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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
The “a” = Direction of Opening
a ＞0 parabola opens UP a ＜0 parabola opens DOWN
26
The “a” = 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’)
27
The “h” = Horizontal Translation
(x-h)^2 translates ‘h’ units to the right (x+h)^2 translates ‘h’ units to the left
28
The “k” = Vertical Translation
+k translates up ‘k’ units -k translates down ‘k’ units
29
In (function notation) y = af [k (x - d)] + c (a is...
a = Direction of Opening (reflection of the x-axis) & Vertical Stretch/Compression
30
In (function notation) y = af [k (x - d)] + c (k is...
k = Reflection of y-axis & Horizontal Stretch/Compression
31
In (function notation) y = af [k (x - d)] + c (d is...
d = Horizontal translation left/right
32
In (function notation) y = af [k (x - d)] + c (c is...
c = Vertical translation up/down
33
Vertical Reflection (of the x-axis) - y = af(x)
If a < 0, all y values become NEGATIVE, all x values remain the same
34
Horizontal Reflection (of the y-axis) - y = f(kx)
If k < 0, all x values become NEGATIVE, all y values remain the same
35
Horizontal Translation (Left or Right) - y = f(x-d)
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
Vertical Translation (Up or Down) - y = f(x)+c
All y values move Up or Down, all x values remain the same y=f(x)+c →Up y=f(x)-c →Down
37
In y = af [k (x - d)] + c
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
Vertical Stretch - y = 2√x
Multiply all Y values by 2
39
Vertical Compression - y = ½√x
Multiply all Y values by ½
40
a is either the ? or the ? factor
vertical stretch or the vertical compression factor
41
If a > 1, there is a ?
vertical stretch
42
If 0 < a < 1 , there is a
vertical compression
43
If A present -
multiply the y values of the parent function y=f(x) by A
44
Vertical stretch (if a is present ...
multiply the y values by a)
45
Horizontal Compression √2x
Multiply all X values by ½
46
Horizontal Stretch √½x
Multiply all x values by 2
47
k is either the ? or the ? factor
Horizontal stretch or the horizontal compression
48
If k > 1, there is a...
horizontal compression
49
If 0 < k < 1, there is a...
horizontal stretch
50
If K present -
multiply the x-values of the parent function y=f(x) by 1/k the reciprocal of ‘k’)
51
The “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
The “k
- 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
In any function y=af(kx):
- 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
Mapping Notation Steps
- 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
Mapping Notation Formula
(1/kx + d, ay + c)
56
The reciprocal function
f(x) = 1/x
57
Values of f(x) = 1/x
-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
Asymtopes;
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
When f(x) =1/x...
Vertical asymptote at x = 0, and a horizontal asymptote at y=0
60
the y = a [k(x - h)] + c in f(x) = 1/x
y = a [1/k (x - d)] + c
61
So when C gets moved the horizontal asymptote changes too!
y = (1/x) + 3 - horizontal asymptote y = 3 and vertical x = 0
62
So when H gets moved the vertical asymptote changes too!
y = (1/x - 2) - vertical asymptote x = 2 and horizontal y = 0
63
Determining Transformed Functions from Graphs Steps:
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
Inverse Relations Definition:
A function and its inverse relation can be described as the "DO" and the "UNDO" functions
65
Generally speaking, the process of finding an inverse is simply...
the swapping of the x and y coordinates.
66
In Inverse Relations - The ? and ? obviously will also be interchanged
The domain and range obviously will also be interchanged
67
This newly formed inverse will be a relation, but may not necessarily be a ?.
Function
68
If the function is f(x) , then the inverse of the function is denoted...
f -1(x) .
69
Let's graph the function and its inverse:
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
Find the inverse of a relation using the equation Steps:
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
Parent Function x^2
X Y (0,0) (1,1) (-1,1) (2,4) (-2,4)
72
Parent Function √x
X Y (0,0) (1,1) (4,2) (9,3)
73
Parent Function 1/x
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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