PreCalc Chapter 1 Test Part 2 Flashcards Preview

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Flashcards in PreCalc Chapter 1 Test Part 2 Deck (84)
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0
Q

a constant function can be horizontal and vertical

true or false

A

false–ONLY HORIZONTAL

1
Q

f(1)=4 really is

A

an ordered pair (1,4)

2
Q

f(x)=x^2

A

parent quadratic function

3
Q

f(x)=square root x

A

parent radical function

4
Q

f(x)=1/x

A

parent rational function (reciprocal function)

5
Q

what is the difference between 1/x and x/1 in terms of functions

A

1/x is the parent rational function and x/1 is a linear function with slope 1/1

6
Q

f(x)= [[x]]

A

greatest integer function (or, step function)

7
Q

what is a piecewise function

A

a function with pieces (normally 2 or 3)

8
Q

the greatest integer function takes the next integer ______

so -3.1 would go to __ and 2.9 would go to ___

A

-4, 2

it is also called the round-down function by some

9
Q

f(x)=x

A

identity function

10
Q

the constant function (f(x)=c) and the identity function (f(x)=x) are two special types of _____ functions

A

linear

11
Q

a piecewise defined function will always have at least one x-intercept or at least one y-intercept
true or false

A

true- defined means that the domain is all real numbers so it will have a y intercept at least

12
Q

what is a defined function

A

one that has a domain of all real numbers

13
Q

when performing a piecewise function, always ______ your solutions and make sure that your functions work

A

test

14
Q

a linear equation will always have an x intercept and a y intercept
true or false

A

false-constant functions will not have an x intercept

15
Q

what are the three types of transformations

what are the four types of translations

A

translation, reflection, dilation

up, down, left, right

16
Q
describe the translation
f(x)+c
f(x)-c
f(x+c)
f(x-c)
A

c units up
c units down
c units left
c units right

17
Q

describe the graph

h(x)=(x+2)^3 +1

A

cubic function shifted two units to the left and shifted up one

18
Q

when is there a reflection in the y-axis for a function?

x-axis?

A

y-axis: h(x)=f(-x)

x-axis: h(x)=-f(x)

19
Q

describe the graph

j(x)=-(x+3)^2 +1

A

quadratic function reflected over the x-axis, shifted three units to the left and up one

20
Q

when the shapes are congruent or unchanged after a transformation, the transformation is a _________
what kinds of transformations are included

A

rigid transformation

translations, reflections

21
Q

transformations that cause shapes to change (horizontal or vertical stretches) are _____________
example?

A

nonrigid transformations

dilation

22
Q

when will a function vertically stretch?

Shrink?

A

n*f(x) when n<-1(-infinty,-1) or n>1(1, infinity)

n*f(x) when -1<n<1 (-1,1)

23
Q

g(x)=(x-1)^3 +2
what is the parent function?
use function notation to write g in terms of f

A

f(x)=x^3
g(x)=f(x-1) +2
***Remember not to include the ^3, as that is implied in f(x)

24
Q

horizontal shifts, vertical shifts, and reflections are called _____ transformations

A

rigid

25
Q

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)=__________

A

-f(x)

f(-x)

26
Q

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

A

vertical stretch, vertical shrink

27
Q

in (f of g)(x), what is the domain?

A

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

28
Q

what is the composition of functions

A

taking one function and plugging it into another function (not commutative)

29
Q

is composition of functions commutative? what does this mean?

A

no- this means that you will get the same answer in reverse (think addition, multiplication [4+2=2+4])

30
Q

is (f of g)(x) muliplication

A

No

31
Q
how do you decompose a composite function?
decompose h(x)=1/(x-2)^2
A

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

32
Q

what are transcendental functions

A

mix of two types of functions (1/x^2) quadratic and rational

33
Q

what are the 4 types of functions

A

polynomial, rational, radical, trigonometric (i dont think we need to know this just in case though)

34
Q

two functions f and g can be combined by the arithmetic operations of ________,_________,_________, and ____________to create new functions

A

addition, subtraction, multiplication, division

35
Q

The _____ of the function f with g is (f of g)(x)=f(g(x))

A

composition

36
Q

f(x)=x^2 +6 g(x) square root (1-x)

divide these

A

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

37
Q

what is (f-g)(0)

A

f(0)-g(0)

38
Q

a set ordered pair (mapping, x/y chart, etc.)

