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Gravity
Terms in this set (182)
Rational Number
-any number that is a real number that can be written ass a terminating or repeating decimal
Irrational Number
-real number that cannot be written as a terminating or repeating decimal (square root of 5)
Quadratic Formula
-b±[√b²-4ac]/2a
Factoring Cubes
a^3+b^3=(a+b) (a^2-ab+b^2)
-if it is A^3-B^3 then change to (a-b)(a^2+ab+b^2)
Discriminant
-for quadratic expression, the discriminant is b^2-4ac
-determines tht nature of the roots when the quadratic equation=0
-if discriminant is greater than 0, 2 real roots
-if discriminant=0, 1 real root
-if discriminant is less than 0, 2 imaginary roots
Discriminant with x intercepts
-greater than 0, 2 x intercepts
-equal to 0, the vertex lies on the x axis
-less than 0, the parabola doesn't intersect the x axis
Given roots to quadratic equation and have to find the original equation
-formula is
X^2-(sum of roots)x+(product of roots)=0
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
-is coordinate set
Circle with center at origin
X^2+Y^2=R^2
Circle with center at (h,k)
(X-h)^2 +(y-k)^2=r^2
-if a line is tangent to a circle, the line is perpendicular to the radius drawn to the point of tangency
Slope intercept form
y=mx+b
-m=slope
-b=y intercept
point slope form
y-y1=m(x-x1)
-passes through point (x1,y1)
standard form
ax+by=c
Parallel Lines
-have equal slopes
Perpendicular Lines
-have slopes that are opposite and reciprocal
Constant of proportionality with K
look at other slides
Y=Kx
-y varies as x
-y varies directly as x
-y is directly proportional to x
Y=K/x
-y is inversely proportional to x
Y=kxz
-y varies jointly as x and z
Y=Kx/z
-y varies directly as x and inversely as z
Y=Kx+b
-y varies linearly as x
relation
- a set of ordered pairs
function
- a relation in which each input has a unique output
domain
-the set of input values
range
-the set of corresponding output values
zeros
-the input values that make the output zero
-values for which f(x)=0
-located on the x axis
Vertical Line test
a relation is a function and passes the vertical line test if when a vertical line is drawn through the graph, it touches the graph only ONE time.
Constant
f(x)=a
Linear
f(x)=ax+b
Quadratic
f(x)=ax^2+bx+c
Cubic
f(x)=ax^3+bx^2+cx+d
Quartic
f(x)=ax^4+bx^3+cx^2+dx+e
Triangle
A=(B)(H)/2
- formula is one half base times hight
Rectangle
A=bh
Parallelogram
A=bh
Trapezoid
A=(1/2)(B+b)(H)
Circle
A=@r^2
@=pie
C=2(pie)r or (pie)(diameter)
Prisms
V=(area of base)(height)
Pryamids
V=(1/3)(area of base)(height)
Cube
V=(edge)^3
S.A.= 6(edge)^2
Rectangular Prism
V=(lenght)(width)(height)
S.A.=2(lenght)(width)+2(lenght)(height)+2(width)(height)
Right circular cylinder
V=(pie)r^2h
S.A.=2(pie)r^2+2(pie)RH
Sphere
V=(4/3)(pie)r^3
S.A.=4(pie)r^2
Right Circular Cone Formula
V=1/3πr^2h
SA=πr√(r^2+h^2 )+πr^2
Distance
D=(rate)(time)
Position Function
-16t^2+v0t+h0
V0=initial velocity
H0=initial hieght
Finding Domain for a rational function
-omit numbers that make the denominator zero
Finding domain for radical functions
-the cadicand must never become negative so find domain by solving radicand> or equal to 0
Difference Quotient
-gives the secant slope to a curve
f(x+h)-f(x)/h
Increasing Function
A function whose output value increases as its input value increases.
