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Math Unit 3 Test Flashcards

1
Q

Relation:

A

Relation: any relationship between two variables (ie. x and y)

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2
Q

Function:

A

Function: a special type of relation where each value of x corresponds with only one value of y. [ie. All values of x are unique]

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3
Q

All functions are ?, but not all relations are ?

A

Relations
Functions

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4
Q

Function: only ONE ? for each ?

A

y for each x

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5
Q

Relation: MULTIPLE ?’s for an ?

A

y’s for an x

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6
Q

Function: For every value of x there is only ? value of y. The x values are ? repeated Note: it does not matter the y values are repeated

A

One
Not

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7
Q

Relation: For the x value of -1, there are three values of y (4,6,7); therefore, it is ? a function. The x values of -1 is repeated

A

Not

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8
Q

Function: ? x values repeated

A

No

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9
Q

Relation: The x value 1 is ?

A

Repeated

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10
Q

To determine if a graph is the graph of a function, principle:

A

for every x value there should be only one y value.

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11
Q

Vertical Line Test:

A

if you draw an imaginary vertical through the graph at any position, it will only ever cross the relation once. If it crosses more than one time, it is not a function.

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12
Q

y = 3x - 1, ?, ?

A

Linear, Function

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13
Q

x^2 + y^2 = 25, ?, ?

A

Circle, Relation

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14
Q

y = -2x2 + 5x - 1, ?, ?

A

Quadratic, Function

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15
Q

Function Notation

A

sub the (x) in the equation for the x and wherever else x is

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16
Q

Domain -

A

all possible input values for the relation - the x values

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17
Q

Range -

A

all possible output values for the relation - the y values

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18
Q

{(-2, 2), (-3, 3), (-4, 4), (-3, 5), (-1, 6)} -

A

D: {-4,-3,-2,-1} R: {2, 3, 4, 5, 6}

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19
Q

If it’s a line -

A

D:{xER} R:{yER}

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20
Q

If it’s a parabola -

A

D:{xER} R:{yER l y≥?}

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21
Q

If it’s a circle -

A

D:{xER l ? ≥ x ≥ ? } R:{yER l ? ≥ y ≥ ? }

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22
Q

In (not function notation) y = a(x - h)^2 + k (a is…

A

the direction of opening, and vertical stretch/compression)

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23
Q

In (not function notation) y = a(x - h)^2 + k (h is…

A

a horizontal translation left or right)

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24
Q

In (not function notation) y = a(x - h)^2 + k (k is…

A

a vertical translation up or down

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25
Q

The “a” = Direction of Opening

A

a ＞0 parabola opens UP
a ＜0 parabola opens DOWN

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26
Q

The “a” = Stretch/Compression

A
  • l a l ＞1 stretched by a factor ‘a’ (multiply y values by ‘a’)
  • 0＜ l a l＜1 compressed by a factor of ‘a’ (multiply y values by ‘a’)
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27
Q

The “h” = Horizontal Translation

A

(x-h)^2 translates ‘h’ units to the right
(x+h)^2 translates ‘h’ units to the left

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28
Q

The “k” = Vertical Translation

A

+k translates up ‘k’ units
-k translates down ‘k’ units

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29
Q

In (function notation) y = af [k (x - d)] + c (a is…

A

a = Direction of Opening (reflection of the x-axis) & Vertical Stretch/Compression

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30
Q

In (function notation) y = af [k (x - d)] + c (k is…

A

k = Reflection of y-axis & Horizontal Stretch/Compression

31
Q

In (function notation) y = af [k (x - d)] + c (d is…

A

d = Horizontal translation left/right

32
Q

In (function notation) y = af [k (x - d)] + c (c is…

A

c = Vertical translation up/down

33
Q

Vertical Reflection (of the x-axis) - y = af(x)

A

If a < 0, all y values become NEGATIVE, all x values remain the same

34
Q

Horizontal Reflection (of the y-axis) - y = f(kx)

A

If k < 0, all x values become NEGATIVE, all y values remain the same

35
Q

Horizontal Translation (Left or Right) - y = f(x-d)

A

All x values move to the Left or Right, all y values remain the same
y=f(x-d) →Right y=f(x+d)→Left

36
Q

Vertical Translation (Up or Down) - y = f(x)+c

A

All y values move Up or Down, all x values remain the same
y=f(x)+c →Up y=f(x)-c →Down

37
Q

In y = af [k (x - d)] + c

A

a = Direction of Opening (reflection of the x-axis) & Vertical Stretch/Compression
k = Reflection of y-axis & Horizontal Stretch/Compression
d = Horizontal translation left/right
c = Vertical translation up/down

38
Q

Vertical Stretch - y = 2√x

A

Multiply all Y values by 2

39
Q

Vertical Compression - y = ½√x

A

Multiply all Y values by ½

40
Q

a is either the ? or the ? factor

A

vertical stretch or the vertical compression factor

41
Q

If a > 1, there is a ?