A

relation

39
Q

Two techniques for fitting models to data are called direct and iverse _______ and least squares ________

A

variation, regression

40
Q

Statisticians use a measure called the ______ of _____ _______to find a model that approximates a set of data most accurately

A

sum, square differences

41
Q

The linear model with the least sum of square differences is called the ______ ______ _______ line

A

line of regression

42
Q

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?

A

correlation coefficient

0, 1

43
Q

direct variation models can be described as “y varies directly as x,” or “y is _______ ________ to x”

A

directly proportional

44
Q

In direct variation models of the form y=kx, k is called the ____ of ____

A

constant, variation

45
Q

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”

A

directly proportional

46
Q

The mathematical model y=k/x is an example of _____ variation

A

inverse

47
Q

The joint variation model z=kxy can be described as “z varies jointly as x and y,” or “z is ________ ________ to x and y.”

A

directly proportional

48
Q

what is mathematic modeling

A

coming up with the equation

49
Q

what is a model?

A

an equation

50
Q

y varies directly as x
y is directly proportional to x
y=kx for some nonzero constant k

A

direct variation

51
Q

what is k in y=kx?

A

constant of variation, also the rate

52
Q

what is the equation for state income tax

what kind of variation does it have

A

state income tax=k(gross income) (T=k*g where T is the dependent variable and g is independent)
direct

53
Q

what must you have in order to find k-the constant of variation

A

initial condition

54
Q

inverse variation says as one gets bigger, _________________________

A

the other gets smaller

55
Q

y=k/x is ______ variation. It is the opposite of _____

A

inverse, direct

56
Q

z=kxy

A

joint variation (z varies jointly as x and y)

57
Q

what is the equation for Interest

A

I=Prt

since k is also rate, you could say I=k(P)(t

58
Q

how do you find a lines equation and graph a scatter plot on your calculator?

A

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)

59
Q

what kind of variation will these ordered pairs have?

5, -3.5)(10, -7)(15, -10.5)(20, -14)(25, -17.5

A

Direct—although technically the numbers are getting smaller, the positive values are increasing

60
Q

z varies directly with the square of x and inversely with y with a constant variation of 2/3
how would you write this?

A

z=2x^2/3y (separate your fraction)

61
Q

S=4pi r^2

how would you use variation terminology to say this aloud

A

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])

62
Q

how would you use variation terminology to say A=1/2bh

A

the area of a triangle is jointly proportional to its base and height

63
Q

Inverse operations _________ each other

A

undo

64
Q

what is f^-1(x)

A

inverse of f (-1 has new mathematical value!)

65
Q

what does a graph’s inverse do

what kind of symmetry do they have

A

switches x and y

reflectional symmetry over the line y=x

66
Q

for a function to have an inverse, it must pass (Horizontal/Vertical) Line Test

A

BOTH- it must pass vertical to be a legitimate function. It must pass horizontal to have an inverse

67
Q

an inverse cannot fail (Horizontal/Vertical) line test

A

I think just Vertical???

68
Q

The domain of a function = the _____ of its inverse.

the range of a function = the _____ of its inverse

A

range, domain

69
Q

how do you prove f(x) and g(x) are inverses of eachother

A
use composite functions (analytically)
plug g(x) into f(x)  [f(g(x))] and you will get x
70
Q

a function is __________ if it passes Horizontal and Vertical line test

A

one to one

71
Q

only __________ have an inverse function

A

one to one

72
Q

how would you make f(x)=x^2 a function with an inverse

A

restrict the domain (x>_ 0)

73
Q

in a piecewise function, the ranges of the starting functions are the _________ of the inverse functions

A

domains

74
Q

If the composite functions f(g(x)) and g(f(x)) both equal x, then the function g is the ______ function of f

A

inverse

75
Q

The inverse function of f is denoted by

A

f^-1

76
Q

The domain of f is the ____ of f^-1 and the, and the ______ of f^-1 is the range of f

A

range, domain

77
Q

The graphs of f and f^-1 are reflections of each other in the line ___

A

y=x

78
Q

A function is _____ when each value of the dependent variable corresponds to exactly one value of the independent variable

A

one-to-one

79
Q

A graphical test for the existence of an inverse function of f is called the _____ Line Test

A

Horizontal

80
Q

To reflect over x axis, make _ values negative

Vice versa

A

y

81
Q

Be careful with distributing negatives in reflection cases

Square root of (x+6) reflected in both x and y axes is…

A

-square root of -x-6

82
Q

Remember when a graph is up 3 and a point is (1,7), it’s technically

A

(1,4)

83
Q

If f(x)=x^2 and you plug in 4t, what do you get

A

(4t)^2=16t