-graph is increasing between (a,b) when F(b)>F(a) when b>a
-graph rises to the right
Decreasing
opposite
-graph falls to the right
constant
as move from a to b, f(b)=f(a)
-graph is horizontal
Basic shapes
look at next ones
Y=absolute value of x
Looks like a V that intersects at the origin
y=square root of x
is a line that comes from the x axis that extends rightward. Gets less steep as it moves along
Y=x^2
Is a parabala im pretty sure. Looks like the square root of x one but intersects at the origin and has two sides
Y=x^3
flip the left side of the x^2 side down
Y=1/x
don't even know how to explain
Y={x}
-don't even know
y=square root of A-x^2
semicircle on top half
f(x)+d
-vertical shift with the sign
F(x+c)
horizontal shift against the sign
Af(X)
-vertical stretch or squeeze
-multiply each y value by A
f(Bx)
-horizontal stretch or squeeze
-f(x)
-reflect graph across x axis
f(-x)
-reflect graph across y axis
absolute value of f(x)
-keep all positive parts of graph
-flip negative parts up
f(absolute value of x)
-keep positive x function value
-mirror those values across the y axis
Even functions
f(-x)=f(x)
Odd functions
f(-x)=-f(x)
(f+g)(x)
F(x)+g(x)
(f-g)(x)
F(x)-g(x)
(F*g)(x)
F(x)*G(x)
(f/g)(x)
F(x)/G(x)
-g(x) can't equal zero
Finding inverse of a functions
switch x and y and solve for the new y
Invertable functions
finverse{f(x)}=x
parabola standard form
y=ax^2+bx+c
-a gives shape
=1 is normal, >1 is fat,<1 is skinny(absolute values)
-- if a >0 opens up, <0 opens down
-c is the y intercept
-b gives movement on the horizontal axis
find the vertex by x=-b/2a, y=f(-b/2a)
Completed square form for a parabola
y=a(x-h)^2+k
a gives shape
(h,k) is the vertex
intercept form for a parabola
y=a(x-r1)(x-r2)
a gives shape
r1 and r2 are the x intercepts
vertex x=(r1+r2)/2 y=f((r1+r2)/2)
discriminant with parabola
b^2-4ac
>0 then the parabola has 2 x intercepts
<0 parabola has 0 x intercepts
=0 the parabola has a vertex on the x axis
Circle formula
(x-h)^2+(y-k)^2=r^2
-center is (h,k)
-radius is r
Ellipses
-set of points(x,y) in a plane, the sum of whose distances from two distinct fixed points(foci) is constant
(x-h)^2/a^2 + (y-k)^2/b^2 =1
-can swap the numerators
-has center (h,k)
-eccentricity=c/a
a^2+b^2=C^2
-c gives the distance to the focus from the center
-eccentricity is less than one
Hyperbolas
-set of all points(x,y) in a plane, the difference of whose distances from two fixed distinct points (foci) is a positive constant
(x-h)^2/a^2 - (y-k)^2/b^2 = 1
-can switch the numerators
-has center (h,k)
c^2=A^2+b^2
-c gives the distance from the focus to the center
-asymptotes pass through (h,k) and have slope (+/-) b/a or (+/-) a/b
-equation is y-k=(+/-)b/a(x-h)
-eccentricity is greater than 1
Parabolas
-set of all points P in a plane such that the distance from P to a fixed point F(the focus) is equal to the distance from P to a fixed line D( the directrix)
y-k=a(x-h)^2 or x-h=a(y-k)^2
-has a vertex (h,k)
a=1/4c or c=1/4a
-absolute value of c is the distance from vertex to focus or vertex to directrix
Polynomials
-continous smooth functions having no corners or gaps
-no horizontal segments
-if the degree is n, the graph has at most n x axis
-maximum number of turning points is n-1
-polynomial of degree n has at most n-1 local extrema
-at zero of even multiplicity, the graph touches but doesn't cross the x axis
-at zero of odd multiplicity, the graph crosses the x axis
-if degree is even, the graph ends in the same direction
-if the degree is odd, the graph ends in opposite directions
-if the leading coeffienct is positive, the graph ends in quadrant 1
-if the leading coefficient is negative, the graph ends in quadrant 4
Chapter 4.2 need help
need blakes help
-use synthetic division
ex: x-1 put opposite of c(1) in box. Then take coefficients of the formula and put them up. Pull first one down and then multiply across.
