Formulas and Key concepts Flashcards
Higher degree polynomial e.g. 3x^4 - 48x^2 = 0
Sec.1.5
- Factorise
- Number in front of bracket equals 0
- Number inside bracket equals 0
- Solve both for x
Higher degree polynomial by grouping
- Split into two groups
- Find similar inside bracket
- Outside both brackets equal to form 1 bracket and then use the similar bracket and multiple together
- Set each bracket to 0 and solve
Can you square root a minus e.g. square root -3
No, it equals no solution
[a,b] means?
a and b are included, a <= x <= b, circles filled in
(a,b) means?
a and b are no included, a < x < b, circles hollow
Bounded is when?
Line together between two points -3 < x < 5
unbounded is when?
Line from point to infinity e.g. x < infinity
What do you do when you multiple an inequality with a minus number
switch the symbol
Solving double inequalities, what do you do?
- Get just x in the middle
- what ever you do on one side you do on the other
- what ever is left on each side of x, put into notation
- graph on number line
Solve absolute inequalities e.g. | x - 5 | < 2
- Put absolute in the middle and then the number 2 on right side and -2 on left
- solve
- notation
- Graph
Solve absolute inequalities e.g. | x + 3 | >= 7
- Make | x + 3 | <= -7 and >= 7
- solve both
- Notation with U
- graph
Test intervals of inqualities
- Find zeros
- write test in order
- for each test make x = a sensible number
- For
Pythagorus
a^2 + b^2 = c^2
Distance formula
SR (x2 - x1)^2 + (y2 - y1)^2
Mid point formula
(x1 + x2 / 2) , (Y1 + Y2 / 2)
Find Y int
Plug 0 into X and solve for Y
Find X int
Plug 0 into Y and solve for X
Testing for symmetry
- x axis = replace y with -y, should equal same equation
- y axis = Replace x with -x, should equal same equation
- Origin = replace both with minus
Equation of circle
(x-h)^2 + (y - k)^2 = r^2
(h,k) is centre
Equation of line and what each represents?
y = mx + b m = slope b = y int
Slope formula
Y2 - Y1 / X2 - X1 = M
Types of slopes
m = + , positive slope M = - , negative slope m = 0 , horizontal m = undefined , verticle
Point slope formula
y - y1 = m(x - x1)
How do we know if lines are parallel or perpendicular
parallel = same slope perpendicular = recipricol
Increasing and decreasing functions
- Label ONLY x values
- First X value, then second
- (x1 , x2) e.g. (-infinity , -3) decreasing
Increasing and decreasing functions
- Label ONLY x values
- First X value, then second
- (x1 , x2) e.g. (-infinity , -3) decreasing
Min or Max value on graph
(x , y) between an increasing and decreasing slope
Odd or even functions
Place -x into the function
- if you return with same function = even
- if you return with minus function = odd
Verticle shift up
f(x) = h(x) + c
Vertical shift down
f(x) = h(x) - c
Horizontal shift right
f(x) = h(x - c)
Horizontal shift left
f(x) = h(x + c)
Reflection over x axis
f(x) = -h(x)
Multiple all y coordinates with -1
Reflection over y axis
f(x) = h(-x)
Multiple all x coordinates with -1
Nonrigid transformation vertical stretch and shrink
f(x) = ch(x)
c > 1 = stretch
c is 0. = shrink
Multiple c value with y coordinate
Nonrigid transformation horizontal stretch and shrink
f(x) = h(cx)
c > 1 = stretch
c is 0. = shrink
Multiple c value with x coordinate
(F o G) = ?
f( g(x) )
(G o F) = ?
g( f(x) )
h(x) = f ( g(x) ) h(x) = (2x + 3)^3 , Identify two functions
f(x) = x^3 g(x) = 2x + 3
Vertex form steps
- (x^2 + 2x + ______) +7
- (b/2)^2 = 1
- (x^2 + 2x + 1 ) +7 - 1
- (x + 1)^2 + 6
- Vertex = (-1 , 6)
Vertex short way
(-b/2a) = x
f( (-b/2a) ) = y
Leading co efficient test - right and left hand behaviour
An = Leading co efficient x^N = highest degree
When n(degree) is odd and An > 0
Falls left
Rises right
When n(degree) is odd and An < 0
Rises left
Falls right
When n(degree) is even and An > 0
Rises left
Rises right
When n(degree) is even and An < 0
Falls left
Falls right
Polynomial division
- remember to add stop gaps like 0^2
- If you get remainder put in final form =
f(x) / d(x) = Q(x) + r(x) / d(x)
Rational zero test steps
- Use leading coefficient
- Find factors of those numbers
- plug one of the factors into f(x) until it equals 0
- if factor number plugged in is for example 1, then do long division of f(x) by x-1
- factorise or quad formula of Q(x)
- Represent all zeros including the one factor at the beginning
whats the inverse function of f(x) = y
f^-1(y) = x
verifying inverses steps
- Solve f ( g(x) ) = x
- Solve g ( f(x) ) = x
If both equal x then they are inverse of each other
Finding inverse of function steps
- Replace f(x) with y
- switch x and y
- Solve for y
- Replace y with f^-1(x)
- Verify f ( f^-1(x) ) & f^-1( f(x) )
Horizontal line test tells us what
If the graph has an inverse
Exponential functions such as x or e - D, R, H.A, V.A, Y Int
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
Log functions such as log or ln - D, R, H.A, V.A, Y int
D (h, infinity)
R (-infinity , infinity)
H.A None
V.A y = k
Log functions snail method f(x) = log2 (32)
- replace f(x) with y
- put log base number and y together, 2^y
- make it equal 32, 2^y = 32
- solve for y
Properties of log
- LogA 1 = 0
- LogA A = 1
- LogA A^x = x
- A^loga^x = x
- log functions are inverse of exponential functions
LogE x = ?
ln(x)
Properties of ln
Ln(1) = 0 Ln(e) = 1 Ln(e^x) = x e^log(x) = x
Additional properties of log with base of 10 and e
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
Using remainder theorem
- Do long divison
- Get remainder
- plug given value into f(x)
- should equal remainder
