Unit 1 Transformations:
6basic graphs
Inverse: asymptotes at x=0 and y=0
Y=2x y-int at (0,1) asymptote at y=0
Horizontal: With x (inside the function) acts opposite of expected
Translation: (added or subtracted inside to x)
Reflection: negative multiplied inside by x (reflect over y)
Dilation: number multiplied by x inside function (affects x-coordinates opposite of expected)
Vertical: Outside the Function. Acts as expected
Translation: added or subtracted outside the function (acts as expected)
Reflection: negative outside(in front of) the function: reflects over x axis
Dilation: number multiplied by outside of function (affects y-coordinates) as expected
Trig: y=asin[b(x+c)] +d
Amplitude: “a” vertical dilation-distance from highest or lowest point to center of graph/could cause
reflection if negative
Period: “b” length of time to complete one cycle(period). To find period
Phase Shift: “c” Acts opposite-horizontal
Vertical Shift: “d” acts as expected-vertical
Remember to transform the 5 main points of either sine or cosine
2𝜋
𝑏
Vectors:
Addition=tip to tail
Subtraction= Set up the addition of the negative vector, or set up addition to find missing vector
Dot product: u ∙ v = x1x2 + y1y2 where u = <x1,y1> and v= <x2,y2>
Magnitude: Pythagorean theorem
Direction angle: tan-1 but then remember to consider which quadrant the vector falls in
Component form: <magnitude cos𝜃 , magnitude sin𝜃>
Factoring: If you factor and your leading coefficient is not +1, you have to use factor sum tree
ax2 + bx + c what multiplies to be a times c and adds to be b. Then split up the middle term into those
factors and factor by grouping
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
use to solve for x in a quadratic function.
If you use quadratic function and you end up with a negative under radical, simplify and change into
complex numbers
Complex numbers: i=√−1 , i2 = -1
Multiply complex numbers: foil and simplify
Divide complex numbers: multiply numerator and denominator by the conjugate of the denominator,
foil and simplify.
The absolute value of complex number- use Pythagorean theorem
Rational Functions
x-intercepts: set numerator =o and solve for x (If any other function set y=o and solve for x)
y-intercepts: set all x’s to be zero and simplify
VA: set denominator=0 and solve -> related to the domain
HA: deg num > deg den: none (could be a slant if it is exactly 1 more) use long division to find
Deg num = deg den : divide leading coefficients
Deg num < deg den: y=0 (x-axis)
To simplify: factor top and bottom and cancel common factors
To add/subtract: get a common denominator DO NOT CANCEL UNTIL THE END!
Multiply: factor and cancel- multiply straight across
Divide: Flip second fraction and multiply
Solve: set= and cross multiply. Set results EQUAL to each other and then solve for x. OR find common
denominator and set numerator = 0
Symmetry:
x-axis; set all y’s to (-y) and simplify Check to see if it is same as original
y-axis: set all x’s to (-x) and simplify Check to see if it is same as original
origin: set both x and y to (-x) and (-y) and simplify Check to see if it is same as original.
*If you multiply a fraction by -1, only multiply numerator OR denominator NOT BOTH!