i
square root of negative one
i^2
-1
i^3
-i
i^4
1
rationalizing i in the numerator
1/(a+bi)
-multiply by (a-bi)/(a-bi)
intermediate value theorem
-since polynomial functions are continuous, if f(a) and f(b) are of opposite sign, there is at least one zero of f between a and b
chapter 4.8 need blakes help
need blakes help
Angel measurements
angels can be measured in degrees, minutes, seconds or decimal degrees
-1 degree is equal to 60 minutes (60')
-1 minute is equal to 60 seconds (60'')
Converting between radians and degrees
multiply by (pie)/180 degrees or 180/(pie)
positive and negative angles
-positive angles are measured counterclockwise
-negative angles are measured clockwise
Coterminal angles
-have same initial and terminal sides and look identical
-find by adding or subtracting 360 degrees, 2pie radians or 1 revolution
Sin&
=y/r
Cos&
=x/r
Tan&
=y/x
in unit circle
r=1
Csc&
=1/sin
sec&
1/cos
cot&
1/tan
tan&
sin&/cos&
Cot&
cos&/sin&
Pythagorean identities
sin^2+cos^2=1
tan^2+1=sec^2
1+cot^2=csc^2
Negative angle formulas
sin(-x)=-sin(x)
cos(-x)=-cos(x)
tan(-x)=-tan(x)
Complementary angle identities
sin[(pie/2) -&]=cos&
tan[(pie/2) -&]=cot&
sec[(pie/2)-&]=csc&
Sum and difference formulas
sin[x(+/-)y]=sin(x)cos(y)[+/-]cos(x)sin(y)
cos[x(+/-)y]=cos(x)cos(y)[-/+]sin(x)sin(y)
----notice that for cos, if it is positive in the first part, is negative in the second part
tan[x(+/-)y]=tan(x)[+/-]tan(y)/1[-/+]tan(x)tan(y)
---notice that the top and bottom are opposite of each other
Double angle formulas
sin(2x)=2sin(x)cos(x)
cos(2x)=cos^2-sin^2
=1-2sin^2
=2cos^2-1
tan(2x)=2tanx/1-tan^2
half angle formulas
chapter 8.3
arc length
s=r&
area of sector
A=(1/2)r^2
linear velocity
v=s/t
v=r&/t
-measure of how fast a point moves around the circle
angular velocity
w=&/t
-is the measure of how fast the central angle is changing
relating linear and angular velocity
v=rw
graphing y=Asin[B(x-c)]+D
-absolute value of A is the amplitude. if less than 0, the graph flips
-B produces a period change( the new period is the normal period divided by B)
-C produces a horizontal shift. Sin(x-pie) moves to the right pie units
-D is the vertical shift. goes with sign
normal period for sin, cosine, secant and cosecants
2(pie)
normal period for tangent and cotangent
pie
Sin(x)
Domain-all reals
Range [-1,1]
Asymptotes= none
cos(x)
Domain- all reals
range [-1,1]
Asymptotes= none
tan x
Domain x cannot equal (2n+1)(pie)/2
Range- all reals
Asymptotes= what domain cannot equal
cotx
Domain- x cannot equal n(pie)
Range-all reals
Asymptotes = what domain cannot be
sec(x)
Domain-same as tangent
Range=(-infinity,1] in union with [1,infinity)
asymptotes= same as tangent
csc(x)
Domain- same as cot
Range- same as sec(x)
Asymptotes- same as cot(x)
Solving trig equations
sine is positive choose from quadrants 1 and 2
cosine is positive choose from quadrant 1 and 4
tangent is positive, choose between quadrant 1 and 3
sin inverse
domain[-1,1]
range [-(pie)/2,(pie)/2]
cos inverse
Domain[-1,1]
Range[0,pie]
tan inverse
domain (- infinity, infinity)
Range (-Pie/2,pie/2)
LogaB=k
a^k=b
Following rules hold for x,y>0 but don't equal 1
Following rules hold for x,y>0 but don't equal 1
