
(4t)^2=16t
