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Math 8 - College Algebra > Formulas and Key concepts > Flashcards

Formulas and Key concepts Flashcards

1
Q

Higher degree polynomial e.g. 3x^4 - 48x^2 = 0

Sec.1.5

A
  1. Factorise
  2. Number in front of bracket equals 0
  3. Number inside bracket equals 0
  4. Solve both for x
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2
Q

Higher degree polynomial by grouping

A
  1. Split into two groups
  2. Find similar inside bracket
  3. Outside both brackets equal to form 1 bracket and then use the similar bracket and multiple together
  4. Set each bracket to 0 and solve
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3
Q

Can you square root a minus e.g. square root -3

A

No, it equals no solution

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4
Q

[a,b] means?

A

a and b are included, a <= x <= b, circles filled in

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5
Q

(a,b) means?

A

a and b are no included, a < x < b, circles hollow

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6
Q

Bounded is when?

A

Line together between two points -3 < x < 5

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7
Q

unbounded is when?

A

Line from point to infinity e.g. x < infinity

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8
Q

What do you do when you multiple an inequality with a minus number

A

switch the symbol

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9
Q

Solving double inequalities, what do you do?

A
  1. Get just x in the middle
  2. what ever you do on one side you do on the other
  3. what ever is left on each side of x, put into notation
  4. graph on number line
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10
Q

Solve absolute inequalities e.g. | x - 5 | < 2

A
  1. Put absolute in the middle and then the number 2 on right side and -2 on left
  2. solve
  3. notation
  4. Graph
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11
Q

Solve absolute inequalities e.g. | x + 3 | >= 7

A
  1. Make | x + 3 | <= -7 and >= 7
  2. solve both
  3. Notation with U
  4. graph
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12
Q

Test intervals of inqualities

A
  1. Find zeros
  2. write test in order
  3. for each test make x = a sensible number
  4. For
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13
Q

Pythagorus

A

a^2 + b^2 = c^2

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14
Q

Distance formula

A

SR (x2 - x1)^2 + (y2 - y1)^2

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15
Q

Mid point formula

A

(x1 + x2 / 2) , (Y1 + Y2 / 2)

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16
Q

Find Y int

A

Plug 0 into X and solve for Y

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17
Q

Find X int

A

Plug 0 into Y and solve for X

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18
Q

Testing for symmetry

A
  1. x axis = replace y with -y, should equal same equation
  2. y axis = Replace x with -x, should equal same equation
  3. Origin = replace both with minus
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19
Q

Equation of circle

A

(x-h)^2 + (y - k)^2 = r^2

(h,k) is centre

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20
Q

Equation of line and what each represents?

A 
y = mx + b
m = slope
b = y int
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21
Q

Slope formula

A

Y2 - Y1 / X2 - X1 = M

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22
Q

Types of slopes

A 
m = + , positive slope
M = - , negative slope
m = 0 , horizontal
m = undefined , verticle
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23
Q

Point slope formula

A

y - y1 = m(x - x1)

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24
Q

How do we know if lines are parallel or perpendicular

A 
parallel = same slope
perpendicular = recipricol
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25
Q

Increasing and decreasing functions

A
  1. Label ONLY x values
  2. First X value, then second
  3. (x1 , x2) e.g. (-infinity , -3) decreasing
26
Q

Increasing and decreasing functions

A
  1. Label ONLY x values
  2. First X value, then second
  3. (x1 , x2) e.g. (-infinity , -3) decreasing
27
Q

Min or Max value on graph

A

(x , y) between an increasing and decreasing slope

28
Q

Odd or even functions

A

Place -x into the function

  1. if you return with same function = even
  2. if you return with minus function = odd
29
Q

Verticle shift up

A

f(x) = h(x) + c

30
Q

Vertical shift down

A

f(x) = h(x) - c

31
Q

Horizontal shift right

A

f(x) = h(x - c)

32
Q

Horizontal shift left

A

f(x) = h(x + c)

33
Q

Reflection over x axis

A

f(x) = -h(x)

Multiple all y coordinates with -1

34
Q

Reflection over y axis

A

f(x) = h(-x)

Multiple all x coordinates with -1

35
Q

Nonrigid transformation vertical stretch and shrink

A

f(x) = ch(x)
c > 1 = stretch
c is 0. = shrink

Multiple c value with y coordinate

36
Q

Nonrigid transformation horizontal stretch and shrink

A

f(x) = h(cx)
c > 1 = stretch
c is 0. = shrink
Multiple c value with x coordinate

37
Q

(F o G) = ?