A

vertical stretch

42
Q

If 0 < a < 1 , there is a

A

vertical compression

43
Q

If A present -

A

multiply the y values of the parent function y=f(x) by A

44
Q

Vertical stretch (if a is present …

A

multiply the y values by a)

45
Q

Horizontal Compression √2x

A

Multiply all X values by ½

46
Q

Horizontal Stretch √½x

A

Multiply all x values by 2

47
Q

k is either the ? or the ? factor

A

Horizontal stretch or the horizontal compression

48
Q

If k > 1, there is a…

A

horizontal compression

49
Q

If 0 < k < 1, there is a…

A

horizontal stretch

50
Q

If K present -

A

multiply the x-values of the parent function y=f(x) by 1/k the reciprocal of ‘k’)

51
Q

The “a”

A
  • Vertical Reflection (Direction of Opening)
    a ＜0 reflection of the x-axis
  • Vertical Stretch/Compression
    l a l ＞1 stretched by a factor ‘a’ (multiply y values by ‘a’)
    0＜ l a l＜1 compressed by a factor of ‘a’ (multiply y values by ‘a’)
52
Q

The “k

A
  • Horizontal Reflection
    k ＜0 reflection of the y-axis
  • Horizontal Stretch/Compression
    l k l ＞1 compressed (multiply x values by the reciprocal of ‘k’→ ½)
    0＜ l k l＜1 stretched (multiply x values by the reciprocal of ‘k’ → ½)
53
Q

In any function y=af(kx):

A
  • a represents the vertical stretch/compression and reflection Action: multiply y-values by ‘a’
  • k represents the horizontal stretch/compression and reflection Action: multiply x-values by ‘1/k’ (reciprocal of ‘k’)
54
Q

Mapping Notation Steps

A
  • Create a Table of Values using a minimum of 3 key points from the parent function y = f(x)
  • Apply the transformations to the x-values and y-values in the table to create a new Table of Values
  • Create the equations to use on the table of values using your a, k, d, and c
  • Graph the new transformed graph
  • Mapping Notation Formula
    (1/kx + d, ay + c)
55
Q

Mapping Notation Formula

A

(1/kx + d, ay + c)

56
Q

The reciprocal function

A

f(x) = 1/x

57
Q

Values of f(x) = 1/x

A

-3 and -⅓
-2 and -½
-1 and -1
-½ and -2
-⅓ and -3
0 and DNE
⅓ and 3
½ and 2
1 and 1
2 and ½
3 and ⅓

58
Q

Asymtopes;

A

An asymptote is a line that a graph approaches without touching. Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value.

59
Q

When f(x) =1/x…

A

Vertical asymptote at x = 0, and a horizontal asymptote at y=0

60
Q

the y = a [k(x - h)] + c in f(x) = 1/x

A

y = a [1/k (x - d)] + c

61
Q

So when C gets moved the horizontal asymptote changes too!

A

y = (1/x) + 3 - horizontal asymptote y = 3 and vertical x = 0

62
Q

So when H gets moved the vertical asymptote changes too!

A

y = (1/x - 2) - vertical asymptote x = 2 and horizontal y = 0

63
Q

Determining Transformed Functions from Graphs
Steps:

A
  1. If it opens down, a is negative
  2. If there’s a reflection in the y axis, k is negative too
  3. The vertex (x,y) determines the d (=x) and c (=y)
  4. To figure out the a-value is how many moves down/up between each point so over 1 down 2 is a value of -2
  5. To figure out the k-value is how many moves over between each point so over 4 down 1 is a k-value of 4
  6. Plug the identified values into the equation y = af [k(x - d)] + c
64
Q

Inverse Relations Definition:

A

A function and its inverse relation can be described as the “DO” and the “UNDO” functions

65
Q

Generally speaking, the process of finding an inverse is simply…

A

the swapping of the x and y coordinates.

66
Q

In Inverse Relations - The ? and ? obviously will also be interchanged

A

The domain and range obviously will also be interchanged

67
Q

This newly formed inverse will be a relation, but may not necessarily be a ?.

A

Function

68
Q

If the function is f(x) , then the inverse of the function is denoted…

A

f -1(x) .

69
Q

Let’s graph the function and its inverse:

A

1) First plot the points of the original function. (or use mapping notation to change a
parent function and then plot those points)
2) For each point, swap the x and y values, then plot the inverted point
3) Draw a mirror line to check

70
Q

Find the inverse of a relation using the equation
Steps:

A
  1. Write the original equation as y=_____
  2. Switch the x-value and y-value in the equation
  3. Solve the new equation for y
  4. Graph the new equation if required
71
Q

Parent Function x^2

A

X Y
(0,0)
(1,1)
(-1,1)
(2,4)
(-2,4)

72
Q

Parent Function √x

A

X Y
(0,0)
(1,1)
(4,2)
(9,3)

73
Q

Parent Function 1/x

A

X Y
(- 3, -⅓)
(-2, -½)
(-1, -1)
(-½, -2)
(-⅓, -3)
(0, DNE)
(⅓ , 3)
(½, 2)
(1,1)
(2, ½)
(3, ⅓)

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