logb (xy)
=logb(x)+logb(y)
logb(x/y)
=logb(x)-logb(y)
logb(x^k)
=k*logb(x)
logb(x^b)
=x
b^log(x)
=x
logb1
=0
common logs
log(x)=log10(x)
Natural Logarithms
ln(x) refers to loge(x) where e=2.718
ln(x)=k
=loge(x)=k so e^k=x
logb(a)
log(a)/log(b) = ln(a)/ln(b)
Characteristics of even, power functions
f(-x)=f(x)
-graph has even or y axis symmetry
-domain is all reals
-range is nonnegative real numbers
-graph always contains (0,0), (-1,1) and (1,1)
-as the exponent grows in magnitude, the graph flatens toward the origin and becomes steeper as the absolute value of x grows
-both ends go in the same direction
Characteristics of odd, power functions
f(-x)=-f(x)
-graph has odd or origin symmetry
-domain and range are all reals
-graph contains (0,0), (-1,-1), and (1,1)
-as the exponent increases in magnitude, the graph flattens toward the origin and becomes steeper as the absolute value of x increases
-the ends go in opposite directions
exponential functions
f(x)=a*b^x
-a and b are constant and the variable is the exponent
-A gives the initial value
-if B>1 then we have exponential growth
-if B<1 then we have exponential decay
If quantity is growing at a rate of r
f(x)=a(1+r)^t
quantity is decaying at a rate of r
f(x)=a(1-r)^t
Finding the value of an investment
A=P(1+r/n)^nt
-P=amount invested
-r=annual rate
-n=number of times compunded per year
-t=number of years
Value of investment that grows continuously
A=Pe^rt
graph of f(x)=loga(x)
-has x intercept(1,0)
-contains the point (a,1)
-contains the point (1/a,-1)
-has vertical asymptote of x=0
Law of Cosines
c^2=a^2+b^2-2abcos(C)
-can be used if you know 2 sides and the included angle or 3 sides
Law of Sines
sinA/a=sinB/b=sinC/c
Area of a triangle
A=(1/2)ab*sin(c)
Heron's Formula
[sqrt][s(s-a)(s-b)(s-c)] when s = (a+b+c)/2
Polar coordinate system
-the pole is a point at the origin of the coordinate system (0,0)
-the polar axis is a ray along the positive x axis
-(r,&) gives the location of a point by giving a distance and an angle
Convert from rectangular to polar use
r=[sqrt](x^2+y^2)
tan&=y/x
convert from polar to rectangular use
x=rcos(&)
y=rsin(&)
finding the distance between (r1,&1) and (r2,&2)
-as long as r1 and r2>0 then you can use the law of cosines
d=[sgrt]r1^2 +r2^2-2r1r2cos(&2-&1)
Rules for absolute values
[sqrt]x^2= absolute value of x
absolue value of x=absolue value of -x
absolute value of a*b=(absolute value of a)(absolute value of b)
graphing y=absolute value of (ax-b)
rewrite as y=absolute value of [a(x-b/a)]
distance formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
d/dx(c)
=0
d/dx(x^n)
=nx^n-1
d/dx(sin[x])
=cos(x)
d/dx(cos[x])
=-sin(x)
d/dx(tan[x])
=sec^2
d/dx(cot[x])
=-csc^2
d/dx(sec[x])
sec(x)tan(x)
d/dx(csc[x])
=-csc(x)cot(x)
d/dx(e^x)
=e^x
d/dx(a^x)
=a^xln(a)
d/dx(ln[x])
=1/x
d/dx(loga[x])
=1/(x*ln(a))
d/dx(sin inverse(x))
=1/[sqrt](1-x^2)
d/dx (cos inverse(x))
=-1/[sqrt](1-x^2)
d/dx(tan inverse(X))
=1/(1+x^2)
d/dx(cot inverse(x))
-1/(1+x^2)
d/dx(sec inverse(x))
=1/(absolute value)[sqrt](x^2-1)
d/dx(csc inverse(x))
=-1/(absolute value)[sqrt](x^2-1)
inverse functions
if (a,c) is on f and (c,a) is on f inverse,
then f inverse(c)=1/f(a)
;