A

f( g(x) )

38
Q

(G o F) = ?

A

g( f(x) )

39
Q 
h(x) = f ( g(x) )
h(x) = (2x + 3)^3 , Identify two functions
A 
f(x) = x^3
g(x) = 2x + 3
40
Q

Vertex form steps

A
  1. (x^2 + 2x + ______) +7
  2. (b/2)^2 = 1
  3. (x^2 + 2x + 1 ) +7 - 1
  4. (x + 1)^2 + 6
  5. Vertex = (-1 , 6)
41
Q

Vertex short way

A

(-b/2a) = x

f( (-b/2a) ) = y

42
Q

Leading co efficient test - right and left hand behaviour

A 
An = Leading co efficient
x^N = highest degree
43
Q

When n(degree) is odd and An > 0

A

Falls left

Rises right

44
Q

When n(degree) is odd and An < 0

A

Rises left

Falls right

45
Q

When n(degree) is even and An > 0

A

Rises left

Rises right

46
Q

When n(degree) is even and An < 0

A

Falls left

Falls right

47
Q

Polynomial division

A
  1. remember to add stop gaps like 0^2
  2. If you get remainder put in final form =
    f(x) / d(x) = Q(x) + r(x) / d(x)
48
Q

Rational zero test steps

A
  1. Use leading coefficient
  2. Find factors of those numbers
  3. plug one of the factors into f(x) until it equals 0
  4. if factor number plugged in is for example 1, then do long division of f(x) by x-1
  5. factorise or quad formula of Q(x)
  6. Represent all zeros including the one factor at the beginning
49
Q

whats the inverse function of f(x) = y

A

f^-1(y) = x

50
Q

verifying inverses steps

A
  1. Solve f ( g(x) ) = x
  2. Solve g ( f(x) ) = x
    If both equal x then they are inverse of each other
51
Q

Finding inverse of function steps

A
  1. Replace f(x) with y
  2. switch x and y
  3. Solve for y
  4. Replace y with f^-1(x)
  5. Verify f ( f^-1(x) ) & f^-1( f(x) )
52
Q

Horizontal line test tells us what

A

If the graph has an inverse

53
Q

Exponential functions such as x or e - D, R, H.A, V.A, Y Int

A 
D (-infinity , infinity)
R (k , infinity)
H.A x = k (MAKE SURE TO DRAW IT ON THE GRAPH)
V.A NONE
Y int , replace x with 0 and solve for y
54
Q

Log functions such as log or ln - D, R, H.A, V.A, Y int

A

D (h, infinity)
R (-infinity , infinity)
H.A None
V.A y = k

55
Q

Log functions snail method f(x) = log2 (32)

A
  1. replace f(x) with y
  2. put log base number and y together, 2^y
  3. make it equal 32, 2^y = 32
  4. solve for y
56
Q

Properties of log

A
  1. LogA 1 = 0
  2. LogA A = 1
  3. LogA A^x = x
  4. A^loga^x = x
  5. log functions are inverse of exponential functions
57
Q

LogE x = ?

A

ln(x)

58
Q

Properties of ln

A 
Ln(1) = 0
Ln(e) = 1
Ln(e^x) = x
e^log(x) = x
59
Q

Additional properties of log with base of 10 and e

A

Log10 (ab) = Log10 (a) + Log10 (b)
Log10 (a/b) = Log10 (a) - Log10 (b)
LogA^x = Xlog(a)
logA A^x = x

Ln(ab) = same above
Ln(a/b) = same above
LnA^x = same above
60
Q

Using remainder theorem

A
  1. Do long divison
  2. Get remainder
  3. plug given value into f(x)
  4. should equal remainder

Math 8 - College Algebra (1 decks)

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